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Abstract
Using spectral interference from a highly chirped and high-repetition-rate optical frequency comb, a technique for dead-zone free one-shot three-dimensional (3D) imaging was developed. We generated continuously emitted pulse train (dead-zone-free pulse) using a highly chirped pulse from a 750-MHz repetition mode-locked Yb fiber comb. The measurement performances of the system were evaluated using an optical spectrum analyzer, and a longitudinal delay length resolution of 31 µm was obtained at arbitrary position for more than 400 mm range. Using an imaging spectrometer capable of multipoint spectroscopy, a delay length resolution of 81 µm was demonstrated. Finally, for a proof-of-concept measurement of 3D imaging, line profiles of three mirrors located in the middle of 200-mm-distance, corresponding to pulse-to-pulse interval, were measured without any scanning with an uncertainty of 0.7 mm. The uncertainty could be further improved by optical alignment and analysis to the level of 10 ∼ 100 µm at arbitrary position over the range not limited to pulse-to-pulse interval.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recently, industrial technology has developed and advanced measurement technology is in high demand. For example, nondestructive, noncontact, and noninvasive three dimensional (3D) shape measurement with high accuracy has been in demand to measure large-scale structures such as exceeding 10 m or high-speed moving objects. Although the traditional method of laser 3D shape measurement is highly accurate [1–3], it is limited to stationary or rather slowly moving objects since scanning the beam position at the target is required. Several attempts have been made to increase the measurement speed and to extend the measurement range [4–6]. Measurement technique using a fiber-based optical frequency comb (OFC), which has excellent stability and practicability is promising to simultaneously satisfy these requirements. An OFC possesses highly evenly spaced frequency modes in the optical frequency domain and an ultrashort pulse train with high controllability and coherence in the time domain. The pulse-to-pulse time interval and phase evolution are precisely controlled by two frequency parameters, namely, repetition frequency (frep) and carrier-envelope offset frequency (fCEO). It has demonstrated high accuracy and large dynamic range using direct output of OFC and continuous wave lasers referenced to OFC for various point-by-point distance measurements [7–10] and surface measurements using LiDAR [11], interferometry [12–14], holography [15], and time-resolved measurements [16].
Our group has developed an instantaneous 3D shape measurement method with ultra-high speed and high depth resolution using chirped pulses to change spatial information into temporal information and detect it as frequency information [17–20]. We used the spectral interference of chirped and chirp-free ultrashort pulse trains from an OFC to generate spectral images encoding ultrafast time and space information. Using OFCs with high coherence and high-precision pulse interval, pulse-to-pulse interference can significantly extend the measurement range. Moreover, the measurement uncertainty is not enhanced by the number of pulses since the pulse-to-pulse separation is precisely determined, making it an attractive measurement method for large scale objects. Since 3D shape information is obtained in a single pulse, it is possible to obtain 3D information in one shot. Using an Er-doped fiber OFC with a 50-MHz repetition frequency, we demonstrated fully nonscanning 3D measurement of 3 m separated surface images with sub-µm uncertainty [20]. However, in a previous study, there was a certain area between pulse-to-pulse interval in a mode-locked pulse train where no pulse interference occurred, meaning a dead-zone, in a simultaneous measurement. Although repetition frequency scanning can eliminate the dead-zone [21], extending the simultaneous measurement range without scanning is preferable to fully utilize the one-shot measurement technique.
In this study, we developed a technique for dead-zone-free one-shot 3D measurements using a high repetition-rate comb and a highly chirped pulse train. Specifically, three items were conducted as follows: generation of dead-zone-free pulses, application of the generated pulses to the instantaneous 3D imaging method, and demonstration of simultaneous measurement of line profiles of three mirrors positioned over the entire pulse-to-pulse interval. These studies demonstrate the proof-of-principle of simultaneous 3D imaging at an arbitrary distance.
2. Method
This section describes the principle of 3D imaging technique using chirped spectral interferometry and the method for achieving dead-zone-free measurement (Fig. 1). First, we describe the principle of 3D measurement using spectral interferometry. To get spectral interference, a light source from an OFC was divided into two paths, and one of the pulses was highly chirped. The interference fringe signal was then obtained in the spectrum by combining two pulses. Due to the characteristic of the chirped pulse, a nonuniform interference fringe appeared with a minimum fringe frequency at a characteristic wavelength corresponding to the timing when the chirped-free pulse overlapped with the chirped pulse (Fig. 1(a), upper right) [18–20]. This allows the relative position of both pulses to be determined from the characteristic wavelength of the interference fringes.
Fig. 1. (a) Schematic of chirped spectral interferometer. When the chirped and chirp-free pulses overlapped in time, interference fringe with the minimum fringe frequency, i.e., the broadest fringe, appeared in spectrum. When dead-zone-free pulses are generated, two characteristic fringe patterns should appear simultaneously at some delay. (b) Two methods of generating dead-zone-free pulses. Dead-zone-free pulses are generated by chirping the pulse and/or increasing the pulse repetition-rate.
