A laser-frequency-based displacement measurement system with sub-nanometer uncertainty using an optical frequency comb generator is developed. In this method, the optical frequency of a tunable laser is locked to the resonance of a Fabry-Perot cavity. One of the two mirrors of this Fabry-Perot cavity is connected to the element whose displacement is to be measured. Wide range optical frequency and displacement measurements were realized by using an optical frequency comb generator, which consists of an electro-optic modulator placed inside of an optical resonator. We demonstrate a displacement measurement of up to 10 μm with 220 pm uncertainty under the stable condition.
© 2006 Optical Society of America
Displacement measurement is one of the key technologies in nanotechnology systems, such as those used in semiconductor manufacturing, scanning tunneling microscopy (STM), and atomic force microscopy (AFM). The demand for displacement measurement resolution and accuracy is becoming increasingly higher owing to the progress of nanotechnology. The requirements for the measurement accuracy of displacements in such systems are reaching sub-nanometer level. Heterodyne or homodyne interferometers are directly used to measure displacements, or used to calibrate displacement sensors, such as capacitive sensors, inductive sensors, and holographic scales. Although heterodyne or homodyne interferometer systems can guarantee traceability of measurement, it is difficult to achieve sub-nanometer accuracy because of nonlinear errors [1,2]. Special efforts must be made to remove periodic nonlinearity [3–6] and a system with periodic nonlinearities around 10 pm was demonstrated .
The laser-frequency-based displacement measurement method has no periodic errors and it further has the potential of measuring displacements with picometer resolution and accuracy. In the method, the optical frequency of a tunable laser is locked to one of the resonances of a Fabry-Perot (FP) cavity with one of its mirrors connected to the element whose displacement is to be measured. The displacement is then converted into an optical frequency change, and this frequency change is measured using a reference laser. This method has been used in displacement sensor calibration systems [8–11] and STMs . However, the dynamic ranges of these systems are limited to the measurement range of the optical frequency change. Usually, the measurement range of the optical frequency change is limited to several GHz around the reference laser, which limits the displacement measurement range to typically less than 1 μm. To expand the displacement measurement range, the mode change scheme of the FP cavity was utilized [9–11].
In this study, we developed a new laser-frequency-based displacement measurement system using an optical frequency comb generator, which consists of an electro-optic modulator (EOM) placed inside of an optical resonator. The optical frequency measurement range can be expanded to several hundreds of GHz using such an optical frequency comb generator [13,14]. We succeeded to measure displacements of more than 10 μm without changing the mode of the FP cavity by combining the optical frequency comb and a widely tunable cw-laser. In addition, we show with a theoretical analysis that the first order effects of the dispersion due to the wide-range-optical-frequency-tuning can be neglected, and therefore, the displacement measurement range could be extended without compromising the measurement accuracy. In the next section, the measurement theory including the dispersion effects is described in detail.
2. Measurement principle and optical layout
Figure 1 shows a schematic layout of the laser-frequency-based displacement measurement system. The optical frequency of a tunable laser diode (ECLD: external cavity laser diode) tracks the resonance frequency of a FP cavity using EOM1 and a lock-in amplifier. If one of the mirrors of the FP cavity is displaced, the resulting optical frequency change of the ECLD is measured by the optical frequency measurement system using a Rb-stabilized diode laser and the optical frequency comb generator.
In the conventional laser-frequency-based displacement measurement method, the dispersion effect due to the optical frequency tuning was ignored because the frequency tuning range was small and the displacement ΔL was given as 
where f is the optical frequency of a given longitudinal mode of the FP cavity, Δf is the optical frequency change of this mode caused by a change ΔL of the length L of the FP cavity, and n is the refractive index of the air at f. Knowing the optical frequency f and its change Δf, the refractive index of air, and the initial optical length of the cavity, we can determine the displacement ΔL. To obtain all values in the Eq. (1), the measurement procedure of the laser-frequency-based displacement measurement method consists of two steps: In the first step we determine the initial optical length L of the FP cavity by measuring the free spectral range (FSR) of the cavity and secondly we measure the optical frequency and its change caused by the displacement.
