Thermally compensated common-path differential interferometer with reduced long-term zero-drifts

Laser interferometers have served as the workhorses in the metrology of length for several decades. Their broader application brings further challenges, especially for longer measurement time-frames or outside the laboratory environment with strictly controlled conditions. As a part of our team’s activities aiming at characterizing and eliminating the effects of unstable temperature on interferometric length measurements, we report on successfully remodelling a differential interferometer’s optical arrangement focused on increasing resilience against temperature changes. The experimental characterization under constant temperature and subsequently under thermal load proved a tenfold decrease in short-term fluctuations and reduced sensitivity to temperature changes by a factor of 100.


Introduction
The development of nano-and macro-scale dimensional measurement brings new challenges in extending high-precision measurement procedures from a well-controlled experimental environment with a typically single measurement cycle to a manufacturing and industrial environment with constant load.
In the area of laser interferometry, which is both a cornerstone of length metrology [1,2] and an essential part of nanometrology, we have focused on the long-term stability of measurements in interferometry-based measurement methods and applications.With resolution and accuracy at a level smaller than the size of single atoms, there are naturally a number of effects that must be dealt with that negatively affect * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. the measurement result.Many of these aspects have already been well described in the literature [3] and are still being actively investigated-both in terms of causes and dealing with the consequences.In particular, however, the issue of longterm measurement stability, usually expressed as so-called zero-drift, e.g. in the ANSI/ASME B89.1.8standard [4], has either been overlooked or has yet to be a focus of interest to date.
Well-characterized and optimally compensated zero drift of measurement apparatus generally becomes more critical with use in environments with less controlled parameters and with increasing measurement timeframes.Typical examples of such requirements are found in applications that require precise and stable positioning (long scans in nanometrology, long exposures in electron lithography, positioning of reference objects in optical metrology), in applications where the interferometer(s) serve as a length reference (large volume metrology, calibration with more extensive than usual number of measurement points or repetitions [5]) and of course research areas (e.g.Gravitational wave research [6] or unit kilogram realization using Kibble balance [7]).
Our goal was to develop an interferometer that would integrate state-of-the-art approaches to eliminate the errors that burden the interferometric measurement (e.g.Abbe and cosine errors, guide rail and guidance errors from the translation mechanisms, dead-path, thermo-mechanical effects, vibrations).We aimed to bring the ultimate accuracy of the interferometric measurements performed in the dedicated metrology laboratories to the industrial metrology environment.We were mainly focused on the area of 1D dimensional measurements where interferometers serve as length references for length calibrations, testing or characterizations.
Our previous work presented an innovative optical arrangement that realizes a four-beam, two-pass differential planar interferometer with co-planar beams and co-axial arms [8].Later on, it was identified that the arrangement is sensitive to ambient temperature changes to a greater extent than is suitable for particular applications.It turned out to be a significant deficiency, especially considering the differential commonpath interferometers are made to excel in this discipline by design.
In this paper, we report on redesigning the optical path layout of our design's differential common path interferometer to achieve balanced path lengths of the two interferometer arms within the bulk optics.The subsequent characterization with a common fixed mirror for the interferometer's co-planar and co-axial arms over several weeks revealed a significant reduction of temperature sensitivity and drifts.