In a previous 3D imaging study [18], there was a dead-zone where pulse-to-pulse interference did not occur because of the wide interval between pulses. The repetition frequency was 50 MHz, i.e., a pulse interval of 6 m, and the full width at half maximum of the chirped pulse was 5.7 ps, i.e., 1.7 mm in space. In other words, there were some cases where the distance measurement could not be performed successfully because no signal could be observed. Although interference could be obtained by rapidly scanning the delay or repetition frequency of the comb, the capability of the instantaneous measurement has not been fully utilized. Therefore, dead-zone-free one shot 3D imaging is achieved by employing a pulse train with no dead-zone, i.e., dead-zone-free pulses, allowing pulse-to-pulse interference to occur at an arbitrary distance without scanning.
As shown in Fig. 1(b), there are two methods to obtain a dead-zone-free pulse. One is using a highly chirped pulse so that the pulse overlaps with neighboring pulses. A highly chirped pulse can be generated using high-dispersion optics, such as grating pairs, prism pairs, highly dispersive fibers, or fiber Bragg gratings (FBGs). However, due to extreme chirp, wavelength change per time, i.e., per distance, becomes very small in this method. As distance information corresponds to spectral information, the corresponding distance resolution should be degraded when the spectral resolution is practically limited by measurement equipment. The other method is using a high repetition frequency OFC. However, such extremely high repetition frequency OFC that allows pulse-to-pulse overlap is not practically available, especially with sufficient power for broad application. This study applied combination of both methods to generate dead-zone-free pulses to mitigate both requirements. In this way, a dead-zone-free 3D imaging method with extremely wide dynamic range, i.e., both high resolution and large measurement range without principle limitation was achievable.
In the case that the reference pulse is a dead-zone-free pulse, the spectral interference fringe should always show the characteristic fringe pattern, indicating that both pulses overlap in the range of the detected spectrum, when the delay between the probe and reference was changed. In this case, at a certain delay position, two characteristic fringe patterns appeared simultaneously in the spectrum, as shown in the Fig. 1(a) (bottom right). In this experiment, we measured the spectral interference fringe by changing the delay and showing the spectrum with two fringes to demonstrate that a dead-zone-free pulse is generated. Therefore, in this way, instantaneous 3D shape measurement at an arbitrary distance is achievable.
3. Experiments
An experimental system for obtaining the amount of chirp necessary to achieve a dead-zone-free pulse was constructed. Two setups were developed to add a high degree of dispersion to the pulse as follows: one that involves using a grating pair and the other, which uses the FBGs. With both setups, the characteristics of the generated dead-zone-free pulse were evaluated. Subsequently, we constructed an imaging system to enable the simultaneous measurement of the two-dimensional (2D) profile. Finally, we measured a shape with steps over the pulse interval and demonstrated a method for instantaneous 3D shape measurement using dead-zone-free pulses.
3.1 Dead-zone-free pulse generation with a grating pair
Figure 2 shows an experimental setup using a grating pair to provide the chirp required to generate dead-zone-free pulses. The light source was a lab-made high repetition Yb:fiber comb. The basic performances of the oscillator are described in the references; a center wavelength of 1044 nm, a repetition-rate of 746 MHz, a spectral half width of 19 nm, pulse width at half maximum of 118 fs, and an average power of approximately 100 mW [22,23]. Each parameter changed depending on the experimental condition. In this work, both the repetition frequency and the carrier-envelope offset frequency were free running. The Yb comb was divided into two, namely, the probe and reference paths, to construct a Mach–Zehnder interferometer. The probe pulse was highly chirped using a grating pair (LightSmyth, T-1000-1040, 1000 lines/mm), combined with the reference pulse using a beam splitter, and the interference signal was detected using an optical spectrum analyzer (OSA). A mechanical stage was inserted in the reference path to vary the delay between the probe and reference pulses to evaluate the interference signal. Additionally, a retro-reflector was mounted on the mechanical stage to add the delay to the optical path length by up to pulse interval of 400 mm.
Fig. 2. Experimental setup of chirping a pulse using a grating pair. The left plot shows an example of the OFC spectrum at the input of the interferometer. Both the repetition frequency and the carrier-envelope offset frequency were free running. NPBS: Nonpolarized beamsplitter.
First, the amount of chirp required to achieve a dead-zone-free pulse was estimated. Here, we set the goal of the pulse duration of the chirped pulse to 1.33 ns, which could fill the pulse interval of 400 mm, corresponding to the comb repetition-rate of ∼750 MHz. Then, using the entire spectral range that can be used for measurements with sufficient power, that is approximately 60 nm, the grating pair setup was designed according to the amount of dispersion necessary to achieve the chirp as described in Section 4.1. Finally, the required separation of the grating pair was estimated to be about 1 m for the dead-zone-free measurement.
3.2 Dead-zone-free pulse generation with FBGs
With a view for practical use, a setup with a grating pair is bulky, and free-space optics caused instability. Therefore, we constructed a setup with FBGs, as illustrated in Fig. 3. In this setup, three FBGs were placed to introduce the chirp in the reference path rather than the probe path to avoid optical power loss due to the FBGs. Additionally, a 10-m long polarization maintaining fiber (PMF) was inserted after the OFC to transmit the pulse to the interferometer setup, resulting in an extended pulse width of 15.2 ps at the input of the interferometer setup. The amount of chirp caused in the FBGs was designed according to the results of the previous section. The details of the design are described in Section 4.3.
