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Pulse-to-pulse alignment technique based on syntheticwavelength interferometry of optical frequency combs for distance measurement Guanhao Wu,1,2,* Mayumi Takahashi,2,3,5 Hajime Inaba,2 and Kaoru Minoshima2,4 1
State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instrument, Tsinghua University, Beijing 100084, China 2 National Metrology Institute of Japan (NMIJ), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8563, Japan 3 Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan 4
The University of Electro-Communications (UEC), 1-5-1 Chofugaoka, Chofu 185-8585, Japan 5
Currently at Fujikura Ltd., 1-5-1,Kiba, Kouto-ku, Tokyo 135-8512, Japan *Corresponding author:
[email protected] Received May 16, 2013; accepted May 17, 2013; posted May 20, 2013 (Doc. ID 190707); published June 13, 2013
A synthetic-wavelength interferometry of optical frequency combs is proposed for the pulse-to-pulse alignment in absolute distance measurement. The synthetic wavelength derived from the virtual second harmonic and the real second harmonic is used to bridge the interference intensity peak-finding method and the heterodyne interferometric phase measurement, so that the pulse-to-pulse alignment can be linked directly to single-wavelength heterodyne interferometry. The experimental results demonstrate that the distance measured by the peak-finding method with micrometer accuracy can be improved to the nanometer level by applying the method proposed. © 2013 Optical Society of America OCIS codes: (320.7100) Ultrafast measurements; (320.7120) Ultrafast phenomena; (120.0120) Instrumentation, measurement, and metrology; (120.3180) Interferometry; (120.3930) Metrological instrumentation. http://dx.doi.org/10.1364/OL.38.002140
An optical frequency comb has a large number of stable longitudinal modes with uniform mode spacing and an extremely narrow linewidth. Such inherent advantages have enabled revolutionary progress in absolute distance measurement [1–7]. Especially, a simple but effective scheme based on an unbalanced Michelson interferometer [2] has been widely used in absolute distance measurement [7–10]. By adjusting the repetition frequency f rep of the frequency comb to make the pulses from reference and probe arms overlap, the target distance can be determined by accurate measurement of f rep [11]. The accuracy of such distance measurement depends mainly on the knowledge of relative positions of the two overlapped pulses, i.e., pulse-to-pulse alignment. Usually, the peak position of interferogram envelope of the overlapped pulses is used for the pulse-to-pulse alignment [8–11]. This method is referred to as the peak-finding method in the following text. It can realize micrometer or even submicrometer level alignment, but the accuracy is affected by the intensity noise of the interferogram envelope. Therefore, with only the peak-finding method, it is difficult to ensure an accuracy better than a quarterwavelength to enable further improvement by using interferometry [3,12]. Note that several attempts to improve the accuracy of pulse-to-pulse alignment have been made by using Fourier transformation [10,13,14]. In this Letter, a simple scheme based on syntheticwavelength interferometry of optical frequency combs for pulse-to-pulse alignment is proposed. The synthetic wavelength derived from the virtual second harmonic (half-wavelength of the fundamental) and the real second harmonic (SH, generated by nonlinear crystal) is used to bridge the gap between the peak-finding method and single-wavelength heterodyne interferometry, so that 0146-9592/13/122140-04$15.00/0
pulse-to-pulse alignment can be realized with nanometer accuracy. This technique is effective to improve the accuracy of absolute distance measurement. Here, considering that simultaneous measurements with the fundamental and SH can potentially realize self-correction of the refractive index of air [15], we use the SH for generating the synthetic wavelength instead of the common approaches, e.g., using two appropriate filters to get two close wavelengths. A mode-locked Er:fiber ring laser is used as the frequency comb source (Fig. 1). The f rep and f ceo (carrier–envelope-offset frequency) are stabilized with reference to a hydrogen maser. The f rep of the comb is 54.0 MHz, and it can be varied coarsely up to 900 kHz by a fiber delay line and finely up to 20 Hz by PZT (piezoelectric transducer)-driven fiber stretching. The corresponding optical distance between adjacent pulses is Lp-p 5.55 m, and it can be varied up to ΔLmax 93 mm.