Zero-drift in interferometric measurement
In laser interferometry, the zero drift refers to the gradual (phase) change of the interferometer's output over time [4].The term possibly refers to the conceptual case where an interferometer with a value reset to zero is kept in a steady position, and the value deviates from zero over time.Such a deviation necessarily has a straightforward impact on the measurement uncertainty associated with the interferometric measurement.
The zero drift is not a single influence but rather a superposition of several effects stemming from multiple sources.We categorize these sources according to the principal components of an interferometer-based measurement instrumentation: Interferometer (and optics) is susceptible to thermal effects.
A fluctuating temperature induces changes in the length of the optical paths in the interferometer's optics and also changes the refractive index of the optics [9] and, consequently, the effective wavelength of the laser.These substantial contributions to zero drifts are particularly focused on in this paper, and their mitigation motivates the interferometer's redesign we present in this paper.Mechanics (stress, vibrations) -the mechanical construction induces drifts typically due to a combination of several aspects: temperature fluctuations (again) cause thermal expansion and together with vibration, they can combine with existing material stresses or pre-loads to induce geometrical changes.Apart from isolating the measurement from temperature fluctuations and vibrations, the two fundamental approaches to mitigate these drifts are avoiding them by designing using rigid and low-expansion materials in combination with best practice [10] or introducing redundancy in the measurement [11].Laser noise incorporates fluctuations in polarization state, amplitude and optical frequency.Frequency drifts are proportional to the optical path length difference between the two interferometer's arms.Any drifts in amplitude and polarization might gradually add up to distortion or offsets of the Lissajeous diagram, introducing thus zero drifts to the interferometer's reading.Environment (air) influences the effective wavelength of the laser radiation outside the optics, typically between the interferometer and the reflector that forms the arms of the interferometer.Even with the refraction compensated, typically using indirect refractometry and empirical equations [12,13], any residuals influence due to limited precision or long-term stability of the environmental sensors or inhomogeneities in the refractive index add up to the drifts.Detection electronics are susceptible to noise and thermal effects, especially in the analogue circuitry.Typical examples are analogue-to-digital conversion drifts stemming from temperature-sensitive voltage references or signal convertors drifting with supply voltage from linear stabilizers.The temperature effects are pronounced since electronics are producers of waste heat.Ultimately, these influences can add up to the distortion of the Lissajeous diagram in the same way as the polarization and amplitude noise do.
Also, over time, materials used in interferometer construction may undergo changes in physical properties, contributing to zero drift due to ageing and long-term stability issues; nonetheless, this issue is probably far outside of the operational scope of displacement measuring interferometers for the time being.The drift effects are notably dominated by temperature fluctuations (as discussed, e.g. in [14]).
The presence of zero drifts in interferometric measurements has been mentioned since the advent of commercial interferometric systems [15].All the contributions mentioned in the list have been investigated since then.Nonetheless, to our best knowledge, the zero drift in displacement measurement interferometry has not been comprehensively investigated as a whole.The zero-drift as a phenomenon has been studied in adjacent areas of engineering (e.g.[16,17]) or as a partial aspect of specific methods (e.g.[18,19]) related to interferometry.
In the case of the temperature influences on the interferometer's optics-the aspect we focus on in this work-the magnitude of the drifts is strongly linked to the optical layout of the interferometer, particularly any difference in the length between the reference arm path and object arm path [20].Later works state that a typical value for optics thermal drift is 0, 5 µm K −1 [21]; nonetheless, this value would scale significantly with nominal dimensions of the interferometer and also differ significantly with the optical materials used.

