# A GPS-Referenced Wavelength Standard for High-Precision Displacement Interferometry at λ = 633 nm

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## Abstract

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^{−11}. In this contribution, the application of a He-Ne laser source permanently disciplined to a GPS-referenced frequency comb for the interferometric measurements in a nanopositioning machine with a measuring volume of 200 mm × 200 mm × 25 mm (NPMM-200) is discussed. For this purpose, the frequency stability of the GPS-referenced comb is characterized by heterodyning with a diode laser referenced to an ultrastable cavity. Based on this comparison, an uncertainty of u = 9.2 × 10

^{−12}(τ = 8 s, k = 2) for the GPS-referenced comb has been obtained. By stabilizing a tunable He-Ne source to a single comb line, the long-term frequency stability of the comb is transferred onto our gas lasers increasing their long-term stability by three orders of magnitude. Second, short-term fluctuations-related length measurement errors were reduced to a value that falls below the nominal resolving capabilities of our interferometers (ΔL/L = 2.9 × 10

^{−11}). Both measures make the influence of frequency distortions on the interferometric length measurement within the NPMM-200 negligible. Furthermore, this approach establishes a permanent link of interferometric length measurements to an atomic clock.

## 1. Introduction

_{Rep}, and carrier envelope offset (CEO) frequency f

_{CEO}[8]:

_{Rep}and f

_{CEO}are usually in the rf domain, thus phase locking them to an atomic clock provides a direct link of an optical frequency to a primary frequency standard and consequently, the SI meter definition. In this way, ultraprecise, traceable optical frequency measurements can now be performed with unprecedented small experimental effort and cost. Within the revision of the practical realization of the definition of the meter by the Comité international des poids et mesures (CIPM) in 2002, it was considered

“… that new femtosecond comb techniques have clear significance for relating the frequency of high-stability optical frequency standards realizing the SI second, that these techniques represent a convenient measurement technique for providing traceability to the International System of Units (SI) and that comb technology also can provide frequency sources as well as a measurement technique.”

^{−12}appears to be sufficient [13,14]. Their unique features also turned them into a versatile tool for precise interferometric distance measurements that are playing a crucial role for the advance of precision engineering and nanotechnology [15]. Especially for applications in nanotechnology, length metrology has to face tremendous challenges due to increasing demands on measurement range, accuracy, traverse speed, and traceability (see e.g., [16]). Herein, the combination of state-of-the-art interferometric methods with optical frequency combs has put the term “high-precision” to the next level and created new possibilities for absolute and incremental interferometric length measurements.

^{−8}, that was mainly determined by the uncertainty of the empirical compensation of the refractive index was demonstrated and suggested for traceable, precision distance measurements for mass-production of optoelectronic devices [23].

^{−9}over a 24 h measurement results in a measurement error of 0.4–1.2 nm if a maximum measurement range of 200 mm is considered [35]. Additionally, degradation due to the aging process of the gain medium cause verifiable drifting in the center frequency during lifetime operation of He-Ne lasers [36,37]. Niebauer et al. believe that an effective accuracy of 1 × 10

^{−9}is technically feasible with two calibrations per year [36]. Thus, they have to be periodically measured against a Bureau International des Poids et Measures (BIPM)-compliant frequency standard to guarantee full traceability of interferometric length measurements within the NPMM-200 to the SI meter definition [35].

## 2. Experimental Configuration and Methods

#### 2.1. State of the Art Length Measuring System of the NPMM-200

_{ambient.}Therefore, changes of the laser wavelength are directly considered as length measurement errors ΔL

_{Meas}. The uncertainty in the determination of the laser frequency ν and n

_{ambient}thus finally limit the precision of length measurements in the NPMM-200. For measurements under ambient conditions, the environmental parameters (pressure, temperature, and humidity) are permanently monitored and the refractive index is calculated from an updated version of the Edlén formula [42]. Nevertheless, the compensation of the refractive index due to these empirical equations is usually limited to an uncertainty of 10

