Research Article
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Intensity evaluation using a femtosecond pulse laser for absolute distance measurement HANZHONG WU,1 FUMIN ZHANG,1,* JIANSHUANG LI,2 SHIYING CAO,2 XIANGSONG MENG,1
AND
XINGHUA QU1
1
State Key Laboratory of Precision Measurement Technology and Instruments, Tianjin University, Tianjin 300072, China National Institute of Metrology, Beijing 100013, China *Corresponding author: [email protected]
2
Received 30 December 2014; revised 12 May 2015; accepted 14 May 2015; posted 14 May 2015 (Doc. ID 231650); published 10 June 2015
In this paper, we propose a method of intensity evaluation based on different pulse models using a femtosecond pulse laser, which enables long-range absolute distance measurement with nanometer precision and large nonambiguity range. The pulse cross-correlation is analyzed based on different pulse models, including Gaussian, Sech2 , and Lorenz. The DC intensity and the amplitude of the cross-correlation patterns are also demonstrated theoretically. In the experiments, we develop a new combined system and perform the distance measurements on an underground granite rail system. The DC intensity and amplitude of the interference fringes are measured and show a good agreement with the theory, and the distance to be determined can be up to 25 m using intensity evaluation, within 64 nm deviation compared with a He–Ne incremental interferometer, and corresponds to a relative precision of 2.7 × 10−9 . © 2015 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (320.7160) Ultrafast technology; (120.3180) Interferometry; (120.2650) Fringe analysis. http://dx.doi.org/10.1364/AO.54.005581
1. INTRODUCTION High-accuracy and long-range distance measurement is of significant importance in many fields, including large-scale manufacturing, future space science, etc. Traditionally, optical interferometry with one or several continuous wave lasers allows distance metrology with nanometer precision and large dynamic measurement range [1,2], but with incremental measurement. A technique of absolute distance measurement [3] can solve this problem and simplify the measurement mission dramatically. To improve the measurement accuracy, scientists have made great efforts to obtain sufficiently stable and traceable light sources to perform the distance metrology. In 1983, the meter was related to the second, with the light velocity strictly defined as 299792458 m/s in vacuum, which means distance metrology can be traceable to a time/frequency standard, like Rb clock with 10−12 uncertainty or Cs clock with 10−15 uncertainty. However, in some ultraprecise and ultrafast cases, the microwave frequency is not sufficient to satisfy the demand of the researchers. Optical frequency comb, recognized by the Nobel Prize in 2005, was developed to bridge the optical frequency and microwave frequency smoothly, allowing distance measurement based on optical frequency to be traceable to the microwave atom clock once two parameters, repetition frequency and carrier-envelope-offset frequency, are precisely locked [4,5]. Since the first demonstration by Minoshima and Matsumoto in 2000 [6], frequency-comb-based absolute 1559-128X/15/175581-10$15/0$15.00 © 2015 Optical Society of America
distance measurement has developed rapidly in the past decade, and these approaches can be divided into two categories roughly. First, in the spectral domain, Joo and Kim [7] proposed a method of dispersive interferometry which can determine distances via the slope of the unwrapped phase of spectrograms with 7 nm resolution. Cui [8] extended the measurement range of dispersive interferometry up to 50 m with 3 × 10−8 relative precision. Based on a virtually imaged phase array, van den Berg [9] presented a mode-resolved spectrometer to observe the spectral-interfered patterns, permitting distance measurements with fast response time. Second, in the time domain, Matsumoto [10] developed a heterodyne interference system using an acoustic optical modulator to shift the comb modes with hundreds of kilohertz, which can determine distances up to 403.2 m with reproducibility of 6 μm. Wei [11] proposed an interferometric method of time-of-flight with multiple pulse trains, which can directly determine the fractional part of the distance by estimating the position of the brightest interference fringes with 1 μm accuracy. Balling [12,13] demonstrated a Fourier transform method based on pulse cross-correlation to obtain precise phase variation of the chosen wavelength, determining arbitrary distances with 10 nm uncertainty. On the other hand, Lee and Kim [14] proposed a noninterferometric method by adjusting the repletion frequency to align the reference pulse and measurement pulse precisely, where the measurement range can be up to 0.7 km in
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air. Additionally, frequency comb can be used indirectly to determine distances, operating as a calibrating source [15]. Recently, the dual-comb technique [16–18] has aroused the interest of researchers. These two combs with slightly different repetition frequencies can scan each other with no movable elements to obtain the cross-correlation patterns, and the nonambiguity range can be extended easily by changing one of the repetition frequencies. In addition, to optimize the difference of the two repetition frequencies, Wu analyzed the relation between the difference of the repetition frequencies and the measurement uncertainty, and proposed a two-step guideline to determine the optimal repetition frequency difference [19]. In this paper, we propose a method of intensity evaluation based on the interferometry between the pulses for long-range absolute distance measurement. The pulse cross-correlation is analyzed, and we develop cross-correlation patterns corresponding to different pulse models, including Gaussian, Sech2 , and Lorenz, which can be used for distance metrology. In the experiments, we determine distances in a range of 25 m. The experimental results show that the maximum deviation is −64 nm when the distance is 15 m, and the minimum deviation is 2 nm corresponding to 9 m compared with a reference continuous wave interferometer. 2. MEASUREMENT PRINCIPLE Figure 1 shows the schematic of the experimental setup. The pulse light emitted by the comb source is collimated into a Michelson interferometer, which is split at a beam splitter (Bs). One part goes into the reference beam and is reflected by the reference mirror, and the other part goes into the measurement beam. The aspheric lens pair is set in the measurement beam so that the light spot can be small enough to be reflected by a target mirror long away from the Bs. The two parts finally overlap at the Bs, and are detected by the photodetector. Theoretically, for convenience of the analysis, we neglect the dispersion and the frequency comb can be expressed as E train t Et exp−iωc t iφ0 Δφce t ⊗
∞ X
δt − mT R :
(1)
m−∞
In Eq. (1), Et is the electric field of the pulse, ωc is the center frequency, φ0 is an initial phase of the carrier pulse, Δφce is the carrier phase slip rate due to the difference between the
group and phase velocities, m is an integer, and T R is the time interval between two pulses. T R 1∕f rep , where f rep is the repetition frequency of the comb. f ceo Δφce f rep ∕2π, where f ceo is the carrier envelope offset frequency. The distance to be determined can be expressed as N × l pp d ; (2) 2 where N is an integer. l pp is the pulse-to-pulse length, which can be calculated to be l pp c∕nf rep , where n is the corresponding refractive index of air; n nR inI , where n, nR , and nI denote the complex refractive index, real part, and imaginary part, respectively. nR is the refractive index, nI characterizes the wave attenuation when traveling through the medium, and d is a small length. We can measure the distance by determining N and d . From Eq. (1), one reference pulse can be expressed as L
E ref t Et exp−iωc t iφ0 iN Δφce ;
(3)
where N floor2L∕l pp ; floor denotes the element of 2L∕l pp to the nearest integer less than or equal to 2L∕l pp . Correspondingly, after traveling the distance of L (optical path of 2L), the measurement pulse can be expressed as 2nR L 2nI ωc L exp − E meas t E t − c c 2nR L iφ0 : × exp −iωc t − (4) c When the reference pulse and the measurement pulse overlap in space, the total field at the Bs is E total E ref t E meas t:
(5)
A photodetector with a responding period T d is used to detect the intensity, and the intensity can be expressed as X X hjE total j2 i hE total E total i; (6) I j
j
where j is a positive integer, j floorT d ∕T R . Equation (6) can be calculated as Z Z 1 X 1 X jE total j2 dt E ref t E meas t2 dt I Td Td Td Td Z 1 X E 2ref t E 2meas tdt Td Td Z 2 X ReE ref t E meas tdt Td Td Z 1 2n ω L 2 X 1 exp − I c E 2 tdt Td c Td 2 2n ω L 2nR ωc L N × Δφce exp − I c cos Td c c XZ E 2 tdt; (7) × Td
Fig. 1. Schematic of the experimental setup.