Modified arrangement of the common-path differential interferometer
The optics of the previously designed interferometer [8] were arranged and realized as a separate compact monoblock without the need for an external beam-splitting element.It was assembled with optical contacting (wringing) to increase the thermo-mechanical and adjustment stability of the optical elements.By design, the length of the paths in the individual interferometer arms (the portions of the paths leading through the optical elements) differed significantly.Subsequent applications of the interferometer have revealed that this fact will cause a more pronounced sensitivity to changes in ambient temperature than feasible.
The central part of the new optical arrangement, shown in figure 1(a), is a polarizing beam splitter (PBS) that splits the incident beam into the interferometer's two polarizationseparated arms: object arm ('obj', inner beam pair, emphasized with blue dashed line) and reference arm ('ref'; outer pair).
From the splitting point, the reflected object beam leaves the interferometer (first pass through the PBS) and bounces off the object surface, passing the quarter wave plate twice along the way.Then it passes through the PBS (pass #2), bounces off the backside corner cube (CC), passes PBS again (pass #3) and leaves the interferometer.After the second reflection from the object surface, the object beam enters the interferometer and this time, it reflects from the splitting plane towards the auxiliary retarder QWP 2 (pass #4).Here, it reflects from the coating 1 , passes through the PBS (pass #5), turns around on the auxiliary cube corner CC 3 and makes a final pass through the PBS (#6) to the output.
The optical path of the reference beam is analogic.From the splitting plane, the beam reaches the auxiliary cube corner CC 2 (pass #1), where it turns around and passes to the auxiliary retarder QWP 3 (pass #2).From here, through the PBS (pass #3), it leaves the interferometer and bounces from the reference surface.After re-entering the PBS, the beam turns around in the CC and leaves the interferometer again (pass #4 + #5).After the second return, the beam bounces off on the splitting layer to the output (pass #6).At the output, the recombined (and still polarization-separated) beams continue hand-in-hand towards the integrated four-diode homodyne receiver (figure 1(c)).
The optical components were manufactured from fused silica (bulk components) and crystal quartz (retardation plates), which have low and approximately matching coefficient of thermal expansion.The components, coated with anti-reflection coatings, were assembled using optical cementing.The optical assembly (56 × 51 × 20 mm) was enclosed in a titanium housing (117 × 63 × 25 mm, slightly larger than before), which also integrated the homodyne phase receiver, the optional position-sensitive detection facility (not used in this study) and a collimator slot for a convenient optical fibre light delivery.
The integrated homodyne receiver, principally described, e.g. in [22], used a four-detector arrangement and incorporated an active preamplifier.For the fibre delivery, a customized fibre collimator (1/e 2 beam diameter of 2 mm) with additional anti-reflective coating on the fibre tips to prevent return loss and resonator effects between the fibre end and the interferometer was used.A calcite polarizer (Thorlabs GT5) was inserted between the collimator and interferometer.
With the new layout of the optical path, both beams of the individual arms pass the beam splitter PBS six times, large retro-reflector CC once, the auxiliary CC 2 / CC 3 once and reflect from auxiliary retarder QWP 2 / QWP 3 once.On the contrary, in the original arrangement (figure 1(b)), the reference beam passes the beam splitter four times and the backside CC once, while the object beam has eight PBS passes, one CC pass, two auxiliary CC passes and two reflections at the auxiliary QWP.
As a result, the optical pathlengths of the two arms in the optics differ by four PBS passes (32 mm each), two small CC passes (38 mm each) and four passes through the waveplates (≈0.7 mm each).Omitting the waveplates (for simplicity), the length difference δOPL totals 204 mm.The thermooptic coefficient δn/δt of fused silica is approximately 8, 81 • 10 −6 [9] and the coefficient of thermal C te expansion is 0, 55 • 10 −6 .Considering ideally homogeneous heating/cooling of the interferometer's optics, a change of temperature will result in the change displacement reading δL/δt at the interferometer output: With δt = 1 K, we can estimate the sensitivity of the interferometer assembly to the thermal changes cT to be Following the same approach, we estimated that the thermal sensitivity of the new, compensated arrangement should be close to zero.

Experimental characterization
The experimental characterization was carried out in a temperature-controlled chamber that is currently being developed at ISI [14].We have investigated two interferometers: one with the original arrangement [8], assembled using optical contacting (labelled original) and one with the modified arrangement (presented in this paper, labelled compensated), assembled using optical cementing.Both samples were manufactured of the fused silica (bulk elements) and crystal quartz (retarder plates).The experimental arrangement, depicted in figure 2, comprises the enclosure, a set of electronics (acquisition and control modules, power supplies, sensor interrogators) and several sensors.The enclosure is equipped with thermo-electric cells that enable active temperature control, currently reaching the temperature stability 1σ < 20 mK over days and weeks.A dedicated temperature controller TC (TREGXN, ISI) interrogates the servo thermometer T servo (with 0, 4 mK resolution) and drives the current to the thermo-electric cells (up to 1, 9 A).The environmental interrogation unit E.I.U.(originally an indirect refractometer [23]) handles a set of environmental sensors: three temperature sensors with the resolution of 3 mK were calibrated with U = 0, 08 K (range from 20 • C to 29 • C, k = 2), the pressure sensor (resolution > 1 Pa, range from 85 kPa to 102 kPa, U = 0, 01 kPa) and sensor for the relative humidity (U = 1, 5 %).The triplet of temperature sensors is used to measure the temperature inside the enclosure (reference temperature T ref at the same point as the servo thermometer, which is not calibrated on an absolute scale), external temperature T external and the vertical temperature gradient T gradient .The twochannel interferometric module IFM interrogates the quadrature sine-cosine output from the homodyne optical receivers of the two interferometers under test.The power supply unit PWR feeds the E.I.U.
Two experimental tests were carried out; both were ≈ ten days long.First, the interferometers were tested under nearly constant temperature, and then we applied a thermal load to investigate the response of the interferometers' indication to the thermal changes.In the second test, the temperature was modulated stepwise ±1 K around the nominal temperature with gradual transitions between the steps (heating rate of 0, 2 K•hr −1 ).