Fig. 1. Schematic of experimental setup. BS1 , beam splitter; M1–4 , mirror; I and II indicate the two positions of CR1 . © 2013 Optical Society of America
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The laser pulses are sent to a piece of periodically poled LiNbO3 (PPLN) crystal for SH generation (Fig. 1). Then the residual fundamental (ν) and SH (2ν) are coaxially introduced to an unbalanced Michelson interferometer. In the probe arm, a corner reflector (CR1 ) is used as the target mirror, whose displacement is monitored by a reference interferometer (Agilent 5530). In the reference arm, the fundamental and SH are separated by a dichroic mirror (DM1 ), and then diffracted by two acoustooptic modulators (AOM1 and AOM2 , respectively). The two diffracted beams are reflected back along the same paths by two mirrors (M2 and M3 ) and diffracted by the two AOMs again. The optical path lengths for the two wavelengths in the reference arm are set to be equal. The optical path length difference (round trip) between the probe and reference arms (OPLD) is set to OPLD Lp-p , so that adjacent pulse-to-pulse interferences can occur. The interference signals for the fundamental and the SH are separated by DM2 and detected by two photodetectors (PD1 and PD2 ), respectively. Figure 2 shows the diagram of the electronic system of the experimental setup. A frequency synthesizer (80 MHz) referenced to the hydrogen maser was used as the source of the driven signals for AOMs. The driven signals for AOM1 and AOM2 are 160 and 80 MHz, respectively. Therefore, the frequency shifts generated by the two AOMs (diffracted twice) for the fundamental and SH (f Δ1 and f Δ2 ) are 320 and 160 MHz, respectively. In the frequency domain of the fundamental, the mth mode (m is a positive integer) of the comb in the reference arm is f ref: m mf rep f ceo f Δ1 ;
(1)
and that in the probe arm is f pro: m mf rep f ceo :
(2)
Fig. 2. Diagram of electronic system of experimental setup. LPF, low pass filter; LA1–2 , lock-in amplifier.
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Therefore, when the fundamental beam from the reference arm meets that from the probe arm, the frequency of the beat signal can be expressed by Nf rep − f Δ1 54N − 320 MHz (N is an integer). The lowestfrequency beat signal (6f rep − f Δ1 4 MHz), which is the beat of f ref: m and f pro: m 6, is selected by a low-pass filter as the heterodyne signal for the fundamental [f h1 , Fig. 3(a)]. Accordingly, the heterodyne signal for the SH expressed by f h2 3f rep − f Δ2 2 MHz is obtained [Fig. 3(b)]. The interferometric phases of the fundamental and SH (ϕ1 and ϕ2 ) were measured by two lock-in amplifiers (LA1 and LA2 ). A specific electronic system was designed for the reference signals of the two lock-in amplifiers (Fig. 2). Another photodetector (PD3 ) was used to detect the signals from another branch of the frequency comb, and the output signals from PD3 were mixed with the frequency-doubled signals from the frequency synthesizer by a mixer. In the frequency domain, the output signals from PD3 include the multiples of f rep , and the frequency-doubled signal from the frequency synthesizer exactly matches the frequency shift by AOM2 (f Δ2 ). Consequently, the signal with a frequency of 3f rep − f Δ2 2 MHz can be easily obtained by low-pass filtering the output signals of the mixer. The frequency of this signal exactly equals f h2 . So it can be used as the reference signal for lock-in amplifier LA2 . Moreover, its frequencydoubled signal with a frequency of 6f rep − 2f Δ2 6f rep − f Δ1 4 MHz, which exactly matches f h1 , can be used as the reference signal for lock-in amplifier LA1 . In this system the heterodyne interference involves broadband beams. Therefore the center wavelength should be determined beforehand. The optical spectra of the fundamental and the SH in both the reference arm and the probe arm were measured by a spectrum analyzer (AQ-6315A, Yokogawa Inc.). The spectrum of the fundamental is wide in the probe arm [Fig. 4(a)], but narrow when the beam comes back from the reference arm [Fig. 4(b)], because the diffraction angle of the AOM is sensitive to wavelength, so that only a narrow band of the beam can be reflected back along the same path by M2 . Therefore the center wavelength of the fundamental in the reference arm is tunable by adjusting M2 . For the SH, due to the bandwidth limitation of the PPLN crystal, the spectra in both probe and reference arms are narrow and nearly the same [Figs. 4(c) and 4(d)]. The gravity center of the product of the spectra
Fig. 3. Diagram of heterodyne signals generation for (a) the fundamental and (b) the second harmonic.