^{−7}–10

^{−8}, mainly caused by uncertainty contributors resulting from the determination of ambient conditions and the air composition [43,44]. Therefore, the NPMM-200 can be additionally operated under vacuum down to 1 mbar to reduce the influence of the refractive index. The signal processing unit of the interferometers utilizes 16-bit A/D-converters for a demodulation of the interferometer signals [40]. Their resolving capability was determined according to [45] (see Table 1 in [45]) with a value of 5.72 pm, assuming an analog-to digital quantization of 16 bit and a rounding error of 16 bit for the arctan function. The interferometers are fiber-coupled and deploy frequency-stabilized He-Ne lasers to minimize the influence of frequency fluctuations on the length measurement. Each of the lasers is independently stabilized by a two-mode comparison technique [46]. Frequency changes can be achieved by varying the resonator length due to increasing the current through the silver heating coil around the He-Ne tubes (see Figure 1). Currently, each of the interferometer axes is powered by one He-Ne laser. These lasers are embedded in a compact, enclosed laser module containing the laser control and stabilization electronics. The He-Ne lasers provide a maximum output power of 700–900 µW per interferometer axis after fiber coupling. The laser module is placed outside the NPMM-200 together with the rest of the NPMM-200 control electronics to avoid heat dissipation from the laser sources (maximum electrical power consumption 60 W) into the NPMM-200 [40]. Considering a typical measurement interval of 24 h, the polarization-stabilized He-Ne lasers used within the NPMM-200 showed maximum frequency deviations of 1.2–2.7 MHz (relative frequency deviation Δf/f = 2.5 × 10

^{−9}–5.7 × 10

^{−9}) from the mean value resulting in a measurement error of 0.5 nm–1.1 nm for a maximum measurement range of 200 mm [35]. Those results are in accordance with other long-term measurements of internally stabilized He-Ne lasers [47]. Additionally, changes of the center frequency and the shape of the gain profile directly influence the absolute frequency value of the polarization stabilized He-Ne lasers since the control signals for frequency stabilization are directly derived from the gain profile. Therefore, their lifetime wavelength stability is usually limited to 2 × 10

^{−8}[35,47]. For a measurement range of 200 mm, the corresponding frequency changes of 5–10 MHz would result in a systematic length measurement error of 2–4 nm [35]. To overcome these errors during long-term operation of the NPMM and guarantee full traceability of the length measurement to the SI meter definition, the absolute frequency value of the metrology lasers has to be periodically measured against a BIPM-compliant frequency standard. For the purpose of frequency calibration, the iodine-stabilized He-Ne laser operating at a wavelength of 633 nm with an uncertainty of u = 10 kHz (2.1 × 10

^{−11}) has been mainly used in the visible spectrum for the practical realization of the SI meter definition [48,49]. However, this level of uncertainty can normally only be stated by national metrology institutes (NMI). In Figure 2, a possible configuration for laser vacuum wavelength calibration including iodine-stabilized He-Ne lasers is presented. To demonstrate the hierarchy between the primary realization of time and the realization of the meter definition by using frequency measurements all possible steps with their respective reported uncertainties are included. National metrology institutes (NMIs) realize the unit of time with the highest accuracy. For the best of these primary frequency standards, relative standard uncertainties in the order of 10

^{−16}have been reported [50,51]. The NMIs often hold secondary representations of the second such as optical strontium clocks or rubidium oscillators whose frequencies have to be measured against the primary standards leading to relative uncertainties between 10

^{−14}and 10

^{−16}[50]. Other commercially available frequency standards as cesium (Cs) clocks or hydrogen masers are often redundantly operated from the NMIs for time-service applications [50]. They can furthermore be applied as rf-references for optical frequency combs, determining their uncertainty [52] in case of, e.g., cesium clocks to a level of 5 × 10

^{−13}[50].

^{−11}can increase to 8.4 × 10

^{−11}for systems operating as a secondary standard in industry or research labs [54]. When measuring a polarization-stabilized against an iodine-stabilized He-Ne laser the uncertainty of the frequency calibration is given by the frequency stability of the specific laser under test [47]. For polarization-stabilized lasers, relative standard uncertainties (Typ B, k = 2) between 1.3 × 10

^{−9}and 5.7 × 10

^{−9}for a 24 h-measurement interval have been reported [47]. Based on this uncertainty hierarchy of frequency calibration and the described state-of-the-art configuration of the length measurement system, the following requirements for an ultrastable, traceable next generation metrology laser for displacement interferometry in the NPMM-200 can be deduced:

^{−11}(5.72 pm at 200-mm measurement range). Taking into account a resolution enhancement in A/D conversion and a possible further extension of measurement ranges up to 1 m, there will prospectively be even a frequency stability of 10

^{−12}desirable. Ideally, this frequency stability should be available irrespective of integration time. Finally, to control the absolute frequency of the laser source and thus completely trace back the interferometric length measurements, the metrology laser should ideally provide a permanent link to a primary frequency standard [55]. These requirements cannot be fulfilled with the currently used polarization-stabilized He-Ne lasers. In the following sections, we will thus introduce the new comb-based concept for interferometric distance measurements within the NPMM-200.

