1. Introduction

Optical interferometry for distance measurements is an essential method to calibrate precision machines or control of accurate motions in the industrial applications. For long distance measurement programs such as LIGO and LISA, it has been expected as the important role of a sensor not only to measure the distances but also to detect the disturbances. Until now, well-established homodyne and heterodyne phase measuring interferometers take possession of distance metrology in the industry and science. However, these interferometers have several limitations, namely, their incremental measuring nature i.e. measurement and accumulation of displacements. In the meantime, absolute distance interferometry has been investigated to improve the limitations of displacement interferometry. Absolute distance interferometry aims to determine the distance between the target object and the reference optics with a single operation of instantaneous measurement without accumulation of incremented or decremented displacements.

Several of the most notable techniques for measuring absolute distances are synthetic wavelength interferometry [1], a frequency modulated continuous wave (FMCW) method [2], multiple wavelength interferometry [3] and dispersive interferometry [4]. Recently, a femtosecond pulse laser, which has a large number of monochromatic modes which is phase-locked in the optical frequency domain, has been applied to optical metrology in distance measurements. The femtosecond pulse laser has prompted various efforts to investigate new possibilities of advanced optical interferometry that were not possible with traditional sources such as CW lasers and white light. Minoshima reported the extension of the measurable distance with no periodic ambiguity by means of synthetic wavelength interferometry utilizing the mode spacing of a femtosecond laser, which was carried out in the radio-frequency domain [5]. This system can measure the absolute distance up to 240 m with a resolution of 50 µm, but the mechanical measurement was included to measure long distances beyond the synthetic wavelength and the resolution was limited by the resolution of the phase meter. The superior stability of the optical comb was also used to perform coherent interferometry for absolute distance measurements by extracting the stable modes [6–9] or adjusting the pulse repetition rate [10]. In these techniques, the mechanical or electrical moving mechanism was essential to change the optical frequency.

In this research, we propose a combined interferometer to measure absolute distances with high resolution. This method consists of three measuring principles, spectrally-resolved interferometry (SRI), synthetic wavelength interferometry (SWI) and time of flight (TOF) based on the characteristics of a femtosecond pulse laser, which are a frequency comb and pulse train. They provide distinct measuring resolutions and ambiguity ranges which are complementary to each other. SWI and TOF have the advantage of covering a large measurement range, but their measuring resolutions are limited to about a few tens of micrometers. On the other hand, SRI guarantees a fine measuring resolution down to a few nanometers although its non-ambiguity range is about a few millimeters. These separate principles can be successfully incorporated and implemented simultaneously and the all measurement results are combined so that the absolute distance measurements are produced with a measuring resolution of nanometers over an overall measuring range up to coherence length of the mode.

The proposed combined method has several advantages compared to other distance techniques. First, it needs no mechanical or electrical moving parts to induce measurement errors. Second, it can measure the distance rapidly because the unified result can be calculated with the measurement results determined by three simultaneous measurement techniques. Third, this method uses the only one optical source and almost all optical components are in common, so its optical configuration is simple. Finally, it has the high dynamic range of approximately 10¹⁶ which can be determined by the ratio of the maximum measurement range and the measuring resolution.

2. Principles

Figure 1 shows the optical configuration of the overall system. The optical source is a frequency stabilized femtosecond pulse laser that is a phase-locked summation of discrete quasi-monochromatic light modes of consecutive frequencies, which is seen as an optical comb in the frequency domain. The light from the source propagates to a polarizing beam splitter (PBS₁) where it is split into two, a reference beam and a measurement beam. After the beams are reflected by the reference mirror and measurement mirror respectively, they are recombined by PBS₁ and traverse toward to a beam splitter (BS). The reflected beams at BS are detected by photodetectors (PD_R, PD_M) with PBS₂ to distinguish one beam from the other in SWI and TOF. The lengths between BS and photodetectors are previously adjusted to be same. In the meantime, the beams transmitted in BS are measured by spectrometer after going through 45° polarizer (P) and Fabry-Perot etalon (FPE) to generate the dispersive interference. This dispersive interference between the reference and measurement beams is observed by use of a spectrometer that consists of a line grating and a line array of 3648 photodetectors.

 

Fig. 1. Overall configuration of the interferometer for measuring absolute distances: synthetic wavelength interferometer (SWI) unit, time of flight (TOF) unit and spectrally-resolved interferometer (SRI) unit. FS laser; femtosecond laser, PD_R, PD_M; photodetector, BS; beam splitter, PBS₁, PBS₂; polarizing beam splitter, QWP; quarter wave plate, FPE; Fabry-Perot etalon, M_R; reference mirror, M_M; measurement mirror, P; 45° polarizer, G; diffraction grating.