where we consider that jEtj2 jEt − 2nR L∕cj2 . Since the light source is a pulse train, when the reference pulse and measurement pulse overlap in space, the peak shift is essentially a small length d . In this case, Eq. (7) can be rewritten as
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Z 2n ω L 2 X 1 exp − I c E 2 tdt c Td 2 2nI ωc L 2nR ωc d cos N × Δφce exp − Td c c Z X E 2 tdt: (8) × 1 Td
Td
We find that Eq. (8) consists of two parts, the DC term and AC term. The factor E 2 t is the corresponding power density, and is relatively stable. First, the DC term decreases with increasing L due to the attenuation factor nI exponentially. Second, the AC term is the math expression of the interference fringe when the reference arm is scanned in a fixed range. The amplitude of the interference fringe attenuates when L increases because of the factor nI . For a given distance, N can be easily determined through the pattern peak shift by changing the repetition frequency. For the small length d , we find that the intensity of the interference fringe oscillates as a cosine function with continuously changing d , which means d can be determined by the intensity evaluation precisely. A. Pulse Cross-Correlation of Different Pulse Models
The real pulse is not the ideal pulse model, like Gaussian, Sech2 , or Lorenz. The pulse-shape measurement is a vigorous project in the field of ultrafast optics. To perform absolute distance measurement with intensity evaluation and to determine d with high accuracy, we investigate pulse crosscorrelation of different pulse models in this section, including Gaussian, asymmetric Gaussian, Sech2 , asymmetric Sech2 , Lorenz, and asymmetric Lorenz. The Gaussian pulse can be expressed as E G t E exp−a1 t 2 :
Fig. 2. Gaussian pulse shapes and corresponding cross-correlation patterns. (a) Blue line indicates the Gaussian pulse shape and the corresponding interference fringe. (b) Red line shows the pulse shape and interference fringe of left asymmetric Gaussian pulse model. (c) Green line expresses the pulse shape and interference fringe of right asymmetric Gaussian pulse model.
(9)
The asymmetric Gaussian pulse can be expressed as E exp−a2 t 2 t > 0 E aG t ; (10) E exp−a3 t 2 t < 0 where E is the amplitude of the electric field. a1 , a2 , and a3 are the attenuation factors. When a2 > a3 , the pulse is right asymmetric, and the pulse is left asymmetric when a2 < a3 . Inserting Eqs. (9) and (10), respectively, into Eq. (8), we plot the pulse shape in time domain and the interference fringe [i.e., AC term in Eq. (8)] based on different Gaussian pulse models in Fig. 2, where Fig. 2(a) indicates the ideal Gaussian pulse shape and the corresponding interference fringe, while Figs. 2(c) and 2(c) lines show the pulse shapes and the interference fringes of the left asymmetric and right asymmetric Gaussian pulse model. The Sech2 pulse can be expressed as E s t
E : exp−a4 t expa4 t
(11)
The asymmetric Sech2 pulse can be expressed as E as t
E ; exp−a5 t expa6 t
(12)
where if a5 > a6 , the pulse is left asymmetric; if a5 < a6 , the pulse is right asymmetric. The corresponding pulse shapes and
Fig. 3. Sech2 pulse shapes and corresponding cross-correlation patterns. (a) Blue line indicates the Sech2 pulse shape and the corresponding interference fringe. (b) Red line shows the pulse shape and interference fringe of left asymmetric Sech2 pulse model. (c) Green line expresses the pulse shape and interference fringe of right asymmetric Sech2 pulse model.
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Fig. 5. Experimental setup.
Fig. 4. Lorenz pulse shapes and corresponding cross-correlation patterns. (a) Blue line indicates the Lorenz pulse shape and the corresponding interference fringe. (b) Red line shows the pulse shape and interference fringe of left asymmetric Lorenz pulse model. (c) Green line expresses the pulse shape and interference fringe of right asymmetric Lorenz pulse model.
cross-correlation patterns are shown in Fig. 3, where Figs. 3(a), 3(b), and 3(c) lines correspond to the pulse shapes and the interference fringes of Sech2 , left asymmetric Sech2 , and right asymmetric Sech2 models, respectively. The Lorenz pulse can be expressed as E : (13) E L t 1 a7 t 2 2 The asymmetric Lorenz pulse can be expressed as E t>0 2 2 ; E aL t 1aE8 t t a9 , the pulse is right asymmetric; if a8 < a9 , the pulse is left asymmetric. Figure 4 shows the Lorenz pulse shapes and the corresponding cross-correlation patterns, where Fig. 4(a) shows the Lorenz pulse model and the corresponding interference fringe, while Figs 4(b) and 4(c) indicate the pulse shape and the interference fringes of the left and right asymmetric Lorenz models. In Figs. 2–4, the pulse duration is all about 40 fs. We find that all cross-correlation patterns are symmetric. As shown in Eq. (8), the intensity oscillates as an attenuation cosine function stably, which we will use to determine the distances in the experiments. 3. EXPERIMENTAL SETUP We performed the distance measurements on the 80 m granite underground rail system in the National Institute of
Fig. 6.