#### 2.2. Metrological Basis for Generation of Ultrastable, Traceable Optical Frequencies at 633 nm

#### 2.2.1. GPS-Referenced Frequency Comb (OFC)

_{Rep}= 250 MHz that can be adjusted between ± 1 MHz [60]. A 2f-f interferometer configuration is used for the detection of the CEO that is set to f

_{CEO}= 20 MHz by default [60,61]. For the 2f-f interferometer, one part of the laser output is amplified in an erbium-doped fiber amplifier (EDFA) and afterwards spectrally broadened in a highly nonlinear fiber (HNLF). The frequency comb provides six output ports at 1550 nm with a maximum power output of 18 mW and a spectral width (FWHM) of 20 nm. Additionally, the comb is equipped with a second amplifier unit that generates a 633 nm output by spectral broadening in a HNLF and subsequent frequency doubling. This output provides an average power of 7 mW and a spectral width of 3 nm (FWHM). It can be used to create a beat signal with laser sources at 633 nm, e.g., He-Ne lasers. The data acquisition system of the comb is equipped with four Π-type frequency counters having a gate time of τ = 1 s (model: FXM50) [62] and two spectrum analyzers (model: HMS-X, Rohde & Schwarz). The frequency counters are referenced by the 10 MHz output of the SDU. The comb parameters f

_{Rep}and f

_{CEO}are monitored with two frequency counters. The remaining frequency counters can be used for the counting of beat frequencies with external laser sources. The feedback loops of the comb phase lock the repetition rate and the CEO to a multiple of the 10 MHz-frequency output [60]. A detailed description of the control loops can be found in [35,60]. An undisturbed reception of the GPS signal together with a permanent closed loop operation of the local oscillator are the basic prerequisite for a sufficient stabilization of the comb parameters and thus the generation of stable, traceable optical frequencies [14,35]. To control the locking status of the comb parameters, f

_{Rep}and f

_{CEO}are permanently measured by the frequency counters of the data acquisition system after the respective servo loops have been closed. For a continuous monitoring of the incoming GPS signals we extract data about the number of available satellites and the overall status of the GPS signals from the transmitted data files via the serial interface of the GPSDO. During a typical 24-h operation, we found that the number of available satellites usually varies between 6 and 10 which is sufficient for a distortion-free operation of the GPSDO. When proper operation of the GPDSO is assured, the manufacturer guarantees a relative accuracy better than 8 × 10

^{−12}(τ = 1 s) and a relative frequency stability (relative Allan deviation) A

_{Dev}better than 4 × 10

^{−12}[57]. To determine the absolute frequency of external laser sources, the specific laser under test is superimposed with a comb line and the time dependence of the generated heterodyne signal is analyzed (see Section 2.4). For this purpose, the OFC is equipped with a free-space beat detection unit (FS-BDU) consisting of polarizing beam splitters, half-wave-plates, a diffraction grating and an avalanche photodiode (Model: APD210, Menlo Systems) for spatially overlapping the beams from the frequency comb and the external laser source (see e.g., [35]). For the detection of the heterodyne time traces, one of the counters of the data acquisition system is used.

#### 2.2.2. Optical Reference System (ORS)

^{−9}mbar. The optics required to couple the light from the ECDL into the cavity and the opto-electronical components needed for the implementation of the PDH-Lock (electro-optic modulator for generation of sidebands at 20 MHz, quarter-waveplate, half-wave plate, polarizing beam splitter, photodetector, see Figure 4) are mounted on a breadboard vertically attached to the vacuum chamber. The cavity and all opto-mechanical parts are placed in an acoustic isolation box, mounted on an additional vibration isolation platform [69]. A control unit contains all electronics necessary for operating, tuning and locking the laser as well as the electronics for the temperature stabilization of the cavity [69]. The ECDL, ULE-cavity with optical platform and high vacuum pump and the control electronics are integrated into a compact 19” rack. The system provides an output power of 10 mW that can be accessed via a fiber-coupled, polarization-maintaining output port [69]. The ORS can be used either as stand-alone system or as an optical reference for the OFC (see e.g., [70]).

