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We report on a displacement metrology setup that provides subpm resolution in air. The setup is based on a Fabry-Perot cavity. However, unlike current Fabry-Perot cavity based displacement setups we incorporate a novel fs-laser based arbitrary wavelength synthesizer that provides efficient suppression of atmospheric disturbances while providing very wide and precise tuning of the output wavelength. The wavelength synthesizer provides sub-10 attometer wavelength resolution. The setup provides subpm length stability for integration times of up to one minute and sub-10 pm for up to half an hour without airtight enclosure of the Fabry-Perot cavities.
©2006 Optical Society of America
1. Introduction
Measuring very small displacements with very high accuracies is the foundation for a variety of metrology applications such as for the calibration of scanning tunneling microscopes (STM) or atomic force microscopes (AFM) [1], of holographic scales and other displacement sensors [2] or for the determination of the gravitational constant G [3]. Such measurements are generally based on interferometers, such as two-beam amplitude splitting interferometers (homodyne or heterodyne interferometers) or Fabry-Perot type interferometers. Homodyne and heterodyne interferometers provide the capability to measure large displacements with resolutions reaching sub-nm level. However, substantial efforts have to be undertaken to avoid an intrinsic problem of all two-beam interferometers: That is to avoid nonlinear errors (cyclic errors) in the detection system. These non-linear errors are naturally caused by optical crosstalk or imperfections in the quadrature detection system commonly used in two beam interferometers [4, 5]. Fabry-Perot (FP) cavity based displacement measurements are an interesting alternative to the homodyne and heterodyne two beam interferometers as they can overcome some of the limitations of the two-beam interferometers, that is the aforementioned non-linear errors and the limited sensitivity or resolution of the two-beam interferometers. Former is achieved by tracking the center of the FP resonance rather than the sinusoidal fringe pattern of a two-beam interferometer and latter is achieved by a higher finesse of the FP cavity. Increasing the finesse of the FP cavity does not only increase the resolution of the system, but it also reduces the effect of unwanted optical crosstalk by increasing the frequency selectivity of the cavity without increasing the unwanted crosstalk. However, Fabry-Perot type displacement measurements pose several technical difficulties, such as limitations in the displacement ranges and the requirement for widely and precisely tunable lasers. Former can be addressed by a nice method presented in Ref. [6]. However, that method requires a vacuum system to achieve the required stability. To achieve very wide tuning ranges we recently developed a frequency measurement system based on an optical frequency comb [7]. With that former system we achieved ~100 pm resolution under stable atmospheric condition.
In this paper we present a displacement measurement setup based on two Fabry-Perot cavities and a widely tunable wavelength generator. Using a wavelength stabilized laser instead of a frequency stabilized laser provides a powerful means to suppress environmental perturbations, such as air-pressure or air-temperature induced changes of the refractive index of air. Our tunable wavelength generator is based on a wavelength-stabilized optical fs-comb. It provides wide tuning ranges with sub-10 attometer wavelength resolution. With this novel displacement setup we achieve sub-pm resolution for displacements as large as 12 µm.
Fig. 1. Experimental setup. Solid and dashed lines represent optical and electrical signals respectively. AOM: acousto-optic modulator; EOM: electro-optic modulator; PBS: Polarizing beamsplitter; PZT: Piezo-electric transducer; FP1/2: Fabry-Perot cavities with a finesse: ~156 on common CLEARCERAM-Z base. MISER: monolithic isolated single-mode end-pumped ring oscillator; PDH: Pound-Drever-Hall error-signal. The ‘Enclosure’ is not airtight but it reduces airflow and rapid temperature changes. Mode-matching optics for coupling to the FP cavities are not shown in this figure.
