A.

Principles of the measurement

Phase measurements can be done with high resolution, but the accuracy of the phase measurements is limited by systematic errors due to

In [8], contribution -i) was kept below 5µm by measuring a reference target every minute. If perfectly constant, contribution -ii) can be removed [9] by numerical processing. Otherwise, it will remain as a systematic error.

As shown in [7], shifts and drifts, altogether with cyclic errors, can be removed by use of

Fig. 1.

Two-mode amplitude modulation: experimental set-up. C1, C2, C3: fibred collimators. CC: hollow corner cube. PBS: polarising beam splitter. PhD: high frequency photodiodes. Φ: XOR phase-meter. RMS: RMS-to-dc converter, to remove the contribution due to amplitude-to-phase coupling in the phase-meter [10]. Red: laser beams. Black, HF signals. Blue, intermediate frequency signals. Purple: low frequency signals.

If the LIA output is locked to zero through a feedback to the VCO, then the reference and measurement modulations are in phase when they recombine at the PBS cube, and in that case, not only contributions -i) are totally eliminated, but also cyclic errors (contribution -ii)), provided that the signal amplitude is not affected in the measurement/reference switching. Then the path length difference ΔL is recovered through

The frequency F(t) can be measured with high accuracy, and K is an integer, that can be determined at the start of the measurements by K = F₁/(F₂ – F₁) where the frequencies F₁ and F₂ correspond to two successive zeroes of the LIA output [10].

B.

Implementation

In [10] the high-frequency modulated beam was obtained by phase-locking two single-mode lasers as shown in Fig. 1. The polarization switch was obtained by a polarization controller followed by a fibred polarising beam splitter. The Allan deviation of the length measurements is plotted in dark blue in Fig. 2 and shows a bump for frequencies of 1 or a few Hz. In [10], interference effects, between the photodiode (return loss 15dB) and the fibred collimator, were considered as a possible origin for this noise. This has been confirmed using low return loss (typ. 35dB) photodiodes: the noise is roughly 10 times lower (yellow curves in Fig. 2), as expected if the noise is due to an interference. The fact that a further reduction is obtained by inserting optical isolators (light blue and purple curves in Fig. 2) also confirms this interpretation.

Fig. 2.

13.6 GHz amplitude modulation: stability of the distance measurements, obtained using photodiodes with 15 dB return loss (dark blue points) or with 35 dB return loss (all other data). Yellow points: without optical isolators. Blue and purple points: with optical isolator. The amplitude modulation was obtained by phase locking two single-mode lasers, except for the black line, obtained with a broadband, ASE (amplified spontaneous emission) source and a fibred, integrated amplitude modulator.

The lack of stability at long time scales, however, indicates the presence of systematic effects.

C.

Optical cross-talk and interference

Among other sources of systematic effects [10], optical cross-talk has to be considered. At first sight the finite extinction ratio of the PBS cube, used to separate between the measurement and reference paths, would seem to be a serious problem. In our measurement, however, the phase of the measured signals is not affected. Let us consider, for instance, that the PBS cube reflects, not only the S polarisation, but also a small fraction (typ. in optical power) of the P polarisation. Then the expected signal to be measured in the P (measurement path) channel, I_P(t) = (I / 2) × cos[δ(t –L / c)] in the case of a ideal PBS cube, is replaced by

where δ = 2πF is the pulsation of the HF modulation, and ω_av =2π(v_MasterLaser + F / 2) is the average optical pulsation. The third term (the second in size) results from the interference term between the two paths.

In range meters that measure the phase of the modulated beam, the third term is a significant source of systematic error, since it contributes to the signal with a phase δ(L + l) / 2c ≠δ L / c. In our measurement, these contributions are present. But since we operate at a frequency such that δ(L – l) / c is locked to zero (mod. 2π), the phases, δ(L + l) / 2c for the third term (or δl / c for the second) are identical to δL / c (mod.π). Hence the phases of the optical beat-note remains unaffected by this, otherwise large, systematic effect.

For any other interference the same conclusion does not hold, since the lengths involved do not share the same relation. For instance, reflexion at the entrance and exit sides of the PBS cube are expected to give rise to systematics. This can be checked by scanning the frequency of the master laser, and recording the open loop output of the LIA, all other parameters being kept constant. As shown in Fig. 3, the telemeter signal wobbles around its average value, in regular pattern. Two obvious free spectral ranges can be observed,

Fig. 3.

Scan of the frequency of the master laser (the scan, done manually, is not strictly regular) in the situation where the PBS cube in Fig. 1 is set at normal incidence and gives rise to interference effects. Green: open-loop telemeter signal. Blue: amplitude of one of the two photodiodes, demodulated with the same signature as the telemeter signal. The absence of correlation between the two signals shows that the contribution that affects the telemeter signals is not due to amplitude-to-phase coupling. The open loop telemeter signal can be kept <1µm by preventing interference due to normal incidence reflexions.

These two contributions are absent when the PBS cube is replaced by a thick, 45° plate polariser [11] to prevent interference effects. In that case the open loop signal is smaller than 1μm peak-to-peak.