_{Dev}= 3 × 10

^{−15}at τ = 1 s [71]. Within the frame of the present work, we will apply the ORS as an ultrastable reference laser to characterize the short-term frequency instability of the OFC. This allows for the first time to directly resolve the instability behavior of our GPS-referenced frequency comb. For this purpose, the output of the ORS is heterodyned with one of the comb modes in a fully fiber-coupled beat detection unit (FF-BDU) [69].

#### 2.3. Comb-Referenced, Fiber-Coupled He-Ne Laser Source

^{−9}(see Section 3.2).

#### 2.4. Analysis of Frequency Instability

_{i}averaged over a specific sampling time τ that can be obtained as an integer multiple m of a basic measurement interval τ

_{0}[77,78]:

_{0}= 1 s. For practical calculations of the Allan variance, usually overlapping samples are used to improve the confidence of the stability estimate and the results are expressed as the square root of ${\sigma}_{y}^{2}\left(\tau \right),$ the Allan deviation A

_{Dev}[78,79].

## 3. Results and Discussion

#### 3.1. Frequency Instability of the GPS-Referenced Frequency Comb

^{−8}with f

_{HeNe}= 473.6127 THz) frequency change of the respective comb mode (n = 1,894,861) that is reflected in the frequency drift of the beat frequency between the comb line and the ORS. The opposite sign of the slope is caused by the sign of the beat signal. Since the ORS frequency lies above the considered comb line, an increase in the repetition rate leads to a decrease of the beat frequency as demonstrated in the inset of Figure 7b.

_{Rep}and f

_{CEO}locked to the rf reference are presented. Both comb parameters follow the reference frequency provided by the GPSDO with a maximum frequency deviation from the mean value of 2 mHz (Δf/f

_{Rep}= 8 × 10

^{−12}) for f

_{Rep}and 6 Hz (Δf/f

_{CEO}= 3 × 10

^{−7}) for f

_{CEO}. The respective Allan deviations in Figure 8c present the tracking stability of the comb parameters with a τ

^{−1}-dependence for both parameters. This corresponds to white phase noise characteristics as supported by the modified Allan deviation showing a τ-

^{−3/2}-dependence (see e.g., [78]) for integration times below 10,000 s (not shown Figure 8c).

^{−11}, f = 474 THz) in 24 h and an overall frequency drift of 28.9 kHz within the observation time of 5 days. The frequency drift of the ORS cavity thus dominates the long-term stability of the heterodyne signal as can be deduced from the increase of the Allan deviation for τ > 4000 s in Figure 9c and the data points of the Allan deviation of a linear drift rate introduced as a red line in Figure 9c. Nevertheless, the observed drift rate is much smaller than the drift rate of 150 mHz/s as reported by the manufacturer [69]. This can be attributed to aging effects of the cavity material as reported in [64] and is consistent with prior measurements where a drift rate of 104 mHz/s was obtained [80].

^{−11}) was observed from the drift-corrected frequency values of the beat signal. Over a timeframe of 1 h, a typical maximum frequency deviation of 14.8 kHz (Δf/f = 3.1 × 10

^{−11}, obtained as a mean value of 1-h sections within the 5-day time frame for τ = 1 s) occurred. The measurement data of the unlocked comb presented in Figure 7 demonstrated the influence of small frequency changes of the repetition rate on the overall frequency instability of a comb line. Since the measurement of the comb parameters revealed a high tracking stability (Figure 8), the Allan deviation of the heterodyne signal between the comb line and the ORS thus essentially reflects the frequency instability of the GPSDO. For comparison, Figure 9 additionally contains Allan deviation data provided by the manufacturer (gray line in Figure 9c). This dataset was obtained from a 13-h measurement series of a comb–comb comparison with two GPS-referenced frequency combs [57]. For short integration times, the Allan deviation increases with a slope of τ

^{0.2}, reaching a distinct maximum of 3.6 × 10

^{−12}at τ = 64 s. For longer integration times, the Allan deviation decreases. This behavior is consistent with former results reported in the literature [81,82,83,84]. The frequency output of the OCXO is steered to agree with the signals transmitted by the GPS satellites and can thus take advantage of their high long-term stability [13,84].