2. Experimental Setup
2.1. The FP-cavities
The experimental setup is shown in Fig 1. Two FP cavities (FP1 and FP2) with a finesse of ~156 are constructed on top of a 24×4×3 cm3 low-expansion CLEARCERAM-Z block (Ohara, Inc.). The topside (24×4cm2) of this block is polished to <λ/10 flatness. The mirrors for the two FP cavities are dielectric mirrors with a reflectivity of ~99%. We use 15×15×15 mm3 cubes also made from CLEARCERAM-Z for the mirror substrates. These substrates are polished with <λ/10 for the mirror surface and on the bottom side. The backside of the mirrors is coated with an antireflection coating. Three of the four cavity mirrors shown Fig.1 are optically contacted on the topside of the CLEARCERAM-Z base. The end-mirror of FP2 is placed on a precision flexure stage (P-752 from PI), which provides a tip/tilt angle stability of ~1 µrad and a lateral guiding precision of ~1 nm over a travel range of 15 µm. The cavity lengths are ~6.5 cm and ~8.2 cm for FP1 and FP2, respectively. The cavities are placed in parallel next to each other in a distance of less than 2 cm. Owing to this arrangement of the two cavities thermal drifts and pressure induced volume contraction of the CLEARCERAM-Z base are greatly suppressed. The thermal expansion of the target itself however has to be compensated by other means. A common enclosure around the two cavities reduces airflow and air temperature changes inside of the two cavities. Owing to this enclosure and the close proximity of the two FP cavities, both cavities are exposed to nearly identical atmospheric and thermal conditions, i.e. refractive index of the air inside of each of the two cavities is (nearly) identical for a given wavelength.
The CLEARCERAM-Z base together with the two FP cavities is installed on a 1 by 1 foot optical breadboard, which supports the two photo diodes and the mode matching and coupling optics required to couple the light from the cw-lasers shown in Fig. 1 to the FP cavities. The photo detectors are reverse biased and connected to RF amplifiers over a coaxial line. This is necessary to avoid non-uniform heating of the FP cavities by the amplifiers. For the same reason, the two electro-optic modulators are placed outside from the enclosure mentioned above (see Fig. 1). The light is delivered to the FP cavities by polarization maintaining optical fibers. The fibers have angle-polished fiber-ends to avoid unwanted optical crosstalk that would lead to FM to AM conversion. The 1 by 1 foot optical breadboard is placed on top of a vibration absorbing foam inside of the aforementioned enclosure. The enclosure is not airtight but it helps to reduce airflow and rapid temperature changes in the proximity of the two cavities. The enclosure further helps to reduce temperature gradients over the length of the two cavities.
2.2. Optical wavelength comb
A cw Nd:YAG laser (NPRO: Non-planar ring oscillator; also known as MISER: Monolithic isolated single-mode end-pumped ring oscillator) is locked to the fixed length FP1 cavity using Pound-Drever-Hall (PDH) locking technique [8] (see right side of Fig. 1.). This laser serves as a wavelength reference. A tunable cw fiber laser (Model ‘Boostik,’ Koheras A/S) with 50 GHz tuning range is locked to the second cavity (FP2) also by the PDH locking scheme. Since it is rather difficult to directly observe the beat-note between the wavelength reference laser (that is the MISER locked to FP1) and the tunable laser (locked to FP2) if they differ in frequency by several 10 GHz or more, we constructed a setup similar to our previous work on optical cw frequency synthesis [9]. A fs mode-locked Nd:glass laser produces pulses with ~150 fs duration centered around 1055 nm delivering an average output power of ~25–40 mW at a pulse repetition rate of 56 MHz. The output of this laser is spectrally broadened in a polarization maintaining single mode fiber to generate a smooth and wide optical spectrum covering more than 10 THz. One of the lines of this fs-comb is locked to the wavelength stabilized MISER by adjusting the pulse repetition rate of the mode-locked laser. If the carrier-envelope oscillation frequency (CEO) [10, 11] of the fs-comb is locked to zero we obtain an optical wavelength comb. That means that each optical line in the fs-comb maintains a given wavelength even when the refractive index of air changes due to a pressure or temperature change. The only limitation to this is the pressure dependence of the chromatic dispersion of the refractive index of air. In the following we describe at what level of precision this becomes an issue.