Fig. 3. Experimental setup of 3D shape measurement using dead-zone-free pulses. The left plot shows an example of the OFC spectrum at the input of the delivery fiber. Both the repetition frequency and the carrier-envelope offset frequency were free running. There are two types of targets: a planar mirror and a combination of three planar mirrors. The two types of detector units are an imaging spectrometer and an OSA. PMF: Polarization maintaining fiber, NPBS: Nonpolarizing beamsplitter.
In the interferometer, the target is introduced in the probe path before generating the interference fringe. The interference signal is detected by a detector unit. Two types of detector units were used in this study. One was an OSA and the other was an imaging spectrometer. The OSA was used to evaluate the basic performance of the optical system, and the imaging spectrometer (iHR320, HORIBA, Japan), which is capable of simultaneous multi-point spectra detection, was used for imaging. A Si camera (C13440, Hamamatsu Photonics, Japan) was used for the 2D spectra detection, which was attached to the spectrometer. Since a Yb comb with a center wavelength of 1044 nm was used, we can take advantage that the Si camera can be used in this study. Si camera is inexpensive and has the capability of higher image resolution than the InGaAs camera that was used for the 1.5-µm wavelength region as in the previous study [18–20].
First, with a single planar mirror as the measurement target, the basic performance of the measurement system was evaluated using an OSA and an imaging spectrometer. With an OSA, the delay and wavelength dependence of the system were evaluated to ensure that dead-zone-free measurement is enabled. Subsequently, the detector unit was replaced by an imaging spectrometer to obtain 2D line profiles as an image.
3.3 Shape measurement using dead-zone-free pulses
To demonstrate the capability of the dead-zone-free instantaneous 3D measurement technique, a target (Fig. 3), which consists of three planar mirrors placed along the pulse-to-pulse interval was measured using the imaging spectrometer. The two mirrors in the front are D-cut mirrors, and the other is a planar mirror. Three mirrors were placed to irradiate simultaneously using a probe beam. The first and third mirrors were placed at a distance similar to the pulse interval, while the second was between the two mirrors. Therefore, we demonstrated the capability of a single-shot arbitrary position measurement, which is a dead-zone-free measurement, by measuring this target shape simultaneously.
4. Result
4.1 Design a grating pair for dead-zone-free pulse generation
A First, the amount of chirp required to generate a dead-zone-free pulse using a grating pair was estimated. A negative chirp was given with grating pairs, and the amount of chirp can be controlled by the distance of the grating pairs (Zg). The group velocity dispersion given by the grating pairs is calculated by the following Eq. (1) [24,25], as follows:
The wavelength (λ) was 1040 nm, the grating constant (d) was set to 1/1000 mm, and the grating angle of incidence (θ) was set to 30°. When the separation between the grating pairs (Zg) was set to 84, 180, and 280 mm, the group velocity dispersion was calculated to be –0.56, –1.20, and –1.87 ps2, respectively.
The chirp characteristic of the generated pulse was determined experimentally by changing the delay between the reference and probe pulses and measuring the shift of the spectral interferometer fringe. By changing the grating pair separation (Zg), the relationship between the delay and the characteristic wavelength of the interference fringes was obtained and measured by moving the mechanical delay of the reference path over a 400-mm distance, which was the pulse interval. On the other hand, in distance measurement, since the characteristic wavelength of the interference fringe, i.e., the minimum fringe frequency, indicates the delay when the reference and probe pulses overlap in time, the target distance information was obtained by analyzing the wavelength by a time-of-flight measurement [19]. Therefore, the chirp characteristic of the generated pulse can serve as the calibration curve between the wavelength and the delay, i.e., the distance to be measured.
Next, we demonstrated the process of obtaining the calibration curve when the detector unit was OSA. First, we obtained the spectrum by OSA as shown in Fig. 4(a). The interference fringes appeared in the center of the spectrum, indicating spectral interference. To obtain the center position of the minimum fringe frequency, i.e., the broadest fringe, the convolution was calculated along the wavelength axis (i.e., horizontal direction). Since interference fringes have an almost symmetrical region, the convolution process can highlight the central position of the fringe region. The convolution signal g is given by Eq. (2), where f is the spectral interference and m and n are the indices of the spectral interference and convolution signal data, respectively.
Fig. 4. (a) Interference fringes appearing in the spectra obtained using an OSA when the probe pulse has a chirped amount of –1.87 ps2. (b) Calculated autocorrelation. The wavelength at which the autocorrelation peaks is the center position of the interference fringes. (c) Autocorrelation waveforms of the spectrum were measured at three delays from right to left as follows: 13, 28, and 44 mm, respectively. (d) The relationship between the delay and the fringe center wavelength is shown by plotting the peak positions obtained from the convolution. The dotted points and line represent the interference fringe center position obtained in (c) and their linear fitting.
Figure 4(b) shows the convolution waveform with a sharp peak at the center of the interference fringe. The convolution peaks can be obtained as presented in Fig. 4(c) when the same process was applied to the spectra measured at several delays. Figure 4(d) was obtained by plotting the peak wavelengths obtained by changing the delay, i.e., the calibration curve between the wavelength and delay.