^{−12}at τ = 8 s is observed. The origin of this discrepancy is currently not clear. We did not find any indications that the settings of the comb parameter control loops (e.g., a too high P-gain) or the additional integration of the SDU led to these differences. Therefore, further investigations are needed to clarify the origin of this degradation in short-term-frequency stability compared to the manufacturer data. Since the accuracy of the GPSDO over a given period of time is usually limited by its stability, it is common sense to use twice the Allan deviation for a time duration that orientates towards the duration of frequency calibration to determine the frequency uncertainty of a GPSDO [82]. Covering a worst-case scenario, manufacturer data rather tend to “overestimate” uncertainty by relying on the highest Allan deviation reported (see Section 2.2). Taking these considerations into account and regarding that the frequency instability of a comb line is primarily determined by the frequency instability of the used rf reference (see e.g., [52]), we have to claim an uncertainty of 9.2 × 10

^{−12}(k = 2, τ = 8 s) for the GPS-referenced frequency comb at the TU Ilmenau. Therefore, reaching an uncertainty below 10

^{−12}makes an integration time of at least 8000 s necessary.

#### 3.2. Frequency Instability of the Comb-Referenced He-Ne Source

^{−10}at τ = 8000 s. The first laser serving as a secondary standard laser (SSL) obeys a maximum frequency deviation of 1 MHz. Although both lasers are identical in construction and their long-term frequency instability is comparable as deduced from the intersection of the Allan deviation for τ > 100 s, they obey a different frequency stability behavior within the short-term regime as can be seen from the frequency fluctuations in Figure 10a,b as well as the Allan deviation below 100 s. Nevertheless, the frequency instability of both lasers is consistent with the results we recently reported for the lasers of the He-Ne module of the NPMM-200 [35]. Thus, the heterodyne source can be considered as a suitable representative of the frequency characteristics of the NPMM-200 laser module and an appropriate candidate for the realization of a comb-referenced laser module.

_{Beat}= f

_{Beat}− $\overline{{f}_{Beat}}$ after closing the control loop is shown. Additionally, the changes of the heterodyne signal when the SSL is in its internal stabilization regime are plotted for comparison from Figure 10a. When the control loop between the comb line and the SSL is closed, the frequency deviations of the SSL are eliminated and the beat signal reflects the tracking stability of the SSL in relation to the comb line [75]. In Figure 11c, a 1 h-section of the measurement series is enlarged. The upper graph of Figure 11c additionally contains the changes of the beat signal of the comb line obtained from the measurement data of the beat frequency between a comb line and the ORS as discussed in Section 3.1. Please note that these two measurements were not taken simultaneously and their absolute frequencies correspond to two different comb lines due to the different absolute frequencies of the ORS and the He-Ne lasers.

^{−12}with f = 474 THz) with a standard deviation of 146 Hz for the in-loop heterodyne signal between the SSL and the comb line and of 62 Hz (Δf/f = 1.3 × 10

^{−13}) with a standard deviation of 5 Hz for the in-loop heterodyne signal between the SSL and the ML within a 1 h measurement window (values are obtained as a mean value of 1-h sections within the 10-h measurement window for τ = 1 s). On the other hand, the deviations of the mean value of the measured beat frequencies from the nominal values of the closed-loop control are −47 Hz (Δf/f = 9.9 × 10

^{−14}) at f

_{0}= 62.5 MHz for the SSL and 204 Hz (Δf/f = 4.3 × 10

^{−13}) for the ML at f

_{0}= 4.5 MHz [35]. Comparing this to the observed maximum frequency deviations from the mean value, it can be deduced that the accuracy of the ML control loop is limited by a frequency offset and not the frequency stability of the beat signal.