A fs-mode-locked laser such as the one mentioned above emits an optical spectrum with a comb-like structure. The comb-lines in this optical spectrum are perfectly equidistant in the frequency domain and the spacing between neighboring comb-lines is equal to the inverse of the pulse-repetition rate f rep of the fs-laser [10, 11]. As described in [11] we can express the optical frequency fm of the m’s comb line by
f rep denotes the pulse repetition rate of the fs-laser, and f CEO the carrier-envelope oscillation frequency. In the following we assume the CEO frequency would be forced to zero. How to do this is described in [11]. If we phase-lock the m0’s comb-line to the wavelength stabilized MISER by adjusting f rep we force the comb-line m 0 to the same wavelength λ0 as the wavelength reference laser. Still assuming that we locked f CEO to zero, we can express the wavelength λ0 of the reference laser in terms of the frequency comb:
m 0 f rep denotes the optical frequency of the comb-line m 0 and n λ0 is the refractive index of air at the wavelength λ0. Since the wavelength λ0 of the wavelength reference is independent from the refractive index n λ0 of air, the product n λ0 m 0 f rep between the refractive index of air times the optical frequency of the mode m 0 becomes a constant (the optical frequency m 0 f rep however isn’t). We can now also express the wavelength λ Δm of the comb-line m 0+Δm:
On the right hand side we made a tailor expansion of the refractive index of air around the wavelength λ0 of the reference laser. For clarity we omitted the higher order Tailor coefficients for the dispersion of the refractive index. Further, we expressed the refractive index n λ0 of air at the wavelength λ 0 by the optical frequency and wavelength of the reference laser. The right hand side of Eq. (1) doesn’t contain the refractive index of air but only its derivative, which is much smaller than the index itself. From this we can see that the wavelength of each of the comb-lines becomes insensitive towards small changes in the airpressure or air-temperature as such have only a minor influence on the derivative of the refractive index (∂2 n/(∂p∂f)<10-16 GHz-1 Pa-1). In our system the maximum tuning range Δm max f rep of the tunable laser is limited to about 50 GHz. The center frequencies of the MISER and the tunable laser are around 1064 nm (m 0 f rep≈281 THz), and therefore, the change of the dispersion term Δm max f rep(∂2 n/∂p∂f)Δp max in (1) for a maximum pressure change Δp max of ±10 hPa is always smaller than 5·10–12, corresponding to a wavelength error of <5·10-18 m over the full tuning range of our tunable laser. If we can accept a full-scale error of 0.5 femtometer we may even neglect the influence of the dispersive term in Eq. (1) (∂n/∂f<10-9 nm-1) altogether and write:
We find, that if we lock the comb-line m 0 of a fs-laser to a wavelength reference with the wavelength λ0 by adjusting the fs-laser’s pulse repetition rate, the wavelength of each comb line becomes insensitive towards changes of the refractive index of air. Under the above discussed approximations we can express the wavelength of all the other modes of the fscomb by their integer mode numbers and the wavelength of the reference laser and its associated mode number m 0 even without knowing the refractive index of air.
2.3. Arbitrary optical wavelength generator
In the previous section we described how to realize an optical wavelength ruler that provides a wavelength resolution of sub-5 attometer [based on Eq. (1)]. In this paragraph we describe how to build a widely tunable wavelength generator using such a wavelength stabilized fscomb. Recently we have developed a widely tunable arbitrary optical frequency generator based on a fs-frequency comb [9]. The some of the techniques and tools we described in ref. [9] can be applied to the wavelength generator described in this section. We therefore limit our description here to a few key-points of the former setup.
To generate an arbitrary wavelength one could phase-lock the heterodyne beat note between a tunable laser and a given mode in the wavelength-stabilized fs-comb to a radiofrequency synthesizer. By doing such the tunable laser would become wavelength stabilized. However, in the setup shown in Fig. 1 we require the tunable laser to generate any arbitrary (i.e. selectable) wavelength in order to be able to track a continuous displacement of the cavity length of FP2. This poses the problem that whenever the wavelength of the tunable laser coincides with one of the comb-lines in the wavelength stabilized comb the heterodyne beat note between the tunable laser and the fs-comb would approach zero. This would make it impossible to keep the tunable laser phase-locked to the wavelength comb. To overcome this problem we frequency-shifted one part of the light from the tunable laser by an acousto-optic modulator (AOM, see center of Fig. 1). The first order-deflected beam is frequency shifted by the AOM’s driving frequency f RF. We then generate two heterodyne beat notes; one between the comb and the tunable laser and one between the comb and the frequency shifted component of the tunable laser. When the frequency shift f RF is chosen correctly, i.e. f RF≈f Rep·(2n+1)/4, where n=0,1,2…, we make sure that at all times at least one of the two beat note signals is between 1/8 f Rep and 3/8 f Rep. By selecting the appropriate beat signal is possible to phase-lock the tunable laser to the comb at any arbitrary position throughout the whole comb. The difficulty in phase-locking such a dual beat note system is to construct a servo control that generates the appropriate control signals for the tunable laser based on the two beat note signals. Since at any position only one of the two beat signals is guaranteed to be accurate the servo has to select the appropriate beat note for the locking autonomously. Due to the complexity of this problem we decided to implement the whole servo algorithm in software that is running in real-time on a fast digital signal processor (DSP, TMS320C6713 from Texas Instruments). Details about a similar digital servo can be found in [9].