Experimental photograph.
Metrology, which is calibrated by three continuous wave laser interferometers. The schematic of the experimental setup and the experimental photograph are shown in Figs. 5 and 6, respectively. In our experiments, the light source is Onefive Origami-15 mode-locked femtosecond pulse laser with 18 mW output power; the repetition frequency is 250.01439 MHz, the center wavelength is 1558 nm, and the pulse width is 70 fs, corresponding to the spectrum width of 50 nm. The reference He–Ne laser is Agilent 5519B (633 nm), the nanopositioning platform is PI P621.1, and the oscilloscope is LeCroy WaveRunner 640Zi. As shown in Fig. 5, we propose a new combined system for absolute distance measurement using intensity evaluation, which consists of three Michelson interferometers. The retroreflector Ref1, cube beam splitter BS1, cube beam splitter BS2, and retroreflector Ref2 make up one of the interferometers, which we name MIa . The second one, named MIb , is composed of the retroreflector Ref1, cube beam splitter BS1, cube beam splitter BS2, and retroreflector Target. The last, MIc , consists of the retroreflector Ref2, cube beam splitter BS2,
Research Article and retroreflector Target. The pulse light is collimated into the combined system and split at BS1. One part goes into the shorter beam and is reflected by Ref1, which is fixed on the nanopositioning platform and scanned in an adjustable range constantly. The other part goes through BS1 and is split at BS2. The reference pulse light and the measurement light are reflected by Ref2 and Target, respectively, and overlap with each other finally at BS2, which is detected by PD1. Meanwhile, the pulses reflected by Ref2 and Target interfere with those reflected by Ref1 at BS1, which is detected and recorded by PD2. Two shutters S1 and S2 are set in this system to adjust the operating mode of the system. The intensities from PD1 and PD2 and the control signal of nanoplatform are all displayed on a four-channel oscilloscope. To determine distances with high precision, the environmental conditions (surrounding conditions) are very important. We do experiments in the deep night, where we can consider that there is not extra vibration, but the vibration of the optical tunnel itself. We use the stability of the reference He–Ne interferometer to indicate the tunnel vibration, which is below 100 nm for 10 s with a distance of 30 m, and always below 20 nm at the position of 2 m. We test the environmental conditions in an underground laboratory for a longer time using a sensor network, shown in Fig. 7, which are 22.351°C, 1015.55 hPa, and 13.8% humidity. The environment is stable, the temperature changes are below 0.001°C, the pressure
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variation is less than 0.5 Pa, and the humidity varies by about 0.05% for up to 10 s, while we need about 3 s to finish one single measurement. Based on the Ciddor formula [20], the refractive index of air can be calculated with the corresponding wavelengths ranging from 1495 to 1600 nm, shown in Fig. 8. The group refractive index is then calculated to be 1.0002682. The pulse-to-pulse length can be calculated as l pp
c 299792458 ng f rep 1.0002682 × 250.01439 × 106
1.198779 m:
(15)
In the cases of distance measurement, the laser beam size is not a constant, especially for long-ranging systems. Careful design is needed to guarantee that the beam size is small enough to make sure that all the laser power can be reflected by the target corner cube and gathered by the transmitting-receiving system. In our experiments, we use a simple and equivalent 4-f system (i.e., the aspheric lens pair) to adjust the beam size in the optical path. We focus the laser beam on the position of about 10 m (midpoint of the total optical path), where the beam size is below 1 mm. Therefore, the laser beam is just like a pair of symmetrical cones in space. We measure the beam size at the locations of 0.4 and 20 m, and find that both the outer scales of the beam (i.e., beam diameter) are about 7 mm, indicating a beam divergence angle of about 0.017°. The diameter of the target corner cube (Thorlabs PS976M-C) is 50 mm, >7 mm, and the transmitting-receiving lens size (Thorlabs AL7560-C) is 75 mm, large enough to receive the returning laser. 4. EXPERIMENTAL RESULTS For long-range distance measurement, the pulse propagating in the long arm of the interferometer attenuates due to the absorption effect of the air. Also, the pulse shape will be distorted, and the width of the cross-correlation pattern will be broadened. In the case of intensity evaluation, the intensity values at the distance of zero are of key importance. We measure and plot the autocorrelation pattern and the spectrum of the light source at the distance of zero in Fig. 9. The spectrum is measured by a spectrometer (YOKOGAWA AQ6370D) in the air.
Fig. 7. Environmental conditions in the underground laboratory.
Fig. 8. Relation between refractive index and wavelength.
A. DC Intensity and the Amplitude of the AC Term
From Eq. (8), the DC term can be expressed as 2n ω L 2 I dc ∝ 1 exp − I c : c
(16)
We set S1 as open and S2 as closed to obtain the crosscorrelation patterns corresponding to different distances using PD2, shown in Fig. 10. We can find that, as analyzed in Eq. (8), the AC intensity oscillates stably as an attenuation cosine function with increasing distance. The air dispersion is very important in the long-distance measurement, which can contribute to the distortion and broadening of the cross-correlation patterns. The phase refractive index of air can be precisely measured with a large scanning range of the reference arm using a Michelson interferometer, where we can resolve each individual comb mode by Fourier transforming several continuous fringe packets [12]. In our case, we Fourier transform the crosscorrelation patterns in Fig. 10 and estimate the air dispersion
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Fig. 11. Air dispersion referenced to wavelength of 1560 nm. Green solid line indicates the dispersion measured by Ciddor formula, and pink dashed line expresses the measured dispersion by Fourier transforming the cross-correlation patterns.
through the unwrapped phases of each wavelength. The measurement result, referenced to the center wavelength of 1560 nm, is shown in Fig. 11, where by ppb we mean 1 × 10−9 . We find a good agreement with that measured by Ciddor formula. Based on these cross-correlation patterns, we test the relation between the DC intensities and the distances, shown in Fig. 12, and we find that the DC intensity decreases exponentially as the distance increases, showing a good agreement with Fig. 9. (a) Autocorrelation pattern at zero distance. (b) Spectrum of the light source.