^{−0.8}(τ

^{−0.6}) for integration times between 2 s and 500 s that flattens down to a τ

^{−0.2}–dependence for integration times above 1000 s. The frequency stability of the comb-referenced metrology laser is thus finally determined by the Allan deviation of the comb line. The frequency fluctuations and differences in the Allan deviation of the beat signals between the SSL/comb line and the SSL/ML furthermore indicate that the limited and similar dynamic behavior of the laser control systems produces some lowpass filtering effects on the higher frequency distortions of the comb line. In this case, the locked SSL would not be able to follow fast comb line jitters leading to a “smoothing” out of frequency distortions of the comb line [75]. To prove this thesis, we now prepare direct measurements of the He-Ne-locked system against the ORS and concentrate on more detailed investigations on the frequency noise properties of the comb line and the in-loop-comb-referenced metrology laser. Finally, comparing the Allan deviation of the comb line with the Allan deviation of the SSL that represents the previous status of frequency stability in the NPMM-200, an improvement of nearly two orders of magnitude at τ = 1 s and of up to three orders of magnitude at τ = 10,000 s can be achieved. Thus, the approach of a comb-referenced He-Ne laser source allows a significant improvement of the long-term frequency stability of these lasers and additional changes in absolute frequency due to aging effects can permanently be monitored and avoided once their absolute frequency has been determined as described in [35]. As a result, the traceability chain for the measurement of laser frequency is significantly changed in comparison to Figure 2. The NMIs contribute to the calculation of the UTC by sending data from their local clocks to the BIPM [82]. The GPS satellites carry atomic oscillators and are controlled by the United States Department of Defense (U.S. DoD) [13]. The GPS time is referenced to the UTC time scale maintained at the United States Naval Observatory UTC (USNO) which is the largest contributor to the calculation of the UTC [82]. This way, the GPS receivers generate output signals that agree with UTC (USNO) [82]. Within the GPSDO, the output signals from the GPS receiver are used to discipline the local oscillator providing a frequency output with an uncertainty that is mainly determined by the frequency instability of the locked oscillator [13,82]. The comb technology allows to transfer the accuracy and stability of the GPSDO into the optical domain, where it is finally delivered to the length measuring system of the NPMM-200 by the comb-referenced He-Ne source. Thus, a direct and permanent link to the SI unit second can locally be established as schematically shown in Figure 12.

^{−16}per meter fiber length was retrieved [35]. Compared to the Allan deviation of our GPS-referenced frequency comb, this frequency change is neglectable for the 6-m fiber length for integration times below 10,000 s. Nevertheless, if prospectively longer fiber paths covering several ten meters of fiber length for a further internal distribution of the ultrastable frequencies are considered, the validity of our assumption has to be confirmed within further experiments on the phase-noise characteristics of the comb-referenced fiber-coupled He-Ne laser source. From the current point of view, the overall uncertainty of the comb-referenced metrology laser is thus determined by the frequency instability of the respective comb line for integration times up to 10,000 s.

## 4. Summary and Outlook

^{−13}at τ = 10,000 s. By locking the He-Ne lasers to a comb line, it was possible to improve their long-term frequency stability by three orders of magnitude. This approach allows us to establish a traceable and permanent link of interferometric length measurements to the SI unit second. With the improved frequency stability of the comb-referenced gas lasers, the influence of frequency-related length measurements errors practically drops below the nominal resolution of the signal processing unit of the NPMM-200 for the first time. This allows to neglect the laser frequency as an uncertainty contribution factor within our length measurements.

^{−15}at $\tau $ = 1 s enabled to directly resolve the short-term frequency instability of a single comb line that is limited by the frequency instability of the underlying GPSDO. Based on this comparison, a maximum relative Allan deviation of Δf/f = 4.6 × 10

^{−12}at τ = 8s was obtained demonstrating the typical limitations of a GPS-referenced frequency comb that makes a frequency stability better than Δf/f = 1 × 10

^{−12}only accessible for integration times longer than τ = 8000 s. Furthermore, the currently used He-Ne heterodyne source provides only a limited output power thus lowering the signal-to-noise ratio of the interferometer signals in comparison to the prior interferometer configuration with three independent He-Ne lasers. Our upcoming experiments will thus focus on a characterization of the frequency noise properties of the comb-referenced He-Ne heterodyne source with respect to comb line frequency noise and fiber dissemination to clarify if the high-frequency distortions of the comb line also enhance the frequency noise of the currently used He-Ne lasers. Future experiments will also have to enhance the currently limited output power of the GPS-referenced wavelength standard to feed all of the interferometer axes of the NPMM-200. Although the number of comb-referenced He-Ne lasers is easily scalable within the current control system, it is questionable if in terms of different noise properties of the He-Ne sources, the build-up of an extended laser array is desirable. Therefore, we will concentrate on directly integrating the ORS as an ultrastable, high-power laser source into the length measuring system of the NPMM-200. For this purpose, a suitable procedure for a correction of the cavity drift by means of the GPS-referenced frequency comb has to be established. In perspective, the combination of the ORS and the GPS-referenced OFC will allow to establish a wavelength standard of high output power providing a frequency stability of better than Δf/f = 1 × 10