Experimental results
The thermal conditions inside the temperature enclosure during the first test are depicted in figure 3. The plots indicate the temperature inside the enclosure was stable to σ = 12, 4 mK   and that the coincidence between the servo thermometer and the reference thermometer (σ = 3, 2 mK) is at the level of resolution of the reference.The visible peaks on the reference temperature recording apparently coincide with the temperature outside the enclosure-the magnitude of the temperature changes inside ≈17, 5 times smaller compared to the outside.Also, a vertical gradient is present (−2, 03 K•m −1 ) but remains fairly constant (σ = 0, 17 K•m −1 ).
The measurement of displacement drift under constant temperature, shown in figure 4, revealed significantly reduced fluctuations and drifts with the modified arrangement.Compared to the original one (fluctuations in the range of 45 nm peak-to-peak), the improvement is approximately tenfold with the modified arrangement (fluctuations 4, 5 nm peak-to-peak).However, a small residual drift (0, 43 nm•d −1 ) was observed, which could be attributed to the presence of a mechanical preload in the interferometer's fixture that relaxes at this very slow rate.
The measurement of displacement drift under the temperature load, shown in figure 5, revealed a significant difference in the temperature sensitivity between the interferometers.While the original arrangement exhibited the temperature sensitivity of ≈0.5 µm•K −1 (which corresponds well to the estimation in equation ( 4)), the modified arrangement exhibited only residual drifts in the range of 10 nm peak-to-peak.We attribute this residuals to uneven heating of the optics.The results suggest that the sensitivity to the thermal changes is reduced by a factor of one hundred with the modified arrangement.

Final words
We presented a successful modification of the optical layout of a differential interferometer to suppress temperature sensitivity.Within the subsequent experimental work, we have studied the fluctuation drifts simultaneously in the two variants of the interferometric arrangements.We have demonstrated, and the results prove, a tenfold reduction of the long-term fluctuations and a more than a hundred-fold reduction in the sensitivity of the differential interferometer optics to temperature effects.The results make a small but significant contribution on the way towards stable and accurate interferometric measurements either beyond the walls of the top-grade metrological laboratories or in extended timeframes.

Figure 1 .
Figure 1.The modified arrangement of the differential interferometer with the optical paths in the two arms of an equal length (a), unlike in the original version (b); the interferometer housing contains an integrated homodyne receiver (c).

Figure 2 .
Figure 2. The experimental assembly in a block diagram: the interferometer is used with a single, fixed mirror inside a thermally insulated chamber; the chamber is equipped with refractometric sensors and active temperature stabilization.

Figure 3 .
Figure 3. Temperature conditions around the enclosure, from top to bottom: the temperature inside the enclosure, the coincidence between the servo thermometer and the reference thermometer, external temperature and the vertical gradient.

Figure 4 .
Figure 4.The measurement of displacement drift under constant temperature: the observed drift in the two interferometers under test (top), temperature conditions during measurement (middle) and correlation between the temperature and drifts (bottom).

Figure 5 .
Figure 5.The measurement of displacement drift under thermal load: the observed drift in the two interferometers under test (top), temperature conditions during measurement (middle) and correlation between the temperature and drifts (bottom).