Fig. 12.
Fig. 10. Cross-correlation patterns corresponding to different distances.
DC intensity of the pulse interferometry fringes.
Fig. 13. Amplitude of AC term at different positions.
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2n ω L Aac ∝ exp − I c : (17) c We test the amplitude of the cross-correlation fringes, i.e., the intensity of the brightest fringe of the interference fringes. The results are plotted in Fig. 13, and we find that the measured values attenuate exponentially, showing good agreement with theory. Similar to the DC intensity, we can use the amplitude of the AC term to determine distance roughly. B. Intensity Evaluation for Absolute Distance Measurement
Fig. 14. Corresponding intensity of 13.8 and 20.4 m. Dashed line indicates the intensity of 20.4 m, and solid line expresses the intensity of 13.8 m.
Fig. 15.
Schematic of the measurement process.
the theory. In fact, this can be an approach for distance metrology with precise measurement of nI , very stable environmental conditions, and higher resolution of intensity detection. According to Eq. (8), the amplitude of the AC term can be expressed as
In this section, we set S1 and S2 as open, which means MIc can work well, and we use PD1 to detect the output of MIc . First, we scan retroreflector Ref2 to obtain the cross-correlation patterns, and record the peak intensity, i.e., the intensity of the brightest fringe of the interference fringes. The position corresponding to the peak intensity can be considered as the pulse-to-pulse alignment point, which means in this position the reference pulse and the measurement pulse overlap completely in space, and the distance is the multiple of the pulseto-pulse length l pp . The precision of the measurement of l pp is only determined by the measurement of the repetition frequency and the group refractive index of air. For a given distance, the integer N can be easily determined through the peak shift of the cross-correlation pattern by changing the repetition frequency. The key point is the accurate measurement of d in Eq. (2). We determine d by intensity evaluation using the crosscorrelation patterns of various pulse models discussed in Section 2.A. After locking the point of peak intensity, the reference arm has no need of scanning, which means the system involves no movable elements. We take the point of peak intensity as the reference zero position, move the target mirror with a tiny displacement d , measure the relative intensity continuously for half an hour, and record the average intensity as I ave . Then we pick up the distance values corresponding to I ave in each cross-correlation pattern of each pulse model, and these distance values are the measurement results of d for each pulse model. Using Eq. (2), the distance can be determined.
Table 1. Experimental Results Corresponding to Different Pulse Models Distance (m)
N
Intensity
13.8
23
−0.4807
20.4
34
0.4938
D (μm) Gaussian Left Gaussian Right Gaussian Sech2 Left Sech2 Right Sech2 Lorenz Left Lorenz Right Lorenz Gaussian Left Gaussian Right Gaussian Sech2 Left Sech2 Right Sech2 Lorenz Left Lorenz Right Lorenz
4.975 4.97 4.97 4.955 4.955 4.955 4.97 5.005 5.005 4.043 4.021 4.021 4.013 4.003 4.003 4.062 3.998 3.998
Results (mm)
Reference (mm)
Deviations (nm)
13785.966914 13785.966909 13785.966909 13785.966894 13785.966894 13785.966894 13785.966909 13785.966944 13785.966944 20379.252126 20379.252104 20379.252104 20379.252096 20379.252086 20379.252086 20379.252145 20379.252081 20379.252081
13785.966939
−25 −30 −30 −45 −45 −45 −30 5 5 43 21 21 13 3 3 62 −2 −2
20379.252083
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To give an example, the distances to be measured are about 13.8 m and 20.4 m, respectively, and N can be determined to be 23 and 34, correspondingly. We measure the corresponding intensity for half an hour, shown in Fig. 14. We can find that the intensity is relatively stable in half an hour, and can be used to determine the distances. The average intensities are −0.4807 and 0.4938, respectively, corresponding to distances of 13.8 and 20.4 m. In Section 2.A, we develop cross-correlation patterns of different pulse models, including Gaussian, left Gaussian, right Gaussian, Sech2 , left Sech2 , right Sech2 , Lorenz, left Lorenz, and right Lorenz. For the intensities of −0.4807 and 0.4938, we pick up the corresponding distances in the crosscorrelation patterns of each pulse model. The schematic of the measurement process based on the Gaussian model is shown in Fig. 15, in which we can obtain the distances of 4.043 and 4.975 μm corresponding to intensities of 0.4938 and −0.4807, respectively. Using the same procedure in Fig. 15, we use the left Gaussian, right Gaussian, Sech2 , left Sech2 , right Sech2 , Lorenz, left Lorenz, and right Lorenz models analyzed in Section 2.A to determine different distances, which are shown as D in Table 1. Using Eq. (2), we can determine the distances, and the comparison with the reference incremental interferometer is also listed. From Table 1, we find that the distance can be measured by intensity evaluation. The maximum deviation is 62 nm with the Lorenz model when the distance is about 20.4 m, and the minimum deviation is −2 nm with the asymmetric Lorenz model. Overall, it can be observed that the precision of the asymmetric model is higher than the symmetric model because in most practical cases the pulse is not symmetric, but has some random shapes resulting from distortion and broadening.
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Fig. 16. Displacements corresponding to intensity of 0.2023.