By changing the separation between the grating pairs, the calibration curves were obtained for different chirp parameters. With chirps of –0.56, –1.20, and –1.87 ps2, the slopes of the calibration curves when linear fitting was performed, i.e., the delay to wavelength ratio, were 286, 606, and 948 µm/nm, respectively, and was proportional to the chirp amount. Considering that the entire spectral range that can be used for measurements with sufficient power is approximately 60 nm, the range that can be measured with a single pulse is estimated to be 17, 36, and 57 mm at chirps of –0.56, –1.20, –1.87 ps2, respectively, by assuming a linear calibration curve.
Based on this consideration, the amount of chirp required to generate a dead-zone-free pulse was determined. Since a 750-MHz repetition-rate pulse was used in this study, a delay length measurement range of 400 mm was required for dead-zone-free measurement with a 60-nm spectral range; therefore, the required chirp slope was 400000/60 ∼ 6670 µm/nm. Since there was a linear relationship between the amount of designed chirp and the slope of the calibration curve, the required amount of chirp was estimated to be –1.87 × 6670/948 ∼ –13.2 ps2. From Eq. (1), ϕ2 is –13.2 ps2 when Zg = 1.98 m. In this study, it was estimated that the distance between grating pairs should be Zg/2 ∼ 1 m for the dead-zone-free measurement because of the double path configuration.
4.2 Characterization of dead-zone-free pulses
To confirm the total delay length measurable range with the designed grating pair, placed at 1-m separation, delay was changed by 400 mm with a mechanical stage, and the spectral interference signal was measured using an OSA as the detection unit in the setup (Fig. 2). The detected wavelength position of the interference fringe shifted continuously when the delay was changed from 0 to 176 mm, as shown in Fig. 5. A jump at the delay of 176 mm (dotted line in Fig. 5) corresponds to the spectral edge and the next neighboring pulse appeared from the other edge without a break. When further delay was added, the interference fringes were detected with the next pulse without a break. In other words, this indicates that no dead-zone existed between pulses. From the above, measurement was possible over the entire range for a measurement interval of 400 mm, and the generation of dead-zone-free pulses was confirmed.
Fig. 5. Relationship between interference fringe center wavelength and delay when a chirp of -13.2 ps2 was given.
4.3 Dead-zone-free pulse generation using FBGs and characterization of the performances
Next, we used the optical system shown in Fig. 3, which produces chirp using FBGs. As a first demonstration measurement using dead-zone-free pulses, we measured the target of a single planar mirror, and an OSA was used for the detector units to evaluate the basic performances. The reference pulse delay length was changed by 400 mm with a mechanical stage, which is the pulse interval, and the relationship between the delay and characteristic wavelength was obtained. Since this relationship was constant in the measurement system regardless of the target, it could be used as a calibration curve and applied to further measurements with same chirp parameters.
From the result recorded in Section 4.2, the amount of chirp required to achieve a dead-zone-free pulse is –13.2 ps2. Due to the trade-off between the amount of chirp, the loss, and the transmission spectral bandwidth of FBGs, optimized parameters of a FBG were designed as follows: a spectral range of 60 nm, a center wavelength of 1045 nm, and an amount of chirp of –6.89 ps2. Hence, two FBGs were used to generate a dead-zone-free pulse. However, the pulse was positively pre-chirp and the width was broadened to 15.2 ps after transmitting through a 10 m long PMF, a more negative chirp greater than –13.2 ps2 was required to generate a dead-zone-free pulse. Therefore, three FBGs were used to generate a dead-zone-free pulse.
Figure 6 shows the schematic representation of the analysis to obtain the calibration curve from the OSA measurement data, which is the same scheme described in Section 4.1. Spectral interference fringes appeared at the dotted lines in Fig. 6(a) and 6(d), indicating that the probe and reference pulses are overlapped and interfered. Additionally, since two interference fringes were observed simultaneously in Fig. 6(a), it was confirmed that a dead-zone-free pulse was generated in this optical system. Next, the background light was removed from the detected data to emphasize the position of the interference fringes, as shown in Fig. 6(b) and 6(e).
Fig. 6. (a) Spectra acquired using a OSA in which two fringes appear simultaneously. The dashed line shows the position of minimum fringe frequency in the detected interference fringes. The delay was 380 mm. (b) Interference fringes are emphasized by removing the background in (a). (c) Calculated convolution waveform. The wavelength at which the convolution peaks were the center position of the interference fringes. (d) As in (a), only one fringe was visible in the spectrum acquired using an OSA. The dashed line shows the position of minimum fringe frequency in the detected interference fringe. The delay was 156 mm. (e) Interference fringes are emphasized by removing the background in (d). (f) Calculated convolution of (e). (g) Relationship between delay and interference fringe center wavelength by plotting peak positions calculated by convolution. Blue line: measurement data, red line: fitted curve obtained by polynomial approximation, which was the calibration curve of this optical system. The circled area represents the interference fringe center position obtained in (c) and (f).
To obtain the position of the minimum fringe frequency, a convolution was calculated along the wavelength axis, as shown in Fig. 6(c) and (f), and a sharp peak appeared at the position of minimum fringe frequency, i.e., at the center of the fringe. Furthermore, when two interference fringes were observed simultaneously in the detected data, an additional third convolution peak due to cross-convolution appeared between the two peaks, i.e., ghost, which does not represent the position of individual interference fringes. To obtain accurate fringe positions even in the case of such ghost signals, the peak with the smallest wavelength among the three peaks was used in the analysis.