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In SRI, the total interference can be decomposed into individual interferences of optical comb. The phase ϕ(ν) of SRI varies with the frequency ν and the Fourier-transform analysis allows measuring the phase variation with respect to ν so that the distance L (=L₂-L₁) can be determined by the relation of L=(c₀/4πN)(dϕ/dν), where N represents the group refractive index defined as N=n+(dn/dν)ν and c₀ is the speed of light in vacuum [11]. When sampling the interference signal in the spectrometer of SRI, the non-ambiguity range (L_NAR) is restricted by the Nyquist limit that is given as L_NAR=c₀/4Np, in which p represents the sampling period of the spectrometer. If all the modes of the optical comb could possibly be sampled with one mode per pixel of the spectrometer line CCD, p would be equal to the mode spacing of the comb. In addition, the maximum measurable range L_MAX is far beyond L_NAR up to the temporal coherence length (1.5×10⁷ m), which can be determined by the mode in the comb having a linewidth below 10 Hz when the femtosecond pulse laser is frequency stabilized [12]. However, due to the practical limitation in the spectrometer that cannot resolve whole modes of a femtosecond laser, the original mode density of the optical comb has to be reduced using a FPE filter in Fig. 1. In this circumstance, the sampling period p becomes equal to the free spectral range (F.S.R.) of the FPE, accompanying a reduction in L_NAR. In Ref. [11], the spectrally-resolved interferometry with 7 nm resolution is reported, in which L_NAR is about 1.46 mm with the FPE of 2 mm thickness made of fused silica.

Table 1. Synthetic wavelength corresponding to the mode spacing harmonics of a femtosecond laser

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When measuring distances larger than L_NAR in SRI, an essential procedure is to determine the integer multiple of L_NAR to obtain absolute distances. For the purpose, the scheme of synthetic wavelength interferometry (SWI) based on the mode spacing of the optical comb has been adopted in our configuration.

In SWI, the synthetic wavelength (λ_eq) can be created from the mode spacing (ν_F) of the femtosecond laser or its high harmonic frequencies as the form of λ_eq=c₀/qν_F, where q is an integer which means the order of harmonics. When this synthetic wavelength is used in the interferometer, the measuring range becomes half the synthetic wavelength and the resolution is usually taken as a thousandth of the wavelength by electronic phase measuring technique. For more precise resolution, higher-order harmonics of the mode spacing may be selected as the synthetic wavelength by using appropriate electronic filters and the measurement results are cascaded and incorporated from the first to higher-order harmonics [5]. Table 1 shows the harmonics of mode spacing and their equivalent wavelengths and measuring ranges.

 

Fig. 2. The principle of combined technique. Time of flight (TOF) determines the multiple integer m₁ of synthetic wavelength λ_eq and synthetic wavelength interferometer (SWI) measures the multiple integer m₂ of non-ambiguity range of spectrally-resolved interferometer (SRI). They measures the distance with the same path simultaneously.

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In addition to SWI, time of flight (TOF) method has to be considered for long distance measurements because of the ambiguity of SWI. TOF has the ability of measuring the distance L (=L₂-L₁) by detecting the time interval between two pulses with the time resolution on the order of picoseconds. In principle, TOF is used for determining the multiple integer of synthetic wavelength (m₁) in SWI and SWI plays a role of compensating the number of non-ambiguity range (m2) in SRI as shown in Fig. 2.

In general, the measuring resolution of TOF is smaller than the synthetic wavelength and the resolution of SWI is also much smaller than the non-ambiguity range in SRI. Consequently, these separate measuring principles can be incorporated and implemented simultaneously and the unified output can remove any ambiguities to determine the distance and enhance the dynamic range of the measuring system. The overall measuring distance can be expressed as the form of

where ζ is the measurement result of SRI and it can be determined following the relation of ζ=(c/4πN)(dϕ/dν) above mentioned. Note that m₂ means the integer multiple of L_NAR and can be calculated with

L_s is the measured distance of SWI which includes the multiples of λ_eq/2 with the aids of TOF as a form of

where L_syn is the measured fraction of distance in SWI and m₁ is the integer multiple of λ_eq/2 which can be determined by the similar fashion of Eq. (2) with the measurement result of TOF and λ_eq/2.

The measuring procedure of the system begins with the detection of the time difference between the reference and measurement mirrors from the first pulse of the source. Because the optical source has the consecutive pulse train with the repetition rate, the only first pulse should be measured by PD_R and PD_M which can be implemented by a simple device such as an optical chopper in front of the source. The TOF measurement is then used to approximately the absolute distance. Then, SWI and SRI measure the same distance and the final result is calculated using Eq. (1). When combining the three, the overall system can theoretically measure absolute distances up to 1.5×10⁷ m, which is the coherence length of the stabilized femtosecond laser mode, with the resolution of a few nanometers from SRI.

3. Experimental results and discussion

To verify proposed interferometer, the feasibility experiments were implemented in SWI and SRI at the same time while the measurement mirror was moving along the optical axis. The distance was measured with the step of 500 µm in the range of 100 mm using the motorized stage at an arbitrary position. The stage displacement was obtained from a secondary, heterodyne laser interferometer system (5510A laser measurement system, Agilent). Figure 3 shows the experimental results, comparing this system to a standard industrial interferometer. In Fig. 3(a), the distance measured with the SRI is a triangular wave caused by the L_NAR aliasing. L_NAR was 1.458 mm, which is much larger than the resolution of SWI, 0.154 mm when using the 13^th harmonic as the synthetic wavelength as seen in Fig. 3(b). The measurement results from SWI, Fig. 3(b) was used to compensate for L_NAR ambiguity in Fig. 3(a). The distance obtained has a linear relationship with the displacement as shown in Fig. 3(c). From Fig. 3(c), the measurement range is found from SWI and the measurement resolution is determined by SRI.