We use Hilbert transform to get the envelopes of the crosscorrelation patterns, and then to measure the peak position roughly. Since the real pulse shape is not ideal like Gaussian, Sech2 , and Lorenz, we compare the envelope profile
C. Extension of the Non-ambiguity Range
However, we can find that the non-ambiguity range is limited to λc ∕2, where λc is the center wavelength of the comb source. As mentioned above, the interferometer MIc can determine the distances using intensity evaluation. We use the other two interferometers MIa and MIb to extend the non-ambiguity range. During the measurement process, after getting the corresponding intensity using MIc , we set S1 as closed and S2 as open, and the interference fringe generated by MIa can be recorded. When S1 is open and S2 is closed, we can observe the interference fringe generated by MIb . The relative position between the two interference fringes from MIa and MIb can be easily observed and determined roughly with reference to the control signal of the scanning platform, which means the relative position between Ref2 and Target can be determined. Generally, the simple method of the peak position evaluation of the interference fringe can give errors about 100 nm,
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Intensity evaluation using a femtosecond pulse laser for absolute distance measurement HANZHONG WU,1 FUMIN ZHANG,1,* JIANSHUANG LI,2 SHIYING CAO,2 XIANGSONG MENG,1
AND
XINGHUA QU1
1
State Key Laboratory of Precision Measurement Technology and Instruments, Tianjin University, Tianjin 300072, China National Institute of Metrology, Beijing 100013, China *Corresponding author: [email protected]
2
Received 30 December 2014; revised 12 May 2015; accepted 14 May 2015; posted 14 May 2015 (Doc. ID 231650); published 10 June 2015
In this paper, we propose a method of intensity evaluation based on different pulse models using a femtosecond pulse laser, which enables long-range absolute distance measurement with nanometer precision and large nonambiguity range. The pulse cross-correlation is analyzed based on different pulse models, including Gaussian, Sech2 , and Lorenz. The DC intensity and the amplitude of the cross-correlation patterns are also demonstrated theoretically. In the experiments, we develop a new combined system and perform the distance measurements on an underground granite rail system. The DC intensity and amplitude of the interference fringes are measured and show a good agreement with the theory, and the distance to be determined can be up to 25 m using intensity evaluation, within 64 nm deviation compared with a He–Ne incremental interferometer, and corresponds to a relative precision of 2.7 × 10−9 . © 2015 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (320.7160) Ultrafast technology; (120.3180) Interferometry; (120.2650) Fringe analysis. http://dx.doi.org/10.1364/AO.54.005581
1. INTRODUCTION High-accuracy and long-range distance measurement is of significant importance in many fields, including large-scale manufacturing, future space science, etc. Traditionally, optical interferometry with one or several continuous wave lasers allows distance metrology with nanometer precision and large dynamic measurement range [1,2], but with incremental measurement. A technique of absolute distance measurement [3] can solve this problem and simplify the measurement mission dramatically. To improve the measurement accuracy, scientists have made great efforts to obtain sufficiently stable and traceable light sources to perform the distance metrology. In 1983, the meter was related to the second, with the light velocity strictly defined as 299792458 m/s in vacuum, which means distance metrology can be traceable to a time/frequency standard, like Rb clock with 10−12 uncertainty or Cs clock with 10−15 uncertainty. However, in some ultraprecise and ultrafast cases, the microwave frequency is not sufficient to satisfy the demand of the researchers. Optical frequency comb, recognized by the Nobel Prize in 2005, was developed to bridge the optical frequency and microwave frequency smoothly, allowing distance measurement based on optical frequency to be traceable to the microwave atom clock once two parameters, repetition frequency and carrier-envelope-offset frequency, are precisely locked [4,5]. Since the first demonstration by Minoshima and Matsumoto in 2000 [6], frequency-comb-based absolute 1559-128X/15/175581-10$15/0$15.00 © 2015 Optical Society of America
distance measurement has developed rapidly in the past decade, and these approaches can be divided into two categories roughly. First, in the spectral domain, Joo and Kim [7] proposed a method of dispersive interferometry which can determine distances via the slope of the unwrapped phase of spectrograms with 7 nm resolution. Cui [8] extended the measurement range of dispersive interferometry up to 50 m with 3 × 10−8 relative precision. Based on a virtually imaged phase array, van den Berg [9] presented a mode-resolved spectrometer to observe the spectral-interfered patterns, permitting distance measurements with fast response time. Second, in the time domain, Matsumoto [10] developed a heterodyne interference system using an acoustic optical modulator to shift the comb modes with hundreds of kilohertz, which can determine distances up to 403.2 m with reproducibility of 6 μm. Wei [11] proposed an interferometric method of time-of-flight with multiple pulse trains, which can directly determine the fractional part of the distance by estimating the position of the brightest interference fringes with 1 μm accuracy. Balling [12,13] demonstrated a Fourier transform method based on pulse cross-correlation to obtain precise phase variation of the chosen wavelength, determining arbitrary distances with 10 nm uncertainty. On the other hand, Lee and Kim [14] proposed a noninterferometric method by adjusting the repletion frequency to align the reference pulse and measurement pulse precisely, where the measurement range can be up to 0.7 km in
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air. Additionally, frequency comb can be used indirectly to determine distances, operating as a calibrating source [15]. Recently, the dual-comb technique [16–18] has aroused the interest of researchers. These two combs with slightly different repetition frequencies can scan each other with no movable elements to obtain the cross-correlation patterns, and the nonambiguity range can be extended easily by changing one of the repetition frequencies. In addition, to optimize the difference of the two repetition frequencies, Wu analyzed the relation between the difference of the repetition frequencies and the measurement uncertainty, and proposed a two-step guideline to determine the optimal repetition frequency difference [19]. In this paper, we propose a method of intensity evaluation based on the interferometry between the pulses for long-range absolute distance measurement. The pulse cross-correlation is analyzed, and we develop cross-correlation patterns corresponding to different pulse models, including Gaussian, Sech2 , and Lorenz, which can be used for distance metrology. In the experiments, we determine distances in a range of 25 m. The experimental results show that the maximum deviation is −64 nm when the distance is 15 m, and the minimum deviation is 2 nm corresponding to 9 m compared with a reference continuous wave interferometer. 2. MEASUREMENT PRINCIPLE Figure 1 shows the schematic of the experimental setup. The pulse light emitted by the comb source is collimated into a Michelson interferometer, which is split at a beam splitter (Bs). One part goes into the reference beam and is reflected by the reference mirror, and the other part goes into the measurement beam. The aspheric lens pair is set in the measurement beam so that the light spot can be small enough to be reflected by a target mirror long away from the Bs. The two parts finally overlap at the Bs, and are detected by the photodetector. Theoretically, for convenience of the analysis, we neglect the dispersion and the frequency comb can be expressed as E train t Et exp−iωc t iφ0 Δφce t ⊗
∞ X
δt − mT R :
(1)
m−∞
In Eq. (1), Et is the electric field of the pulse, ωc is the center frequency, φ0 is an initial phase of the carrier pulse, Δφce is the carrier phase slip rate due to the difference between the
group and phase velocities, m is an integer, and T R is the time interval between two pulses. T R 1∕f rep , where f rep is the repetition frequency of the comb. f ceo Δφce f rep ∕2π, where f ceo is the carrier envelope offset frequency. The distance to be determined can be expressed as N × l pp d ; (2) 2 where N is an integer. l pp is the pulse-to-pulse length, which can be calculated to be l pp c∕nf rep , where n is the corresponding refractive index of air; n nR inI , where n, nR , and nI denote the complex refractive index, real part, and imaginary part, respectively. nR is the refractive index, nI characterizes the wave attenuation when traveling through the medium, and d is a small length. We can measure the distance by determining N and d . From Eq. (1), one reference pulse can be expressed as L
E ref t Et exp−iωc t iφ0 iN Δφce ;
(3)
where N floor2L∕l pp ; floor denotes the element of 2L∕l pp to the nearest integer less than or equal to 2L∕l pp . Correspondingly, after traveling the distance of L (optical path of 2L), the measurement pulse can be expressed as 2nR L 2nI ωc L exp − E meas t E t − c c 2nR L iφ0 : × exp −iωc t − (4) c When the reference pulse and the measurement pulse overlap in space, the total field at the Bs is E total E ref t E meas t:
(5)
A photodetector with a responding period T d is used to detect the intensity, and the intensity can be expressed as X X hjE total j2 i hE total E total i; (6) I j
j
where j is a positive integer, j floorT d ∕T R . Equation (6) can be calculated as Z Z 1 X 1 X jE total j2 dt E ref t E meas t2 dt I Td Td Td Td Z 1 X E 2ref t E 2meas tdt Td Td Z 2 X ReE ref t E meas tdt Td Td Z 1 2n ω L 2 X 1 exp − I c E 2 tdt Td c Td 2 2n ω L 2nR ωc L N × Δφce exp − I c cos Td c c XZ E 2 tdt; (7) × Td
Fig. 1. Schematic of the experimental setup.