The delay and the obtained peak wavelengths were plotted as shown in Fig. 6(g). Here we can observe two linear lines corresponding to the neighboring pulses. At any delay, at least one spectral fringe appears in the spectrum, indicating that a dead-zone-free pulse was generated. Additionally, the blue line represents the fitting results with a polynomial function. From the slope of the fitting line, the delay length can be converted to 10.46 mm/nm from the wavelength, and the uncertainty in delay length was obtained as 31 µm by calculating the standard deviation of the fitting residuals. These results indicate that the proposed method is capable of dead-zone-free distance measurement of a target with a delay length resolution of 31 µm over a long range, which is not limited to the pulse-to-pulse interval.
Next, an imaging spectrometer with an entrance slit width of 0.3 mm was used as a detector unit of imaging with dead-zone-free pulses. A 2D profile was obtained by irradiating the probe pulses onto a single planar mirror target and moving the mechanical stage to apply a delay of 400 mm to the reference path for evaluation.
Figure 7 shows the analysis scheme similar to that in Section 4.1. As the imaging spectrometer performed multipoint spectral detection, an image was obtained, as shown in Fig. 7(a), where the wavelength is along its horizontal axis and the transverse spatial position along the slit of the spectrometer is in its vertical axis. When we select a cross-section at a certain spatial position and plot it along the wavelength axis, we obtained a spectrum corresponding to that obtained with an OSA. Figure 7(b) shows such cross sectional spectrum at the spatial position of 750 px, which had the strongest light intensity. After analyzing the convolution signal following the same procedure as in Section 4.1, the characteristic peak of the interference fringes was obtained (Fig. 7(c)). By plotting the characteristic wavelength of the interference fringes and the delay, two linear lines are seen in Fig. 7(d), which are similar as in the case of the OSA measurement. Outliers that did not fit in the linear line were observed near the center of the figure, but this was due to a ghost peak that appeared in the convolution signal between the two peaks generated from the overlapped region. Compared to the OSA, the imaging spectrometer measurements showed a lower signal-to-noise ratio and had more background noise. Therefore, there were more outliers observed than using an OSA.
Fig. 7. (a) Interference fringe image obtained by the imaging spectrometer. The delay was 350 mm. (b) Spectrum at X = 750 px (dotted line in (a)). The background was removed to emphasize the interference fringes. (c) Calculated convolution. The wavelength at which the convolution peaks were the center position of the interference fringes. (d) The relationship between the delay and the fringe wavelength is shown by plotting the peak position calculated from the convolution. Blue dots: measurement data, red line: fitted curve obtained by the 2nd polynomial approximation, which was the calibration curve of this optical system. The circled area represents the interference fringe center position obtained in (c).
Furthermore, we selected a wavelength region of this linear line with no outliers and fitted them with a second polynomial function. The slope of the fitted line was obtained as 0.31 mm/px. Using the relationship between the wavelength and delay length (10.46 mm/nm) from the previously obtained calibration curve, the wavelength resolution determined by the pixel in the measurement using the imaging spectrometer and Si camera was obtained as 0.03 nm/px. The delay length uncertainty determined by the standard deviation of the fitting residuals was 81 µm. These results show that the spectral imaging measurement using an imaging spectrometer with a similar uncertainty as in an OSA was confirmed. Therefore, this demonstrates the capability of single-shot dead-zone-free multi-point delay length measurement with a resolution of 81 µm at an arbitrary position over the pulse-to-pulse interval.
4.4 Shape measurement using dead-zone-free pulses
Finally, we demonstrated the capability of simultaneous multi-point imaging in the longitudinal direction for 3D shape measurement using dead-zone-free pulses. Here, we measured the shape of an object with a three-step structure in the longitudinal direction following the method described in Section 4.2. The entrance slit width was extended to 0.5 mm from 0.3 mm due to the decreasing intensity of reflected light. The three flat mirrors were positioned over the entire pulse-to-pulse interval, as shown in Fig. 8(a). The distance between mirrors was 72 ± 2 and 199 ± 2 mm from the first to the second and third mirrors, respectively. This uncertainty in the nominal value was caused by the manual distance measurement with a hand scale. The optical path lengths, which were twice the distance between mirrors, were 144 and 398 mm, respectively, approximately half and full length of the pulse-to-pulse interval delay length.
Fig. 8. (a) Mirrors of the target simulating steps, placed at a distance of 72 ± 2 mm from the first mirror to the second mirror, and 199 ± 2 mm to the third mirror. (b) Images obtained by the imaging spectrometer. Interference fringe obtained from each mirror is marked. (c) Line profile of each mirror calculated from the interference fringe positions using convolution. The distance between each mirror represents the nominal value.