To make sure that TOF can determine the multiple integer of λ_eq in SWI, a simple experiment to detect the time difference between the reference and measurement arms with PD_R and PD_M was performed. The time delay was obtained from the first pulse into the two arms by an oscilloscope (DSO6012A, Agilent) and was 7.4×10^-9 sec. with a resolution of 50 ps. In these experiments, the distance was calculated to be 1.11 m with a resolution of 7.5 mm. By comparing this to λ_eq in SWI, TOF is sufficient for compensating the measured SWI results, thus making Eq. (1) valid for measuring arbitrarily long distances with resolution on the order of nanometers.

 

Fig. 3. Experimental results of (a) SRI, (b) SWI and (c) compensated result of SRI with (b). The inlets mean the measurement result at the displacement of 35 mm. The result in (c) is better than that of (b) because it was measured by SRI.

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Fig. 4. Experimental result of TOF. The time difference between the first reference and the measurement pulses is used for the distance measurements.

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To extend the measurable range, it is important to determine how many modes of the femtosecond pulse laser exist in every transmission peak of the FPE based on the SRI. This determines the coherence length which is the maximum measurable distance in the proposed method. To extract a single mode for measuring up to 1.5×10⁷ m according to the principle, the transmission peak must become exceedingly sharp and FPE reflectivity should be very high approximately 0.999. Maintaining this reflectivity in the broad spectrum in practice is impossible, thus every mode cannot be filtered. If the linewidth of the transmission peak in the FPE, however, reach the mode spacing of the femtosecond pulse laser and it can be controlled actively, only two modes exist in the transmission peak and they can be sampled at one pixel of the CCD in the spectrometer as shown in Fig. 5(a). Then, the coherence function is decided by the two modes with the mode spacing in the transmission peak and the measurement results of SRI can be expressed as a form of sinusoidal wave with the period of c/2ν_F (roundtrip path) as described in Fig. 5(b). In this case, the measurement results of SRI appear repeatedly at the half of the synthetic wavelength which is created by the optical comb of the femtosecond pulse laser. As shown in Fig. 5(b), the multiple integer m₂ in Eq. (1) is not increased infinitely but has the maximum limit value which can be expressed as ±[(λ_eq/4)/L_NAR] due to the periodicity of the results as the distance is longer. The periods can be also determined by SWI.

 

Fig. 5. Filtered comb of the femtosecond pulse laser by Fabry-Perot etalon (FPE); (a) two modes are filtered by FPE and detected in one pixel of the line CCD and (b) the measurement results of SRI are repeated with the synthetic wavelength determined by mode spacing of the femtosecond pulse laser.

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In this combined interferometry, the overall uncertainty is determined by the uncertainty of L_NAR and the measurement uncertainty of SRI from Eq. (1), excluding the environmental and geometrical errors. Because m2 is a large number for a long distance, the accuracy of L_NAR becomes a more important factor than the other. In SRI, L_NAR is determined by F.S.R. of the FPE filter and it is the half the optical thickness of FPE theoretically [11]. Thus, the accuracy of the proposed interferometer is subordinate to the optical thickness of FPE and its stability. In these experiments, a solid FPE of fused silica in uncontrollable environmental conditions was used. The optical thickness of FPE was determined by the dispersive interference in SRI because the F.S.R. of the FPE is larger than the sampling width of the spectrometer. The relative uncertainty is approximately 10^-5 which is dependent on the accuracy of the spectrometer. To achieve low uncertainty measurements, more research in determining the optical thickness of the FPE and the thermal stability should be investigated.

The optical thickness of the FPE can be precisely determined by the absolute frequency measurements using the comb of a femtosecond pulse laser because the mode spacing frequency and the offset frequency of the comb are measured in the optical frequency synthesizer [12] and the filtered frequency components from the FPE are measured roughly in the spectrometer. The combination of the spectrometer and the frequency comb make it possible to calculate the F.S.R. of FPE which is directly related to the optical thickness of the FPE. To obtain the high thermal stability, the material of FPE should be changed into thermally stable material such as Zerodur^® which has a low thermal expansion coefficient of approximately 0.05 ppm/K·m and temperature stabilization is needed.

4. Conclusion

To summarize, a combined optical method for measuring the absolute distances using a femtosecond pulse laser was proposed. This method can measure distances up to the coherence length of a frequency stabilized femtosecond pulse laser with the resolutions on the order of nanometers by three measurement techniques. These techniques are spectrally-resolved interferometry, synthetic wavelength interferometry and time of flight measurements. These separate measurement principles are incorporated and implemented simultaneously and the unified output can enhance the dynamic range of the measuring system. This combined interferometric technique makes it possible to measure the absolute distance up to 1.5×10⁷ m with the resolution of a few nanometers.