where we consider that jEtj2 jEt − 2nR L∕cj2 . Since the light source is a pulse train, when the reference pulse and measurement pulse overlap in space, the peak shift is essentially a small length d . In this case, Eq. (7) can be rewritten as
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Z 2n ω L 2 X 1 exp − I c E 2 tdt c Td 2 2nI ωc L 2nR ωc d cos N × Δφce exp − Td c c Z X E 2 tdt: (8) × 1 Td
Td
We find that Eq. (8) consists of two parts, the DC term and AC term. The factor E 2 t is the corresponding power density, and is relatively stable. First, the DC term decreases with increasing L due to the attenuation factor nI exponentially. Second, the AC term is the math expression of the interference fringe when the reference arm is scanned in a fixed range. The amplitude of the interference fringe attenuates when L increases because of the factor nI . For a given distance, N can be easily determined through the pattern peak shift by changing the repetition frequency. For the small length d , we find that the intensity of the interference fringe oscillates as a cosine function with continuously changing d , which means d can be determined by the intensity evaluation precisely. A. Pulse Cross-Correlation of Different Pulse Models
The real pulse is not the ideal pulse model, like Gaussian, Sech2 , or Lorenz. The pulse-shape measurement is a vigorous project in the field of ultrafast optics. To perform absolute distance measurement with intensity evaluation and to determine d with high accuracy, we investigate pulse crosscorrelation of different pulse models in this section, including Gaussian, asymmetric Gaussian, Sech2 , asymmetric Sech2 , Lorenz, and asymmetric Lorenz. The Gaussian pulse can be expressed as E G t E exp−a1 t 2 :
Fig. 2. Gaussian pulse shapes and corresponding cross-correlation patterns. (a) Blue line indicates the Gaussian pulse shape and the corresponding interference fringe. (b) Red line shows the pulse shape and interference fringe of left asymmetric Gaussian pulse model. (c) Green line expresses the pulse shape and interference fringe of right asymmetric Gaussian pulse model.
(9)
The asymmetric Gaussian pulse can be expressed as E exp−a2 t 2 t > 0 E aG t ; (10) E exp−a3 t 2 t < 0 where E is the amplitude of the electric field. a1 , a2 , and a3 are the attenuation factors. When a2 > a3 , the pulse is right asymmetric, and the pulse is left asymmetric when a2 < a3 . Inserting Eqs. (9) and (10), respectively, into Eq. (8), we plot the pulse shape in time domain and the interference fringe [i.e., AC term in Eq. (8)] based on different Gaussian pulse models in Fig. 2, where Fig. 2(a) indicates the ideal Gaussian pulse shape and the corresponding interference fringe, while Figs. 2(c) and 2(c) lines show the pulse shapes and the interference fringes of the left asymmetric and right asymmetric Gaussian pulse model. The Sech2 pulse can be expressed as E s t
E : exp−a4 t expa4 t
(11)
The asymmetric Sech2 pulse can be expressed as E as t
E ; exp−a5 t expa6 t
(12)
where if a5 > a6 , the pulse is left asymmetric; if a5 < a6 , the pulse is right asymmetric. The corresponding pulse shapes and
Fig. 3. Sech2 pulse shapes and corresponding cross-correlation patterns. (a) Blue line indicates the Sech2 pulse shape and the corresponding interference fringe. (b) Red line shows the pulse shape and interference fringe of left asymmetric Sech2 pulse model. (c) Green line expresses the pulse shape and interference fringe of right asymmetric Sech2 pulse model.
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Fig. 5. Experimental setup.
Fig. 4. Lorenz pulse shapes and corresponding cross-correlation patterns. (a) Blue line indicates the Lorenz pulse shape and the corresponding interference fringe. (b) Red line shows the pulse shape and interference fringe of left asymmetric Lorenz pulse model. (c) Green line expresses the pulse shape and interference fringe of right asymmetric Lorenz pulse model.
cross-correlation patterns are shown in Fig. 3, where Figs. 3(a), 3(b), and 3(c) lines correspond to the pulse shapes and the interference fringes of Sech2 , left asymmetric Sech2 , and right asymmetric Sech2 models, respectively. The Lorenz pulse can be expressed as E : (13) E L t 1 a7 t 2 2 The asymmetric Lorenz pulse can be expressed as E t>0 2 2 ; E aL t 1aE8 t t a9 , the pulse is right asymmetric; if a8 < a9 , the pulse is left asymmetric. Figure 4 shows the Lorenz pulse shapes and the corresponding cross-correlation patterns, where Fig. 4(a) shows the Lorenz pulse model and the corresponding interference fringe, while Figs 4(b) and 4(c) indicate the pulse shape and the interference fringes of the left and right asymmetric Lorenz models. In Figs. 2–4, the pulse duration is all about 40 fs. We find that all cross-correlation patterns are symmetric. As shown in Eq. (8), the intensity oscillates as an attenuation cosine function stably, which we will use to determine the distances in the experiments. 3. EXPERIMENTAL SETUP We performed the distance measurements on the 80 m granite underground rail system in the National Institute of
Fig. 6.