The digital servo further provides the necessary loop filters for locking the fs-comb to the wavelength stabilized MISER and it also provides means for locking the tunable laser to FP2 while maintaining the phase-lock between the fs-comb and the tunable laser. For this, the digital servo adjusts the locking point of the phase-lock of the double beat note system between the tunable laser and the comb to satisfy the requirement of the simultaneous locking of the tunable laser to two independent error-signals. Recording the changes made to the locking points of the phase-locks between the tunable laser and the wavelength-stabilized fscomb provides the required information to calculate the accumulated wavelength change of the tunable laser. Currently this system provides a wavelength resolution of sub-5 attometer at 1 s averaging time and sub-attometer at averaging times exceeding 10 s. Together with the dispersion effect described in the previous section we reach an over-all wavelength accuracy of better than 10 attometer for wavelength changes of up to 0.2 nm and averaging times longer than 1 s.
2.4. Displacement measurements
The previous sections describe the building blocks of our displacement measurement setup. In the following we use this system to measure length changes of FP2. This is useful for instance to calibrate the displacement of a translation stage with an integrated displacement sensor. This requires the tunable laser to be locked to FP2 while we measure the changes in the wavelength of the tunable laser (using the digital servo) during the displacement of the translation stage. One of the mirrors of FP2 is attached to the moving part of the translation stage as described in paragraph 2.1. On the other hand we could also use the same system to actively control the length of FP2. Active control of FP2 yields a highly linear moving stage and enables the calibration of stand-alone displacement sensors. For such calibration measurement the displacement sensor has to be attached to the moving mirror of FP2. FP2 has then to be locked to the tunable laser and the tunable laser (controlled by the digital servo) has to generate the required wavelength changes in order to force FP2 to the desired length changes.
In the following we describe the measurement procedure for the calibration of a translation stage with an integrated capacitive displacement sensor. During this measurement the stage is controlled in closed-loop, i.e. the stage controller uses the displacement sensor that is integrated in the translation stage to control and calibrate the position of the stage. The stage is attached to the CLEARCERAM-Z base described in paragraph 2.1. and one of the mirrors of FP2 is placed on the stage. With the stage at a fixed position we first measure the free spectral range (FSR) of FP2. For this purpose the digital servo described in paragraph 2.3. alternatively locks the tunable laser to different resonances of FP2. From the optical frequency difference between the individual resonances we calculate FSR of FP2. The digital servo repeats this measurement for a given number of times and then calculates the FSR and its standard deviation. We employ a similar scheme to the one described in [7] to eliminate the first order dispersion terms of the refractive index of air and of the reflectance phase of the cavity mirrors of FP2. Hereby, we measure the mean value FSR mean of FSR over the full tuning range of the laser. For a tuning range of 40 GHz we can typically determine FSR mean of an 8 cm long cavity to better than 1 kHz, which is better than 10-6 of FSR. We also measure the optical frequency f 0 or the wavelength of the tunable laser when it is locked to a given mode of FP2. This can either be done with a wavemeter (typical uncertainty of ~10-6) or by an optical reference (typical uncertainty <10-10) or by determination of the mode-number m of the fs-comb to which the tunable laser is locked to together with the comb spacing, i.e. the repetition rate of the laser. This is done automatically by the digital servo with a current uncertainty of 10-4..10-5. The absolute refractive index of air is determined by measuring the temperature, pressure, CO2 and water content of the air.
During the actual displacement measurement we record the changes of the wavelength during the displacement of the stage while keeping the tunable wavelength reference locked to the same resonance of FP2. Neglecting higher order dispersion terms of the index of air or the reflectance phase of the cavity mirrors, the geometrical length change ΔL of FP2 can be calculated by the simple expression
In a dispersion-free system N is an integer number equal to the longitudinal mode-number of FP2 to which the tunable laser is locked. However, in a system with chromatic dispersion N is generally a fractional number and should therefore not be confused with the longitudinal mode number of FP2. Δλ is the change in wavelength of the tunable laser during the displacement. Δλ is measured by the digital servo as described in paragraph 2.3.