During the measurement, f and Δf are obtained by measuring the optical frequencies of the tunable laser diode, which is locked to one of the longitudinal modes of the FP cavity, before and after the displacement using a reference laser. The refractive index n of air is determined from the environmental data. The initial optical length L of the cavity is determined from L =C/2FSR, where C is the velocity of light. The FSR is obtained from an independent measurement (the first step in the measurement procedure) by measuring the optical frequency difference ΔF between different longitudinal mode numbers mi and mj of the FP cavity while maintaining the initial cavity length. The FSR can then be determined from FSR = ΔF/Δm, with Δm = mj - mi. Usually, the relative measurement uncertainty of f and n is better than 10-6 and negligibly small in the determination of displacement ΔL. The measurement uncertainty of ΔF is similar to that of Δf and does not depend on the value of Δm. Therefore, the relative measurement uncertainty of FSR becomes small as Δm becomes large. In conventional optical frequency measurement systems, the measurable mode number change Δm is typically very limited. In our scheme using the optical frequency comb system, the measurable optical frequency changes Δf and ΔF (Δm) can be expanded, and consequently the measurement range of the displacement ΔL can be extended and the measurement uncertainty of the FSR can be reduced.
However, when the frequency tuning range is wide, the dispersion effect should be carefully considered. There are two dispersion effects, which originate from the changes in the refractive index of the air and the reflectance phase of the mirrors in the FP cavity. These dispersion effects cause an additional optical path length change in both, the measurement of Δf and ΔF. In the following, we describe the theoretical treatment including the dispersion effects and clarify that the dispersion effects can be negligible for the determination of the displacement ΔL.
When the cw-laser is locked to one of the resonances of the FP cavity, the geometrical resonator length L 0 can be written as
where f 0 and λ0 are the frequency and wavelength of the locked cw-laser in vacuum, respectively, n 0 is the refractive index of air at f 0, ϕ0 is the total reflectance phase of the mirrors in the FP cavity at f 0, and m0 is the resonance mode number of the FP cavity for f 0. To measure FSR, the optical frequency is scanned over several longitudinal modes of the FP cavity without changing the geometrical length of the cavity. If we tune the laser to the mode number m0 + Δm, corresponding to the optical frequency f M or wavelength λM, respectively, Eq. (2) can be modified to
where n M is the refractive index of air at f M and ϕM is the total reflectance phase of the mirrors in the FP cavity at f M. The FSR is given as FSR = (f M - f 0)/Δm. As described above, the optical frequency change ΔF = f M - f 0 is set to be close to the maximum frequency tuning range of the laser light source to minimize the measurement uncertainty of the FSR. In our experimental setup, ΔF was around 65 GHz and Δm was 26.
In the displacement measurement, the optical frequency is again locked to the mode number m0 corresponding to the laser frequency f 0. If the cavity length is then changed by ΔL, we can again modify Eq. (2) to
where f d and λd are the frequency and wavelength of the laser after the displacement ΔL. n d is the refractive index of air at f d and f d is the total reflectance phase of the mirrors in the FP cavity for f d. From Eqs. (2) - (4), we can determine the displacement ΔL:
where Δf = f d - f 0. ΔL M and ΔL d are given by
where Δn M = n M - n 0 and Δn d = n d - n 0. The first terms on the right-hand side of Eqs. (6) and (7) express the optical length change of the cavity caused by the dispersion of the refractive index of air. The second and third terms in Eqs. (6) and (7) express the change of the optical length of the cavity due to the dispersion of the reflectance phase of the mirrors in the FP cavity.
In Eq. (5), there are two dispersive contributions ΔL M and ΔL d to optical length change of the cavity. ΔL M is caused in the measurement of ΔF (measurement of FSR) and ΔL d is caused in the measurement of Δf (measurement of the displacement). When the laser frequency fd is tuned to be f M (Δf= ΔF) due to the displacement ΔL, ΔL d becomes equal to ΔL M and Eq. (5) becomes
This equation is equal to Eq. (1) and there is no dispersion effect on the determination of the displacement ΔL. Two dispersive contributions ΔL M and ΔL d cancel each other in the determination of the displacement ΔL. In this condition, the displacement ΔL is around the maximum measurement range ΔL max and ΔL d becomes maximum because ΔF is set to be around the maximum frequency tuning range. The effect of the dispersion in the determination ΔL closes to 0 when ΔL is around 0 and ΔL max.