Experimental photograph.
Metrology, which is calibrated by three continuous wave laser interferometers. The schematic of the experimental setup and the experimental photograph are shown in Figs. 5 and 6, respectively. In our experiments, the light source is Onefive Origami-15 mode-locked femtosecond pulse laser with 18 mW output power; the repetition frequency is 250.01439 MHz, the center wavelength is 1558 nm, and the pulse width is 70 fs, corresponding to the spectrum width of 50 nm. The reference He–Ne laser is Agilent 5519B (633 nm), the nanopositioning platform is PI P621.1, and the oscilloscope is LeCroy WaveRunner 640Zi. As shown in Fig. 5, we propose a new combined system for absolute distance measurement using intensity evaluation, which consists of three Michelson interferometers. The retroreflector Ref1, cube beam splitter BS1, cube beam splitter BS2, and retroreflector Ref2 make up one of the interferometers, which we name MIa . The second one, named MIb , is composed of the retroreflector Ref1, cube beam splitter BS1, cube beam splitter BS2, and retroreflector Target. The last, MIc , consists of the retroreflector Ref2, cube beam splitter BS2,
Research Article and retroreflector Target. The pulse light is collimated into the combined system and split at BS1. One part goes into the shorter beam and is reflected by Ref1, which is fixed on the nanopositioning platform and scanned in an adjustable range constantly. The other part goes through BS1 and is split at BS2. The reference pulse light and the measurement light are reflected by Ref2 and Target, respectively, and overlap with each other finally at BS2, which is detected by PD1. Meanwhile, the pulses reflected by Ref2 and Target interfere with those reflected by Ref1 at BS1, which is detected and recorded by PD2. Two shutters S1 and S2 are set in this system to adjust the operating mode of the system. The intensities from PD1 and PD2 and the control signal of nanoplatform are all displayed on a four-channel oscilloscope. To determine distances with high precision, the environmental conditions (surrounding conditions) are very important. We do experiments in the deep night, where we can consider that there is not extra vibration, but the vibration of the optical tunnel itself. We use the stability of the reference He–Ne interferometer to indicate the tunnel vibration, which is below 100 nm for 10 s with a distance of 30 m, and always below 20 nm at the position of 2 m. We test the environmental conditions in an underground laboratory for a longer time using a sensor network, shown in Fig. 7, which are 22.351°C, 1015.55 hPa, and 13.8% humidity. The environment is stable, the temperature changes are below 0.001°C, the pressure
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variation is less than 0.5 Pa, and the humidity varies by about 0.05% for up to 10 s, while we need about 3 s to finish one single measurement. Based on the Ciddor formula [20], the refractive index of air can be calculated with the corresponding wavelengths ranging from 1495 to 1600 nm, shown in Fig. 8. The group refractive index is then calculated to be 1.0002682. The pulse-to-pulse length can be calculated as l pp
c 299792458 ng f rep 1.0002682 × 250.01439 × 106
1.198779 m:
(15)
In the cases of distance measurement, the laser beam size is not a constant, especially for long-ranging systems. Careful design is needed to guarantee that the beam size is small enough to make sure that all the laser power can be reflected by the target corner cube and gathered by the transmitting-receiving system. In our experiments, we use a simple and equivalent 4-f system (i.e., the aspheric lens pair) to adjust the beam size in the optical path. We focus the laser beam on the position of about 10 m (midpoint of the total optical path), where the beam size is below 1 mm. Therefore, the laser beam is just like a pair of symmetrical cones in space. We measure the beam size at the locations of 0.4 and 20 m, and find that both the outer scales of the beam (i.e., beam diameter) are about 7 mm, indicating a beam divergence angle of about 0.017°. The diameter of the target corner cube (Thorlabs PS976M-C) is 50 mm, >7 mm, and the transmitting-receiving lens size (Thorlabs AL7560-C) is 75 mm, large enough to receive the returning laser. 4. EXPERIMENTAL RESULTS For long-range distance measurement, the pulse propagating in the long arm of the interferometer attenuates due to the absorption effect of the air. Also, the pulse shape will be distorted, and the width of the cross-correlation pattern will be broadened. In the case of intensity evaluation, the intensity values at the distance of zero are of key importance. We measure and plot the autocorrelation pattern and the spectrum of the light source at the distance of zero in Fig. 9. The spectrum is measured by a spectrometer (YOKOGAWA AQ6370D) in the air.
Fig. 7. Environmental conditions in the underground laboratory.
Fig. 8. Relation between refractive index and wavelength.
A. DC Intensity and the Amplitude of the AC Term
From Eq. (8), the DC term can be expressed as 2n ω L 2 I dc ∝ 1 exp − I c : c
(16)
We set S1 as open and S2 as closed to obtain the crosscorrelation patterns corresponding to different distances using PD2, shown in Fig. 10. We can find that, as analyzed in Eq. (8), the AC intensity oscillates stably as an attenuation cosine function with increasing distance. The air dispersion is very important in the long-distance measurement, which can contribute to the distortion and broadening of the cross-correlation patterns. The phase refractive index of air can be precisely measured with a large scanning range of the reference arm using a Michelson interferometer, where we can resolve each individual comb mode by Fourier transforming several continuous fringe packets [12]. In our case, we Fourier transform the crosscorrelation patterns in Fig. 10 and estimate the air dispersion
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Fig. 11. Air dispersion referenced to wavelength of 1560 nm. Green solid line indicates the dispersion measured by Ciddor formula, and pink dashed line expresses the measured dispersion by Fourier transforming the cross-correlation patterns.
through the unwrapped phases of each wavelength. The measurement result, referenced to the center wavelength of 1560 nm, is shown in Fig. 11, where by ppb we mean 1 × 10−9 . We find a good agreement with that measured by Ciddor formula. Based on these cross-correlation patterns, we test the relation between the DC intensities and the distances, shown in Fig. 12, and we find that the DC intensity decreases exponentially as the distance increases, showing a good agreement with Fig. 9. (a) Autocorrelation pattern at zero distance. (b) Spectrum of the light source.