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Fig. 4. Optical spectrum of the fundamental and SH. (a) Fundamental in probe arm, (b) fundamental in reference arm, (c) SH in probe arm, and (d) SH in reference arm.
in the reference arm and the probe arm is used to estimate the effective center wavelengths. It turns out to be 1581.1 nm for the fundamental and 780.8 nm for the SH. Note that the effective center wavelength of the SH (real SH) is not equal to the half-wavelength of the fundamental (virtual SH). In the experiment, the target distance D from position I to position II (Fig. 1) was measured in three steps. First, at both positions, the peak-finding method was applied for the fundamental i.e., f rep was adjusted to maximize the interference signal intensity of the fundamental, ensuring the relative displacement between the overlapped pulses from position I to position II within a micrometer level. Second, the interferometric phase of the synthetic wavelength derived from the virtual SH and real SH was utilized to determine D to find the fringe order of the fundamental unambiguously. Finally, the interferometric phase of the fundamental was used to determine D with nanometer accuracy. Following are the details. After applying the peak-finding method for the fundamental at position I, the ith fundamental pulse from the reference arm (pulse A) overlapped the i − 1th fundamental pulse from the probe arm (pulse B) according to the path length settings mentioned above. Then CR1 was moved by D to position II, resulting in a relative displacement of D between pulses A and B. In the case D < ΔLmax∕4ng1 (ng1 is the group refractive index of air for the fundamental), applying peak-finding method at position II will compensate for this relative displacement to make pulses A and B overlap again. Normally, such compensation cannot be realized perfectly because of the accuracy limitation of the peak-finding method. The relation between the residual relative displacement (δ1 , normally, jδ1 j < 3 μm [8–11]) after compensation and the interferometric phase change of the fundamental can be expressed by δ1 4D − ΔLp-p∕ng1 k1 Δϕ1∕2πλ1 ;
(3)
where ΔLp-p and Δϕ1 indicate the change of Lp-p and ϕ1 from position I to position II, respectively, λ1 is the center wavelength of the fundamental in air, and k1 is the fringe order to be determined. Note that the distance D introduces 4D in path length. Here, ΔLp-p can be obtained accurately by measuring f rep at positions I and II, and ng1 can be calculated accurately by measuring the environmental parameters. Therefore, if the peak-finding
method was able to ensure jδ1 j < λ1∕2 to make k1 0, Δϕ1 could be used to determine δ1 as well as D with very high accuracy. However, this is beyond the ability of the peak-finding method. Our solution is to use the interferometry of the synthetic wavelength derived from the virtual SH and real SH to find 4D with an accuracy better than λ1∕2, so that k1 can be found unambiguously, and δ1 and D can be linked to Δϕ1 directly. To derive the synthetic-wavelength interferometry, besides Eq. (3), we also have to derive the relation between D and the interferometric phase of the real SH. Since the effect of dispersion is not strong over the propagation length of OPLD 5.55 m, when the peakfinding method is applied to the fundamental at positions I and II, the ith SH pulse from the reference arm also overlaps the i − 1th SH pulse from the probe arm. Therefore, as in Eq. (3), we can derive δ2 4D − ΔLp-p∕ng2 k2 Δϕ2∕2πλ2 ;
(4)
where all the parameters are the same as in Eq. (3) except that here the subscript 2 stands for the SH. There is a tiny difference between δ1 and δ2 caused by the dispersion of air. Denoting the wavelength of the virtual SH as λ3 λ1∕2, and the corresponding phase change as Δϕ3 2Δϕ1 , with Eqs. (1) and (2) we can derive δs 4D − C · ΔLp-p ks Δϕs∕2πλs ;
(5)