If we assume that the refractive index of air and the reflectance phase of the mirrors change linearly with the frequency over the optical frequency range of ΔF, ΔL M and ΔL d are given as ΔL M = αΔF and ΔL d = αΔf, respectively, where α is the constant coefficient. In this case, the second term on the right-hand side of Eq. (5) (the dispersion effects on the determination of the displacement ΔL) becomes α(ΔF-Δf)Δf/2n d f d. Here, ΔF is rather fixed because it is set to be around maximum tuning range and only Δf is changed from 0 to ΔF in the displacement measurement. Therefore, this term has a maximum at around Δf = ΔF/2. In this position, ΔL = ΔL max/2 and ΔL d = ΔL M/2, and ΔL can be expressed as
where ΔϕM = ϕM - ϕ0. In our experimental setup, f d was approximately 384 THz (780 nm), ΔF was approximately 65 GHz, and L 0 was approximately 6 cm. One of the mirrors of our FP cavity had a reflectivity of 90 %. The other one was an Al-coated total reflection mirror. The estimated value of ΔϕM is around 5×10-3 rad and the change in the optical length of the cavity due to the dispersion of the reflectance phase of the mirrors in the FP cavity is estimated to be around 600 pm. The estimated value of Δn M is around 1×10-9  and the change in the optical length of the cavity caused by the dispersion of the refractive index of air is estimated to be around 100 pm. It is difficult to know these values with this level of accuracy. Especially, the uncertainty of the estimated value of ΔϕM will be large and the total uncertainty of the estimated value of ΔL M will be larger than 100 pm level. Although ΔL M (the resonator length change due to the dispersion) is as large as 700 pm, Eq. (9) tells us that we can neglect the dispersion effect since the value of the second term on the right-hand side of Eq. (9) is less than 0.1 pm. As described in section 4, the resolution of our experiment is larger than 100 pm. Therefore, the second term on the right-hand side of Eq. (9) can be neglected and the displacement ΔL is given from Eq. (8) over the whole measurement range. Moreover, for the displacement measurement with sub-nanometer uncertainty, we can still use a wider optical frequency tuning range and/or a larger dispersion mirror than those in our setup.
3. Experimental setup
3.1 Optical frequency locking system
The experimental setup for locking the tunable laser to the FP cavity is shown in Fig. 1. A commercially available external cavity laser diode (ECLD) was used as the frequency tunable laser light source (New Focus, Model 6300 Velocity). This laser emits a power of more than 6 mW at approximately 780 nm and the optical frequency can be tuned by approximately 65 GHz. As shown in Fig. 1, the linearly polarized output beam was divided into two beams by a λ/2 plate and a polarizing beam splitter (PBS). One beam was introduced into the optical frequency measurement system and the other beam into the FP cavity. The intensity of the beam for the FP cavity was around 1 mW. Therefore, an intensity of around 5 mW was available for the optical frequency measurement. λ/4 plate was used before the FP cavity to rotate the polarization of the laser beam so that the beam was steered by the PBS.
As described in section 2, the FP cavity was made up of two flat mirrors and the cavity length was around 6 cm. The beam diameter in the FP cavity was around 8 mm. The FSR of the FP cavity was around 2.53 GHz and the finesse was around 17. The FP cavity was set under near-air tight conditions. When one of the mirrors in the FP cavity (the Al-coated mirror) was displaced, the ECLD tracked the resonance frequency of the FP cavity. The measurement range of displacement was determined from the optical frequency tuning range of the ECLD and the FSR. In our setup, the measurement range of the displacement was approximately 10 μm.
For locking the optical frequency of the ECLD to the resonance frequency of the FP cavity, the first-derivative of the resonance curve of the FP cavity was used as the error signal E(f), where f is the laser frequency. The optical frequency was modulated with a frequency of 60 MHz by the EOM1. The intensity modulation signal generated around resonance modes was detected by the photodetector (detector1). The first-derivative signal of the resonance curve was obtained from the intensity modulation signal through a lock-in amplifier. The optical frequency was locked by a PI control loop using the first-derivative of the cavity signal. A fast control signal was fed back to the pump current of the ECLD, while the slow part of the signal was fed to the piezoelectric transducer (PZT) that tuned the grating in the ECLD. Figure 2 shows the observed first-derivative signal (error signal E(f)). In Fig. 2, we can see the symmetrical signals of the first derivative of the resonance curve of the FP cavity.