The measurement results for this target are shown in Fig. 8(b). It also shows the interference image obtained using the Si camera after the imaging spectrometer, as in Section 4.2. The interference pattern corresponding to each mirror is observed in marked area with zoomed view. Although Mirrors 1 and 3 showed interference patterns at approximately the same wavelength, they interfered with the next neighboring pulses. The position of the interference fringes at a certain spatial position on the vertical axis was obtained by calculating the convolution of the cross section along the wavelength axis with the same scheme as the previous measurement for a single planar mirror. According to the previously obtained calibration curve between the wavelength and delay length which is twice of distance (the slope is 10.46 mm/nm), the distance of the mirror was calculated as shown in Fig. 8(c). The distance results in Z calculated from the interference fringe are described as a function of the transverse spatial position X in the vertical axis of the interference image of Fig. 8(b). Table 1 shows the distance between mirrors and their uncertainty. The measurement uncertainty was obtained from the standard deviation of residuals of linear fitting to exclude the tilt of the mirrors. From this result, this method demonstrates the ability of single-shot dead-zone-free shape measurement at arbitrary distance with an uncertainty of 0.7 mm over the 200 mm distance range corresponding to the pulse-to-pulse interval distance.
Table 1. Measured distance between mirror1 and 2 or 3
5. Discussion
5.1 Resolution and uncertainty
Here, we discuss the resolution and uncertainty of the measurement evaluating the characteristics of the system using an imaging spectrometer. In the experiment, convolutions were calculated from the characteristic wavelengths appearing in the spectral interference fringes, and the wavelengths were converted to delay length using a calibration curve. The number of pixels per delay of 1 mm was 0.31 mm/px, or 3.3 px/mm. The uncertainty in the longitudinal direction obtained from the analysis of the calibration curve was 0.081 mm, corresponding to 0.27 px. Due to the nature of the convolution calculation, the calibration curve in Fig. 7(d) shows a plot with a step of 0.5 px. Although the step resolution is larger than the obtained delay length uncertainty, statistical analysis can support better resolution since the standard deviation of the residual between the calibration curve and the measurement data was used.
Considering the principle of the method, which uses chirped spectral interference, the distance resolution is inversely proportional to the amount of chirp and proportional to the spectral resolution. Since the pixel number of the camera limits the current wavelength resolution, using a high resolution camera as the detector unit increases the wavelength resolution per pixel, and higher resolution distance measurement should be possible. Another method to improve the distance resolution regarding wavelength resolution is to expand the spectral bandwidth of the OFCs, such as using a highly nonlinear fiber for supercontinuum generation. If a broader spectrum is used, pulse with less chirp can be used for the same total measurement range, thus, the distance per unit wavelength becomes shorter and the distance resolution is improved.
Additionally, the performance can be improved by the analysis method. Although so far, the fringe analysis was conducted based on the envelope [19], further improvement should be possible using phase information together, which could provide wavelength resolution below 1 nm. Absolute distance measurement using such a combination, i.e., coherent link, between envelope and phase information is well known in OFC distance measurements [7,26].
The other uncertainty source is the background noise, which affects the peak detection uncertainty in the convolution process. Specifically, when we observe two peaks at the region where neighboring pulses appear simultaneously in the spectrum, the accuracy is degraded by false detection of the additional cross convolution peak. When the peak intensity is weak and close to the background level or when a large noise peak appears, an error could occur in the analysis. Increasing the signal intensity and making the background noise relatively small, and flattening the spectrum of the light source could improve the background noise removal accuracy and correctly emphasize the interference fringes.
Notably, the advantage of this method for dead-zone-free measurement is that the spectrum width of the OFCs, the amount of chirp, the resolution of the spectrometer, and the analysis method can be adjusted to obtain suitable characteristics that are enough to match the application requirement.
5.2 Uncertainty discrepancy between the OSA and imaging spectrometer
Here, the discrepancy in uncertainties of the calibration curves obtained using the OSA and imaging spectrometer in the case measuring same target is discussed. The uncertainty of the calibration curve measured using the OSA was 31 µm, and that using the imaging spectrometer was 81 µm. As mentioned in Section 5.1, since the distance resolution is proportional to the wavelength resolution, we speculate that it is due to the wavelength resolution. To investigate this, we measured the spectra of a narrow-line continuous wave laser, Monolithic Isolated Single-mode End-pumped Ring lasers (MISERs), using both the OSA and the imaging spectrometer as detector units, and compared the results. As shown in Fig. 9(a), the spectrum full width at half maximum measured using the OSA was 0.04 nm. Next, that spectrum width measured using the imaging spectrometer, as shown in Fig. 9(b), was 8 px, which was converted to a wavelength of 0.24 nm based on the pixel to wavelength ratio of 0.03 nm/px obtained in Section 4.2. The spectrum shows some fringes at the bottom, which suggests that diffraction at the entrance slit blurred the beam and degraded the wavelength resolution in the imaging spectrometer. These results suggest that the wavelength resolution degrades the distance resolution.
Fig. 9. (a) Spectra of narrow spectrum width laser MISERs measured by the OSA. (b) Spectra of MISERs measured by the imaging spectrometer. The slit width was 0.3 mm. (c) Spectrum widths of MISERs measured by the imaging spectrometer while the slit width is changed.
Therefore, to investigate the possibility of improving the resolution further, we measured the spectra of MISERs while changing the slit width of the imaging spectrometer. The result in Fig. 9(c) indicates that the spectral width can be further decreased; therefore, the wavelength resolution can be further improved by narrowing the slit width. Since narrowing the slit width also decreases the light power entering the spectrometer, amplifying the light source and constructing an efficient focusing optical system would also be needed. By proper optimization of the system, achieving a similar resolution as obtained by the OSA with image detection is possible.