^{−12}independent on integration time to be used for displacement interferometry within NPMMs up to 1 m or a further on-site dissemination of ultrastable optical frequencies for the implementation of comb-based interferometry approaches within NPMMs for, e.g., refractive index correction.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Setup of the NPMM-200 and its He-Ne laser source for three interferometer axes. The abbreviations in the left picture denote: TMFS—two-mode frequency stabilization, PD—photodiode, PBS—polarizing beam splitter, FC—fiber coupling. The numbers in the right picture correspond to: 1—metrology frame, 2—Positioning stage with corner mirror plate, 3—Sensor mount with nano probe system, 4, 5—Interferometer x, y-axis, 6, 7—Autocollimators (angle sensors). The interferometer of the z-axis is not visible. The base plate below the metrology frame additionally carries the guiding and drive system of the x-, y-, and z-axis (see details in [32,40]).

**Figure 2.**Scheme of a possible traceability chain for laser wavelength calibration deploying iodine-stabilized He-Ne lasers. The given uncertainties were taken from the literature as given within the text.

**Figure 3.**Structure of the GPS-referenced frequency comb at the IPMS. The abbreviations denote: OFC—optical frequency comb, GPSDO—GPS disciplined oscillator, SDU—signal distribution unit, PD—photodiode. The numbers 1–4 correspond to the respective frequency counter channels.

**Figure 4.**Structure of the ORS. The abbreviations denote: ECDL—external cavity diode laser, FC—fiber-coupled, NBS—neutral beam splitter, C—collimator, PMF—polarization-maintaining fiber, M—mirror, EOM—electro-optic modulator, HWP—half wave plate, PBS—polarizing beam splitter, QWP—quarter wave plate, L—lens, DDS—direct digital synthesizer, LPF—low pass filter. The red lines indicate optical path lengths, the dashed gray lines electronic signals. The light of the ECDL is sent through a phase-modulator operating at 20 MHz to generate sidebands for the PDH-lock. The 20-MHz reference signal is provided by a DDS. The error signal is created by mixing the signal from the photodetector containing the reflected light from the cavity with the signal from the DDS and subsequent low-pass filtering. The frequency of the laser is stabilized via fast feedback to the current and slow feedback to the piezo-electric actuator controlling the external cavity length of the diode laser [65].

**Figure 5.**Frequency stability of the ORS as retrieved from manufacturer data (M.d.). (

**a**) Time dependent beat frequency between two optical reference systems of comparable frequency stability with linear drift removed, (

**b**) corresponding relative Allan deviation (see Section 2.4).

**Figure 6.**Layout of the comb-referenced He-Ne heterodyne source. The abbreviations denote: GPSDO—GPS disciplined oscillator, SDU—signal distribution unit, APD—avalanche photodiode, PBS—polarizing beam splitter, M—mirror, BSA—beam splitter assembly, FI—Faraday isolator, FC—fiber coupling, UIC—transconductive amplifier (U/I converter), FPGA—field programmable gate array.

**Figure 7.**Frequency instability of the free-running comb. (

**a**) Time traces of the free-running comb parameters measured with the FXM50. The left y-axis depicts the change of repetition rate from its initial value. The right y-axis the CEO. (

**b**) Time trace of the beat note between the respective comb line and the ORS. Since the frequency υ

_{ORS}of the ORS lies above the respective frequency of the comb line υ

_{n}, an increase in the repetition rate f

_{Rep}results in a decrease of the beat frequency f

_{Beat}. To monitor the frequency changes of the beat note, the measurements were performed without the bandpass filter at 60 MHz.

**Figure 8.**In-loop comb parameters of the GPS-referenced frequency comb. (

**a**) Time trace of the locked repetition rate. (

**b**) Time trace of the locked CEO. Both plots depict the frequency deviations from the mean value over a measurement time of 24 h. (

**c**) Calculated Allan deviations. The gray and blue lines present a linear fit to the data as described within the text.