Fig. 12.
Fig. 10. Cross-correlation patterns corresponding to different distances.
DC intensity of the pulse interferometry fringes.
Fig. 13. Amplitude of AC term at different positions.
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2n ω L Aac ∝ exp − I c : (17) c We test the amplitude of the cross-correlation fringes, i.e., the intensity of the brightest fringe of the interference fringes. The results are plotted in Fig. 13, and we find that the measured values attenuate exponentially, showing good agreement with theory. Similar to the DC intensity, we can use the amplitude of the AC term to determine distance roughly. B. Intensity Evaluation for Absolute Distance Measurement
Fig. 14. Corresponding intensity of 13.8 and 20.4 m. Dashed line indicates the intensity of 20.4 m, and solid line expresses the intensity of 13.8 m.
Fig. 15.
Schematic of the measurement process.
the theory. In fact, this can be an approach for distance metrology with precise measurement of nI , very stable environmental conditions, and higher resolution of intensity detection. According to Eq. (8), the amplitude of the AC term can be expressed as
In this section, we set S1 and S2 as open, which means MIc can work well, and we use PD1 to detect the output of MIc . First, we scan retroreflector Ref2 to obtain the cross-correlation patterns, and record the peak intensity, i.e., the intensity of the brightest fringe of the interference fringes. The position corresponding to the peak intensity can be considered as the pulse-to-pulse alignment point, which means in this position the reference pulse and the measurement pulse overlap completely in space, and the distance is the multiple of the pulseto-pulse length l pp . The precision of the measurement of l pp is only determined by the measurement of the repetition frequency and the group refractive index of air. For a given distance, the integer N can be easily determined through the peak shift of the cross-correlation pattern by changing the repetition frequency. The key point is the accurate measurement of d in Eq. (2). We determine d by intensity evaluation using the crosscorrelation patterns of various pulse models discussed in Section 2.A. After locking the point of peak intensity, the reference arm has no need of scanning, which means the system involves no movable elements. We take the point of peak intensity as the reference zero position, move the target mirror with a tiny displacement d , measure the relative intensity continuously for half an hour, and record the average intensity as I ave . Then we pick up the distance values corresponding to I ave in each cross-correlation pattern of each pulse model, and these distance values are the measurement results of d for each pulse model. Using Eq. (2), the distance can be determined.
Table 1. Experimental Results Corresponding to Different Pulse Models Distance (m)
N
Intensity
13.8
23
−0.4807
20.4
34
0.4938
D (μm) Gaussian Left Gaussian Right Gaussian Sech2 Left Sech2 Right Sech2 Lorenz Left Lorenz Right Lorenz Gaussian Left Gaussian Right Gaussian Sech2 Left Sech2 Right Sech2 Lorenz Left Lorenz Right Lorenz
4.975 4.97 4.97 4.955 4.955 4.955 4.97 5.005 5.005 4.043 4.021 4.021 4.013 4.003 4.003 4.062 3.998 3.998
Results (mm)
Reference (mm)
Deviations (nm)
13785.966914 13785.966909 13785.966909 13785.966894 13785.966894 13785.966894 13785.966909 13785.966944 13785.966944 20379.252126 20379.252104 20379.252104 20379.252096 20379.252086 20379.252086 20379.252145 20379.252081 20379.252081
13785.966939
−25 −30 −30 −45 −45 −45 −30 5 5 43 21 21 13 3 3 62 −2 −2
20379.252083
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To give an example, the distances to be measured are about 13.8 m and 20.4 m, respectively, and N can be determined to be 23 and 34, correspondingly. We measure the corresponding intensity for half an hour, shown in Fig. 14. We can find that the intensity is relatively stable in half an hour, and can be used to determine the distances. The average intensities are −0.4807 and 0.4938, respectively, corresponding to distances of 13.8 and 20.4 m. In Section 2.A, we develop cross-correlation patterns of different pulse models, including Gaussian, left Gaussian, right Gaussian, Sech2 , left Sech2 , right Sech2 , Lorenz, left Lorenz, and right Lorenz. For the intensities of −0.4807 and 0.4938, we pick up the corresponding distances in the crosscorrelation patterns of each pulse model. The schematic of the measurement process based on the Gaussian model is shown in Fig. 15, in which we can obtain the distances of 4.043 and 4.975 μm corresponding to intensities of 0.4938 and −0.4807, respectively. Using the same procedure in Fig. 15, we use the left Gaussian, right Gaussian, Sech2 , left Sech2 , right Sech2 , Lorenz, left Lorenz, and right Lorenz models analyzed in Section 2.A to determine different distances, which are shown as D in Table 1. Using Eq. (2), we can determine the distances, and the comparison with the reference incremental interferometer is also listed. From Table 1, we find that the distance can be measured by intensity evaluation. The maximum deviation is 62 nm with the Lorenz model when the distance is about 20.4 m, and the minimum deviation is −2 nm with the asymmetric Lorenz model. Overall, it can be observed that the precision of the asymmetric model is higher than the symmetric model because in most practical cases the pulse is not symmetric, but has some random shapes resulting from distortion and broadening.
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Fig. 16. Displacements corresponding to intensity of 0.2023.
We use Hilbert transform to get the envelopes of the crosscorrelation patterns, and then to measure the peak position roughly. Since the real pulse shape is not ideal like Gaussian, Sech2 , and Lorenz, we compare the envelope profile
C. Extension of the Non-ambiguity Range
However, we can find that the non-ambiguity range is limited to λc ∕2, where λc is the center wavelength of the comb source. As mentioned above, the interferometer MIc can determine the distances using intensity evaluation. We use the other two interferometers MIa and MIb to extend the non-ambiguity range. During the measurement process, after getting the corresponding intensity using MIc , we set S1 as closed and S2 as open, and the interference fringe generated by MIa can be recorded. When S1 is open and S2 is closed, we can observe the interference fringe generated by MIb . The relative position between the two interference fringes from MIa and MIb can be easily observed and determined roughly with reference to the control signal of the scanning platform, which means the relative position between Ref2 and Target can be determined. Generally, the simple method of the peak position evaluation of the interference fringe can give errors about 100 nm,