3.2 Optical frequency measurement system
The experimental setup for the optical frequency measurement is shown in Fig. 3. As a reference laser we used a second commercially available ECLD (Newport, Model 2010). The optical frequency of this laser was locked to the saturated absorption line in the 87Rb D2-line using a radio-frequency (RF) sideband technique . The wavelength of the Rb-stabilized ECLD was approximately 780.2 nm and its uncertainty was sufficiently better than 1×10-10 . The output of the Rb-stabilized ECLD was amplified by a tapered amplifier diode system (TUIOPTICS, TA100) and introduced to the optical frequency comb generator. The input power for the optical frequency comb generator was around 50 mW. The amplifier could be omitted if one would use a high-power ECLD.
The optical frequency comb generator consists of a linear optical cavity containing an EOM. A commercially available EOM (EOM2: New Focus, Model 4423) was used and driven by a 2 GHz RF-synthesizer. The EOM2 was placed in the middle of the optical cavity, which was formed by two curved mirrors with a 500 mm radius of curvature, separated by approximately 123 mm. Both cavity mirrors had a reflectivity of approximately 95 %. Taking into account the index of refraction of the EO crystal, the free spectral range of the cavity was half of EOM2-RF frequency (1 GHz). One of the curved mirrors was attached to a PZT. With an RF power of around 1 W and the correct cavity length adjustment to match optical frequency of the reference laser to one of the longitudinal modes of the cavity, the cavity transmitted around 3.5 % of the incident light. Figure 4 shows the spectrum of the resulting optical frequency comb observed by an optical spectrum analyzer. The resolution of the optical spectrum analyzer was 0.1 nm. A comb span of several hundred GHz was obtained around the reference laser.
The optical frequency f FP (f FP = f 0, f M, or f d) of the ECLD locked to the measurement FP cavity (see previous chapter) was determined from the beat-note signal between the optical frequency comb and the first ECLD. Figure 5 shows a schematic explanation of the relative frequency location of the optical frequency comb and the optical frequency of the ECLD that is locked to the FP cavity. If the ECLD tracks one of the resonances of the FP cavity generating the beat notes f beat with the k-th comb sideband, one can calculate the optical frequency f FP of the ECLD:
where f EO is the RF frequency driving EOM2 in the comb generator and k is the number of comb-lines between the Rb-stailized laser and the tracking ECLD. To determine f FP, we have to determine the integer of k and the signs of k and f beat. In our setup, the center of the optical frequency tuning range for the tracking ECLD was set to be close to f Rb and f 0 was set to be close to the limit of the tuning range of the ECLD. The sign of k can be determined by monitoring the direction of the tuning of the beat note f beat while tuning the ECLD. The integer k and the sign of f beat can also be determined by varying the modulation frequency f EO by Δf EO and observing the change Δf beat in the beat frequency f beat. In the experimental setup a frequency variation Δf EO of 2 MHz was used.
In order to be able to observe the beat note frequency even close to the comb lines, we adapted a technique previously reported in Ref 18. The tracking ECLD light was split in an acousto-optic modulator (AOM, see Fig. 3). The un-deflected beam (AOM’s zero’s order) reveals the optical frequency f FP, whereas the first order deflected beam is shifted by f AOM = f EO/4 (500 MHz). Each of these two components were brought to interference with the optical comb generator producing two beat signals as shown in Fig. 3. The heterodyne beat-notes signals f FP ± f Rb ± kf EO and f FP ± f Rb ± kf EO + f EO/4 were measured using two avalanche photodiodes (APDs) followed by two frequency counters through band-pass filters. The passband of the band-pass filters was from 230 MHz to 770 MHz. The optical frequency f FP can be determined from the beat frequency measured by the frequency counters 1 and 2. Figure 6 shows an example of the measured beat signal fbeat after amplification with a 300 kHz resolution bandwidth. When length changes are applied to the FP cavity, the optical frequency f FP will also change accordingly. Over the full displacement range of the FP (about 10 μm) the S/N ratio of the beat notes varied between 45 dB ~ 53 dB due to the finite spectral width of the optical comb generator. This is a sufficient S/N to accurately measure the optical frequency and its change. The S/N ratio became maximum when f FP was around f Rb.