**Figure 9.**Frequency instability of an in-loop GPS-referenced frequency comb. (

**a**) Time trace of the heterodyne signal between the OFC and the ORS. The red line indicates a linear fit to the data with f = 62.384 MHz–0.067 Hz/s. (

**b**) A 1 h section of the frequency fluctuations of the heterodyne signal after the linear drift was removed from the measured data. (

**c**) The respective relative Allan deviation of the heterodyne signal obtained by dividing the Allan deviation by f

_{HeNe}≈ f

_{Comb}≈ 474 THz. The abbreviation M.d. denotes “manufacturer data”.

**Figure 10.**Frequency instability of the He-Ne heterodyne source used as a starting point for the realization of a comb-referenced laser module. (

**a**) A 24 h-measurement series of the secondary standard laser (SSL) against a comb line, (

**b**) 24 h measurement of the metrology laser (ML). In each of the plots, the frequency deviations from the mean value derived over the whole measurement series of 24 h are shown. (

**c**) Relative Allan deviations obtained by dividing the frequency fluctuations by a nominal value of 474 THz.

**Figure 11.**Frequency instability of the comb-referenced-He-Ne heterodyne source. (

**a**) 24-h measurement series of the secondary standard laser (SSL) prior and after closing the control loop (closed-loop operation reproduced from [35]). (

**b**) Relative Allan deviations. The Allan deviation of the OFC is obtained by combining the Allan deviation as calculated from the measurement of the beat signal between the ORS and the OFC for integration times below 4000 s and the Allan deviation given by the manufacturer for integration times above 4000 s. (

**c**) The 1 h measurement sections of the 24-h timetrace. The upper graph depicts the SSL locked to the comb and the frequency fluctuations of the drift-removed beat signal between a comb line and the ORS (taken from Figure 9b) for comparison. Please note that in contrast to the in-loop measurements of the heterodyne source, those measurements were not taken simultaneously. The graph in the middle enlarges the frequency scale to illustrate the frequency fluctuations of the SSL when locked to the comb line. The lower graph further enlarges the frequency scale to depict the fluctuations of the beat signal between the ML and the SSL, when locked to the comb line. All three graphs show the same 1 h section and present the frequency deviations from the respective mean value of the whole measurement series.

**Figure 12.**Scheme of the reduced traceability chain for laser wavelength calibration of the metrology lasers of the NPMM-200.

**Figure 13.**Influence of frequency distortions on the interferometric length measurement within the NPMM-200. (

**a**,

**b**) transform the frequency fluctuations and uncertainty of the comb line into a theoretical length measurement error for a maximum measurement range of L = 200 mm. (

**c**,

**d**) experimentally demonstrate the influence of frequency changes on the interferometric length measurement within the NPMM-200 at a constant distance of L = 100 mm using the comb-referenced He-Ne laser as a tunable laser source, (

**c**) equidistant frequency steps of Δf = 1.25 MHz and (

**d**) repeated frequency jumps of Δf = 0.625 MHz height. Integration time 0.5 s [35].

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## Share and Cite

**MDPI and ACS Style**

Blumröder, U.; Köchert, P.; Fröhlich, T.; Kissinger, T.; Ortlepp, I.; Flügge, J.; Bosse, H.; Manske, E.
A GPS-Referenced Wavelength Standard for High-Precision Displacement Interferometry at λ = 633 nm. *Sensors* **2023**, *23*, 1734.
https://doi.org/10.3390/s23031734

**AMA Style**

Blumröder U, Köchert P, Fröhlich T, Kissinger T, Ortlepp I, Flügge J, Bosse H, Manske E.
A GPS-Referenced Wavelength Standard for High-Precision Displacement Interferometry at λ = 633 nm. *Sensors*. 2023; 23(3):1734.
https://doi.org/10.3390/s23031734

**Chicago/Turabian Style**

Blumröder, Ulrike, Paul Köchert, Thomas Fröhlich, Thomas Kissinger, Ingo Ortlepp, Jens Flügge, Harald Bosse, and Eberhard Manske.
2023. "A GPS-Referenced Wavelength Standard for High-Precision Displacement Interferometry at λ = 633 nm" *Sensors* 23, no. 3: 1734.
https://doi.org/10.3390/s23031734