5.3 Uncertainty in step shape measurement
Here, we discuss the uncertainty of the simultaneous three-step mirror structure measurement in Section 4.3. We also discuss the probable cause of the degradation of the distance resolution in the case of the three-step structure distance measurements, which is 0.7 mm, whereas it was 0.04 (= 0.081/2) mm in the case of the flat mirror measurement, where the factor 2 comes from double path. In the case of the three-step structure measurement, we focused the probe beam reflected from the target and broaden the entrance slit of the imaging spectrometer to collect light effectively. Consequently, the wavefront of the probe and the reference beams were not perfectly matched, thereby causing a distortion in the interference fringe patterns. It is evident in Fig. 8(b) that the fringe patterns are distorted and even truncated after subtracting the background. Therefore, we speculate it is the cause of the degradation of the distance resolution, thus uncertainty. In the fringe analysis, we picked up the cross-section along the horizontal, i.e., wavelength axis, and applied the convolution to determine the center position of the fringe. However, due to these distorted and truncated shapes, the central position of the fringe could not be precisely determined since the convolution uses the whole fringe shape, including fine fringes at the bottom. Therefore, to investigate the possibility of improving the resolution in the fringe position determination, we repeated 30 consecutive measurements. Due to environmental fluctuation, the fringe pattern changed between bright and dark fringes according to the interference phase fluctuation. By doing so, we naturally obtained the differential of the fringe shape and determined the fringe peak position more precisely. We re-calculated the longitudinal position Z shown in Fig. 8(c) for each vertical position X. As a result, the standard deviation of the distance Z of each mirror was improved and obtained as 0.15 mm. Considering that the slit was broadened from 0.3 to 0.5 mm, which could cause degradation of the wavelength resolution, we can conclude that the obtained resolution of 0.15 mm was similar level to that obtained in the case of the flat mirror, i.e., 0.04 mm. Therefore, by improving the optical alignment and analysis technique, we expect to achieve better distance measurement uncertainty in the level of 10 ∼ 100 µm with the developed method.
6. Conclusions
We demonstrated the proof-of-principle of a one-shot dead-zone-free 3D shape measurement technique with a high repetition 750-MHz Yb:fiber comb and highly chirped spectral interferometry. We constructed an interferometric imaging system and showed that distance information was obtained simultaneously for multiple targets positioned over the full range of the pulse interval of approximately 200 mm in distance, i.e., 400 mm in delay length. First, the amount of chirp required to generate a dead-zone-free pulse was investigated using a grating pair and found to be –13.2 ps2. Next, the interferometer delay between the two pulses was changed by moving the mechanical stage, and it was confirmed that the dead-zone-free pulses were generated by measuring the spectrum of the interference fringe signal with an OSA, which could achieve a longitudinal length resolution of 31 µm. Furthermore, using an imaging spectrometer capable of multi-point detection and a Si camera, multi-point delay length measurement was demonstrated with a flat mirror, and a length resolution of 81 µm was obtained. Finally, to demonstrate the applicability of simultaneous multi-point imaging, line profiles were obtained using an object with a step structure as a model where three mirrors were placed over the full range of a pulse interval, 73.5 and 198.3 mm from the first mirror, respectively. The distance measurement uncertainty of the three mirrors was 0.7 mm, which could be further improved by optical alignment and analysis to the level of 10 ∼ 100 µm. Moreover, the current line profile measurement technique can be directly extended to simultaneous 3D imaging using 2D spectroscopy technique such as a fiber bundle as demonstrated in our previous work [20]. Notably, these results demonstrated simultaneous imaging of the targets at the middle of a pulse interval, which had not been achieved in previous measurements, therefore, confirming the capability of the dead-zone-free single-shot 3D measurement. Although the current distance image uncertainty is still relatively large, it is limited by the resolution of the spectroscopy equipment and alignment of the optical system, thus, it could be further improved to achieve a micrometer uncertainty at an arbitrary position over the pulse-to-pulse interval. It is expected to be applied to shape measurement of various targets, such as vibrating objects and/or large-scale structures.
Funding
Japan Science and Technology Agency (JST) through the Exploratory Research for Advanced Technology (ERATO) Minoshima Intelligent Optical Synthesizer (IOS) Project (JPMJER1304); Japan Society for the Promotion of Science (JSPS) KAKENHI (21H05014, 21F21363).