Higher order dispersion terms of the refractive index of air and of the reflectance phase of the cavity mirrors are neglected in Eq. (3). In the current system this results in a maximum error of less than 1 pm (see [7]), which is comparable to the resolution of the present system. Therefore, this approximation does not compromise the actual accuracy of the measurement.
3. Experimental results
3.1. Resolution and long-term stability
In the following we discuss the resolution and the long-term stability of the system described in the previous paragraphs. We also investigate the benefit of using a wavelength-stabilized comb (see paragraph 2.2.) over a frequency-stabilized comb. For this purpose, we placed all four mirrors of FP1 and FP2 on a common CLEARCERAM spacer to avoid drifts and noise caused by the displacement mechanism. The results of such a stability measurement over 50 hours of continuous operation are shown in Fig. 2. The traces labeled with “Wavelength comb” are taken with the system shown in Fig. 1 using FP1 and the MISER as wavelength reference to generate the wavelength comb discussed in paragraph 2.2. The traces in Fig. 2 labeled with “Frequency comb” are taken with a system that uses a frequency-stabilized fscomb instead of the wavelength-stabilized comb. For this purpose, we replaced the wavelength-stabilized MISER by an iodine-stabilized Nd:YAG laser. The iodine-stabilized laser serves as an optical frequency reference with a fractional frequency stability of a few parts in 10–14 over many hours of operation. Locking the fs-comb to the iodine-stabilized laser instead of locking it to the wavelength-stabilized laser yields a frequency-stabilized comb instead of a wavelength-stabilized comb. During the measurements with the frequencystabilized fs-comb FP1 and the MISER shown in Fig. 1 are unused.
Fig. 2. Left: Stability of the displacement measurement setup over 50 hours of operation when using a ‘Wavelength stabilized comb’ (blue) and using a ‘Frequency comb’ (red). The inset shows the same data for the wavelength-stabilized case on a 300 times magnified scale. All data was taken with a 1s averaging time. Right: Stability (Allan deviation) obtained from the data shown on the left side. The wavelength-stabilized configuration (blue) provides sub-pm stability for up to 1 minute, and sub-10 pm stability for over 30 minutes. The standard deviation over the full 50 hours is 47 pm. The trace labeled with “Servo drift” shows the approximate systematic drift caused by imperfections of locking the cw-lasers to the FP cavities.
The effect of the wavelength stabilization of the fs-comb is shown in Fig. 2: The suppression of the environmental fluctuations occurs over nearly three orders of magnitude. The residual drifts in the wavelength-stabilized case (see inset in Fig. 2, left) might be caused by deformations of the CLEARCERAM base plate or by imperfections of the attachment of the cavity mirrors to the base. The trace labeled with “Servo drift” (right hand side of Fig. 2) shows the approximate drift of the system caused by phase-noise in the phase-locking servos and by imperfections of locking the two lasers to the FP cavities. Latter could be caused by the lock-in amplifiers or by beam-pointing instabilities or temperature changes in the electrooptic modulators. This drift was determined by locking both cw-lasers to the same cavity at different longitudinal modes.
From the data shown on the right side of Fig. 2 we determined the resolution of the system using the wavelength-stabilized fs-comb (blue trace). We achieved sub-pm resolution for measurement times of up to 1 minute and sub-10 pm for over 30 minutes. The standard deviation of the full data over 50 hours is 47 pm.