To determine the optical frequency changes ΔF and Δf, the change Δk in the integer k of the comb sideband has to be measured. Δk was measured by counting the RF power signal which was obtained from the beat signal through an RF detector, as shown in Fig. 3. Figure 7 shows an example of RF power signal variation when f FP was tuned as a result of displacements in the FP cavity. By counting the numbers of the rising and falling edges of the RF power signal, we can determine Δk.
4. Experimental results and uncertainty analysis
From Eq. (8), the relative standard uncertainty u(ΔL)/ΔL of the displacement measurement is given as
Figure 8 shows the time variation of the measured beat-note signal f beat for a fixed length of the FP cavity when the environmental condition (temperature and air pressure) was stable. The integration time of frequency counters was 0.1 s. The standard deviation of the beat signal was around 690 kHz and the repeatability of lock-to-lock beat frequency was around 280 kHz. Under this condition, the uncertainty u(f d) for the optical frequency measurement was calculated to be around 750 kHz and the uncertainty u(Δf) for the optical frequency change was calculated from u 2(Δf) = 2u 2(f d) to be approximately 1.05 MHz. The relative uncertainty u(f d)/f d of optical frequency measurement was approximately 2×10-9 and negligible small compared with the relative uncertainty u(Δf)/Δf of optical frequency change. The fluctuation of the beat frequency can be reduced using a longer integration time in the frequency counters at the cost of an increased measurement duration.
In our setup, the result of this beat frequency deviation could not always be obtained. The standard deviation of the beat frequency increased when the environment was unstable. For practical applications, it is desirable to improve the stability of the FP cavity using a low-thermal-expansion material in vacuum [9,12], or using an additional reference FP cavity to compensate for the environmentally induced influence . In addition, the fluctuation of the beat frequency would also be reduced when using a FP cavity with a higher finesse at the cost of a more delicate alignment to the cavity.
The measured value of the optical frequency change ΔF was around 65 GHz and the measurement repeatability of ΔF was around 1 MHz. The relative uncertainty u(ΔF)/ΔF of the FSR measurement was estimated to be around 1.5×10-5. The refractive index of air, n d, was determined by measuring air temperature, pressure, and humidity . The relative uncertainty u(n d)/n d of the refractive index of air was estimated to be around 2×10-7 and was negligibly small. The total uncertainty u(ΔL)/ΔL of displacement measurement was mainly determined by u(Δf)/Δf and u(ΔF)/ΔF.
Figure 9 shows the beat frequency change Δf when displacements ΔL of around 1, 5, and 10 μm were sequentially given. The standard deviations of the beat signals of each displacement position were less than 690 kHz. The mode number changes Δk of comb-lines were 3, 16, and 32 when ΔL = 1, 5, and 10 μm, respectively. The optical frequency change Δf was around 65 GHz when ΔL = 10 μm. One mirror of the FP cavity was moved by a ball-bearing linear stage pushed by a PZT. If the movable mirror would misalign during the displacement, the optical length of the FP resonator would be changed resulting in a cosine error. The tilting angle of the mirror of the FP cavity was measured by an autocollimator and was less than the resolution of the autocollimator (1 second). Therefore, the cosine error due to the tilting of the mirror of the FP cavity was negligible and a change in the error signal E(f) due to the displacement could not be observed. From the uncertainty analysis discussed above, we can evaluate the standard uncertainty u(ΔL) of the displacement for this measurement to be around 160 pm for ΔL = 1 μm and 220 pm for ΔL = 10 μm. For practical applications, the uncertainties of the cosine error and the Abbe error, which are caused between the movable mirror and the measurement target, have to be added. These uncertainties are expected to be sub-nanometer level [9–12].
An accurate and wide-range laser-frequency-based displacement measurement system using an optical frequency comb generator was developed. Using the optical frequency comb generator, the measurement range of the optical frequency change and the displacement can be expanded. In our scheme, the dispersion effects due to wide-range optical frequency tuning are negligibly small. We demonstrated a displacement measurement of up to 10 μm with sub-nanometer uncertainty. The measurement range of 10 μm was not limited by the optical frequency comb generator but rather by the tuning range of the ECLD. In addition, the measurement uncertainty could be improved by using an FP cavity with a higher finesse and better environmental stability.
The authors would like to thank Dr. F.-L. Hong and Dr. S. Telada for useful discussions. This work was supported by Mitutoyo association for science and technology (MAST) and the Grants-in-Aid for Scientific Research, No. 17360034, JSPS.
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