Acknowledgments
The authors thank Drs. Bo Xu, Yuxuan Ma, Yoshiaki Nakajima, Isao Matsushima, and Zhigang Zhang for their contributions in early stage to develop the laser used in this study, and Asahara Akifumi for the help in preparation of the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31(7), 919–925 (1992). [CrossRef]
2. W. Carter, R. Shrestha, G. Tuell, D. Bloomquist, and M. Sartori, “Airborne laser swath mapping shines new light on Earth’s topography,” Eos Trans. AGU 82(46), 549 (2001). [CrossRef]
3. K. Omasa, F. Hosoi, and A. Konishi, “3D lidar imaging for detecting and understanding plant responses and canopy structure,” J. Exp. Bot. 58(4), 881–898 (2006). [CrossRef]
4. Y. Y. Cheng and J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23(24), 4539 (1984). [CrossRef]
5. A. G. Podoleanu, G. M. Dobre, and D. A. Jackson, “En-face coherence imaging using galvanometer scanner modulation,” Opt. Lett. 23(3), 147–149 (1998). [CrossRef]
6. Z. Wang, B. Potsaid, L. Chen, C. Doerr, H. C. Lee, T. Nielson, V. Jayaraman, A. E. Cable, E. Swanson, and J. G. Fujimoto, “Cubic meter volume optical coherence tomography,” Optica 3(12), 1496–1503 (2016). [CrossRef]
7. K. Minoshima and H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. 39(30), 5512–5517 (2000). [CrossRef]
8. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29(10), 1153–1155 (2004). [CrossRef]
9. S.-W. Kim, “Photonics metrology: Combs rule,” Nat. Photonics 3(6), 313–314 (2009). [CrossRef]
10. G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correlation of air refractive index using two-color optical frequency combs,” Sci. Rep. 3(1), 1894 (2013). [CrossRef]
11. E. Baumann, F. R. Giorgetta, J.-D. Deschênes, W. C. Swann, I. Coddington, and N. R. Newbury, “Comb-calibrated laser ranging for three dimensional surface profiling with micrometer level precision at a distance,” Opt. Express 22(21), 24914–24928 (2014). [CrossRef]
12. W. Zhang, H. Wei, H. Yang, X. Wu, and Y. Li, “Comb-referenced frequency-sweeping interferometry for precisely measuring large stepped structures,” Appl. Opt. 57(5), 1247 (2018). [CrossRef]
13. S. Hyun, M. Choi, B. J. Chun, S. Kim, S.-W. Kim, and Y.-J. Kim, “Frequency-comb-referenced multi-wavelength profilometry for largely stepped surfaces,” Opt. Express 21(8), 9780–9791 (2013). [CrossRef]
14. H. Zhao, Z. Zhang, X. Xu, H. Zhang, J. Zhai, and H. Wu, “Three-Dimensional Imaging by Frequency-Comb Spectral Interferometry,” Sensors 20(6), 1743 (2020). [CrossRef]
15. E. Vicentini, Z. Wang, K. Van Gasse, T. W. Hänsch, and N. Picqué, “Dual-comb hyperspectral digital holography,” Nat. Photonics 15(12), 890–894 (2021). [CrossRef]
16. D. Kim, Y. Lu, J. Park, B. Kim, L. Yan, L. Yu, K. N. Joo, and S. W. Kim, “Rigorous single pulse imaging for ultrafast interferometric observation,” Opt. Express 27(14), 19758–19767 (2019). [CrossRef]
17. K. Minoshima, H. Matsumoto, Z. Zhang, and T. Yagi, “Simultaneous 3-D imaging using chirped ultrashort optical pulses,” Jpn. J. Appl. Phys. 33(Part 2, No. 9B), L1348–L1351 (1994). [CrossRef]
18. T. Kato, M. Uchida, and K. Minoshima, “No-scanning 3D measurement method using ultrafast dimensional conversion with a chirped optical frequency comb,” Sci. Rep. 7(1), 3670 (2017). [CrossRef]
19. T. Kato, M. Uchida, Y. Tanaka, and K. Minoshima, “High-resolution 3D imaging method using chirped optical frequency combs based on convolution analysis of the spectral interference fringe,” OSA Continuum 3(1), 20–30 (2020). [CrossRef]
20. T. Kato, M. Uchida, Y. Tanaka, and K. Minoshima, “One-shot three-dimensional imaging using a two-dimensional spectrometer with a fiber bundle,” Opt. Express 29(26), 43778–43792 (2021). [CrossRef]
21. Y. Nakajima and K. Minoshima, “Highly stabilized optical frequency comb interferometer with a long fiber-based reference path towards arbitrary distance measurement,” Opt. Express 23(20), 25979–25987 (2015). [CrossRef]
22. Y. Ma, B. Xu, H. Ishii, F. Meng, Y. Nakajima, I. Matsushima, T. R. Schibli, Z. Zhang, and K. Minoshima, “Low-noise 750 MHz spaced ytterbium fiber frequency combs,” Opt. Lett. 43(17), 4136–4139 (2018). [CrossRef]
23. B. Xu, Y. Ma, H. Ishii, I. Matsushima, Z. Zhang, and K. Minoshima, “Nonlinear amplification based on a tightly phase locked 750 MHz Yb:fiber frequency comb,” Appl. Phys. Lett. 118(3), 031101 (2021). [CrossRef]
24. E. B. Treacy, “Optical pulse compression with diffraction grating,” IEEE J. Quantum Electron. 5(9), 454–458 (1969). [CrossRef]
25. F. Druon, M. Hanna, G. Lucas-Leclin, Y. Zaouter, D. Papadopoulos, and P. Georges, “Simple and general method to calculate the dispersion properties of complex and aberrated stretchers–compressors,” J. Opt. Soc. Am. B 25(5), 754–762 (2008). [CrossRef]
26. G. Wu, M. Takahashi, H. Inaba, and K. Minoshima, “Pulse-to-pulse alignment technique based on synthetic wavelength interferometry of optical frequency combs for distance measurement,” Opt. Lett. 38(12), 2140–2143 (2013). [CrossRef]