Thus, reducing interference effects is critical for reaching high accuracy. The fact that, in the scheme presented above the HF amplitude modulation is obtained through the beat-note of two single-mode lasers is very useful in tracking the origin of systematic effects. But once they are minimised by choosing optics that minimise reflexion at normal incidence, then the importance of systematic effects can be further reduced by using a source with a coherence length shorter than the cavity involved in the interference.

Fig. 4 shows a typical set-up for a telemeter that would exploit modulation of a broadband source. A stability of about 7nm at 1 minute duration has been obtained with such a set-up (black curve in Fig. 2), using a ~5nm bandwidth, ASE (amplified spontaneous emission) source. The fact that stability is somewhat worse for shorter and longer time scales seems to be due to technical noise in the ASE source, and to drifts in the polarisation controller, respectively. It seems reasonable to expect further improvements at long and short time scales.

Fig. 4.

Setup for a telemeter that exploits an amplitude modulated broadband source. OI: optical isolator. The amplitude balance lock aims at keeping the measurement/reference optical balance, whatever the distance, hence cancelling systematics related to amplitude-to-phase coupling.

A.

Principles of the measurement

Although a remarkably simple set-up such as Fig. 4 is certainly of interest for space applications, it is expected to be limited to a ≈1nm√Hz noise floor due to photon statistics for 1mW detected light power. The search for better resolution implies that optical interference lies at the heart of the measurement. In the set-up in Fig. 5, the signal delivered by the “measurement photodiode” PhD1 results from the interference between the measurement beam (path length L) and the reference beam (path length l) and is exploited using the amplitude and phase reference provided by the “reference photodiode” PhD0.

Fig.5.

Set-up for detection of the mixed signal. PhD0 and PhD1:reference and measurement fast photodiodes. λ/2: fixed half-wave plate. (P)BS: (polarising) beam splitter.

The data acquisition records the values of the amplitudes A₀, A₁ and phases φ₀, φ₁ of the HF modulations detected by PhD0 and PhD1. Considering the different contributions to the signal at frequency F = δ / 2π, and writing ε² the fraction of power (of the order of one or a few %) that propagates along the reference path, one gets, for the HF signal from photodiode PhD1, the expression

in which constant factors have been omitted. The first two terms correspond to the beams that have followed, respectively, the measurement and reference path. In the “phasor” representation, they correspond to two (one large and one small) vectors, constant as long as L, l and δ are constant. Only the third term, the interference term, involves the optical wavelength, through ω_av =2πc / λ. When the optical wavelength is scanned over more than c /(L–l), the amplitude of the third term oscillates, and the corresponding vector in the phasor representation of Fig. 6, ranges from one to the other end of the segment drawn in dashed line.

Fig. 6.

Phasor representation of the detected PhD1 high frequency signal. a) shows the “segment”drawn by the resultant vector when the optical wavelength is scanned. b) shows the curve drawn by the resultant vector when the measurement path length L is scanned over more one synthetic wavelength Λ = c / F. For the clarity of the figure, an unrealistic value of 25 has been assumed for the ratio Λ / λ. The actual HF frequency is 20GHz and λ = 1.5 μm so that Λ / λ ≈ 10⁴ and the actual pattern in b) is actually made of 10⁴ “spikes”.

When the optical wavelength is kept constant, and the measurement path length L is scanned, not only the oscillating behavior mentioned above is observed for the third term of (3), but at the same time,

B.

Exploitation of the mixed signal

This complex shape may seem difficult to exploit for laser ranging. However, a time-of-flight measurement can be done beforehand, yielding ΔL_TOF ≡ (L – l)_TOF with an accuracy better than 0.1mm [12]. This preliminary information can be used to measure the mixed signal for three values λ, λ’ and λ’’ of the optical wavelength that differ by c/(2ΔL_TOF) and c/(4ΔL_TOF). Then, appropriate exploitation of Ae^iφ as a function of λ provides -i) the middle (M, in Fig. 6-a) of the segment, from which ΔL_synth, the intermediate precision ranging can be retrieved and -ii) the interferometric phase 2πΔL / λ, from which the high resolution ranging ΔL_Interf can be obtained. This last value will be a high accuracy (sub-nm) ΔL determination, provided that

Preliminary mixed signal data (Fig. 7), recorded with no optics at normal incidence, show deviations (3×10^-3peak-to-peak) that mainly originate in the non-linearity in the commercial oscilloscope used for data acquisition. More work is needed to bring the accuracy to better than 10^-4 and 10^-4 cycle in the amplitude and phase measurements of the HF signal from PhD1.

Fig. 7.

Recordings of the “segments” (see Fig. 6-a)) on the PhD1 signal (using PhD0 as a phase and amplitude reference) for 18 different positions of the target corner cube at a distance of 1.5m with 0.5 mm increments. Analog to digital conversion is made at the output of a mixer that shifts the frequency of the signal from 20 GHz to 20 MHz. The signal is kept low enough that the mixer is far from saturation.