where δs λ3 δ2 − λ2 δ1 ∕λ3 − λ2 , the coefficient C ng1 λ3 − ng2 λ2 ∕ng1 ng2 λ3 − λ2 , the phase change of the synthetic wavelength Δϕs Δϕ2 − Δϕ3 , the fringe order of synthetic wavelength ks k2 − 2k1 , and the synthetic wavelength λs λ3 λ2∕λ3 − λ2 (nearly 40 times λ1 ). Such derivation is similar to general synthetic-wavelength interferometry [3,12]. Note that jδs j is close to jδ1 j and jδ2 j, which are much smaller than λs∕2; it will not cause the fringe order of λs to change, i.e., ks 0. With Δϕs we can easily obtain 4D with an accuracy better than λ1∕2 according to Eq. (5). Substituting 4D into Eq. (3), k1 can be rounded unambiguously. Finally, resubstituting the k1 obtained and Δϕ1 into Eq. (3), the accuracy of D can be improved. Such a concept linking Δϕs to Δϕ1 is also commonly used [3,12]. To demonstrate the efficacy of the method described above, the target mirror CR1 was moved by seven steps at a increment of 250 μm. During the measurement, the temperature, humidity, and air pressure were recorded for calculation of the refractive index of air. The distances were measured by using the present method and compared with the results of the reference interferometer. For each position, the phase ϕ1 was recorded for 50 s. Their average was used for distance calculation, and their standard deviation is smaller than 6.3°, corresponding to 6.9 nm. The final results [Fig. 5(a)] are in good accordance with the results of the reference interferometer. By application of linear fitting [Fig. 5(b)], the slope is 0.99999, and the correlation coefficient (R2 ) is 1.00000. The residuals range from −14 to 24 nm [Fig. 5(c)]. Please note that the optical path length of the probe arm is longer than 5.55 m, and the 10−8 level instability caused
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In conclusion, we proposed a synthetic-wavelength interferometry of optical frequency combs for the pulseto-pulse alignment in absolute distance measurement. The small difference between the wavelength of the second harmonic and the half-wavelength of the fundamental are utilized to generate a synthetic wavelength, which is used to bridge the peak-finding method and the heterodyne interferometric phase measurement for pulse-topulse alignment. The experimental results demonstrate the efficacy of the present method for improving the accuracy in absolute distance measurement. Fig. 5. Experimental results versus the data obtained by the reference interferometer. (a) Distance data measured, (b) linear fitting of the data measured, (c) residuals of final results, (d) intermediate residuals obtained by applying the phase of the synthetic wavelength for the measurement, and (e) intermediate residuals obtained by applying the interference intensity peak-finding method.
We thank K. Arai (AIST) for his contribution for development of the comb. This work was supported by JST, JSPS (25286076), the Special-funded Program on National Key Scientific Instruments and Equipment Development of China (2011YQ120022) and the Tsinghua University Initiative Scientific Research Program (2011Z02166).
by air’s refractive index will cause an error of tens of nanometers in distance. To demonstrate the advantages of the present method, the intermediate results are also shown. By applying the phase of the synthetic wavelength for pulse-to-pulse alignment, the residuals are ranged from −95 to 72 nm [Fig. 5(d)]. Such accuracy is enough to bridge the peak-finding method and the heterodyne interferometric phase measurement. The residuals of the measurements obtained by only applying the peak-finding method are also shown [Fig. 5(e)]. Obviously, the accuracy of measurement by the peak-finding method can be greatly improved by our method. Here, we have demonstrated only the case in which the same pulses are involved in the overlapping at positions I and II, so the measurement range of D is related to OPLD; e.g., when OPLD jLp-p (j is an positive integer), the measurement range is jΔLmax∕4ng1 . Consider the fixed carrier–envelope-offset phase (ϕceo ) relations in the pulse train of the frequency comb [16]: this method is also applicable in the case of different pulses involved in the overlapping at positions I and II. In this case, the measurement range can be very long, and the change in ϕceo caused by f ceo and the dispersion of air should be considered in Eqs. (3)–(5). Another advantage of this method is that interferometry is used for the pulse-to-pulse alignment, so the fringe order k1 is small. Therefore, the accuracy of the wavelength is not a severe factor in the measurement, since this error will not be greatly accumulated by k1 .
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