3.2. Application: Calibration of a metrology stage with integrated capacitive sensor
In this last section we discuss the experimental results from the calibration of a nanopositioning stage with integrated capacitive displacement sensor (P-752.1CD combined with an E-750.CP stage controller from PI). In this example we evaluate the overall performance of the displacement system, i.e. stage and controller combined. For this purpose we operate the translation stage in closed-loop operation in which the stage controller uses the information read from the capacitive sensor to calibrate the motion and positioning of the stage. The controller hereby applies a linearization polynomial to the values read from the capacitive sensor to improve the linearity of the overall system. Before taking any data we keep the whole system, including the stage and its controller running for more than 24 hours to ensure thermal equilibrium of all components. We then perform the initial calibration as discussed in paragraph 2.4 to determine FSR mean, N, the refractive index of air and the initial wavelength of the wavelength generator. We then apply small continuous displacements to the stage and record the resulting wavelength changes Δλ of the arbitrary wavelength generator. From this we determine the actual motion of the stage using expression (8). The results of three consecutive measurements are shown in Fig. 3. Each measurement took about 20 minutes (12 µm displacement at a rate of 10 nm/s). The averaging time per data point is 1 s. All three measurements shown in Fig. 3 were taken within ~2 hours. The full-scale repeatability was within 1 nm, which fits to the specification of the stage. The measured nonlinearity of ~8.6 nm or ±0.036 % is in good agreement with an independent measurement on a similar capacitive sensor published in [2]. We have reason to belief that the abrupt change in the nonlinearity at displacements around 6.5 µm is caused by a software issue in the stage controller (E-750.CP, PI) as we found several cases where possible rounding errors in the same stage controller firmware lead to displacement errors of the order of a few 100 pm.
Fig. 3. Nonlinearity measurements of a P-752.1CD nanopositioner from PI for 12 µm displacements when operated in closed-loop mode using an E-750.CP (PI) stage controller. The three traces correspond to three independent measurements. The measurement time per trace was about 20 minutes (12 µm displacements at a rate of 10 nm/s). The averaging time per data point is 1 s.
Table 1 shows the systematic errors of our displacement measurement setup. The accuracy for the optical frequency measurement depends on the measurement time. The error decreases with increasing measurement times due to the phase-lock condition between the fs-comb and the measurement laser. The accuracy with which we can lock the lasers to the FP cavities was determined from the re-locking accuracy to a given mode and from long-term stability measurements in which we locked two lasers independently to the same cavity (see trace “Servo drift” on the right hand side of Fig. 2). Spurious reflections at optical components between the EOMs and the FP cavities could lead to additional nonlinear errors. This error has to be minimized by careful alignment and the use of anti-reflection coated optics. As described in paragraph 2.1 it is essential to use angle-polished fiber ends (such as FC/APC connectors on both fiber ends) for the fibers that deliver the light to the FP cavities. Estimates for the effect of the higher order terms of the dispersion of the cavity mirrors and the refractive index of air are taken from [7]. The Abbe-error will largely depend on the target to be measured.
Table 1. Estimated systematic errors for 12 µm displacements excluding Abbe error.
Table 2 shows the calibration errors that occur during the initial calibration described in paragraph 2.4. These errors largely depend on the method used for measuring the initial wavelength (or optical frequency) of the wavelength synthesizer and on the environmental noise that limits the accuracy with which the free spectral range of FP2 can be determined. The indicated uncertainty of <1 kHz is a typical value. Under stable environmental conditions uncertainties of the order of 100 Hz can be achieved, resulting in sub-pm full-scale displacement errors.
Table 2. Uncertainties caused by the initial calibration.
4. Conclusion
In conclusion we have introduced a displacement measurement setup based on a wavelengthstabilized femtosecond-comb. The use of the wavelength-stabilized comb combined with a digital servo that maintains a phase-lock between the measurement laser and the comb even during displacements of the measurement cavity enables very small systematic errors and a very high long-term stability. The system further employs two Fabry-Perot cavities to suppress the influence of atmospheric fluctuations. We have shown that with the current system the suppression of air-pressure and temperature induced errors reaches nearly three orders of magnitude. All in all we have shown that at the expense of a slightly higher complexity a 100 times improvement of the measurement resolution and stability under uncontrolled atmospheric conditions can be achieved. Last but not least it should be noted that system can be operated in a rather straightforward manner mainly due to the simplicity of the fs-comb and the ease-of-use of the DSP-based digital servo. Large parts of the optical setup are implemented in polarization maintaining fiber optics, which greatly improves the longterm stability and reliability of the system. The current system enables continuous operation over months without user intervention.
Acknowledgments
This research was gratefully supported by the ‘Grants-in-Aid for Scientific Research’, No. 17360034, from the Japan Society for the Promotion of Science (JSPS). T. R. Schibli was supported by the JSPS postdoctoral fellowship program for foreign researchers.
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