1. Introduction

For practical applications in various science and technology fields, a highly accurate one-shot three-dimensional (3D) imaging method is desired. Existing 3D imaging methods, on the other hand, find it difficult to achieve one-shot 3D imaging with a wide dynamic range, i.e., high accuracy and a wide measurement range, such as micrometer-level and meter-range, at the same time. Although the traditional method of laser distance measurement can provide high accuracy [1–3], it is limited to stationary objects because the beam position at the target must be scanned, preventing real-time measurement. Previously, we proposed a one-shot 3D measurement method based on the ultrafast conversion of time, frequency, and spatial axis information encoded in chirped ultrashort pulses to solve this problem [4]. The previous method, however, has a limited dynamic range due to the tradeoff between the degree of chirp and the spectral and temporal resolutions. Furthermore, because this method depends on optical nonlinearity to generate a broad supercontinuum and a nonlinear time gating method, high-power ultrashort pulses generated by a bulky optical source were required. Therefore, not only is the method impractical but also its measurement performance is limited by the laser’s pulse-to-pulse instability. It is necessary to develop a high-performance and simple measurement method with a practical light source.

An optical frequency comb (OFC) is high controllability and coherence ultrashort pulse train that can measure distance with high accuracy and a wide dynamic range [5–8]. Using fiber-based OFCs with excellent stability and practicability, we developed a one-shot 3D imaging method with greatly improved performance [9]. In the previous study, 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. The measurement range was significantly extended without sacrificing accuracy and the high-power laser requirement by using OFC's pulse-to-pulse interference as a linear time gating method to capture the time-sliced image, which greatly improved the method's applicability. A non-uniform pattern in the interference spectrum with a minimum fringe frequency at a characteristic wavelength indicates depth, i.e., distance information. Then, to obtain the characteristic wavelength with high accuracy without a complicated and time-consuming analysis, we developed the method using the convolution of the spectral interference fringe pattern and demonstrated sub-µm uncertainty [10]. However, a problem remained in previous demonstrations: only two-dimensional (2D) profiles could be detected in a one-shot using a conventional spectrometer with a one-dimensional (1D) diffraction grating, although 3D information had already been captured in a single shot of a pulse. This is because, in the case of the 1D grating, one of the space axes was used to detect wavelength. Therefore, previous research demonstrated 3D imaging with additional mechanical scanning because line scanning was required to decode the 3D information. To capture the 3D information by one shot, the imaging method must use a 2D spectrometer to capture the spectrum at each spatial point on the 2D image plane.

Many research teams have investigated 2D spectroscopy methods, also known as hyperspectral imaging or integral field spectroscopy. The motivation for using 2D spectroscopy in this study, as stated previously, is to capture the spectrum at each spatial point on an image plane by one shot without scanning. The 3D data, known as the hyperspectral cube, is obtained by using a 2D spectrometer. Generally, there are three strategies for capturing the hyperspectral cube: the first is to obtain the spectrum by each point to point on the image; the second is to obtain the image by scanning the wavelength; the third is to retrieve the hyperspectral image from the measurement data using a computer algorism. The first method divides the image using a slice mirror [11] or fiber bundle [12] and obtained the spectrum of each point using a 1D spectrometer. These methods produced a one-shot measurement with high wavelength resolution, but the spatial resolution was limited by the number of divisions. Alternatively, by combining 1D spectroscopy at a single spatial point with a raster scan, 2D spectral information is obtained [13]. This method can obtain the spectrum with high wavelength and spatial resolutions, but cannot perform a one-shot measurement. Next, in the second method, the image is captured by passing it through a tunable filter [14] or by performing Fourier-transform spectroscopy with a full-field display [15]. These methods produce images with high spatial resolution, but they necessitate scanning the spectral band to obtain wavelength information or a long delay to produce the interferogram. As a result, neither method can take a one-shot measurement. Finally, in the third method, spatial modulation is used as a coded aperture for the image to obtain the coded image passing through a dispersive element [16] or to capture the spectral image using a 2D grating [17]. Through a complex calculation, this method retrieves the hyperspectral image from the coded image. However, because the algorithm assumes an ideal experimental condition, it is difficult to obtain quantitative results.

Our motivation, in this study, is to realize 2D spectroscopy applied to one-shot 3D imaging, which necessitates reliable and quantitative data, the ability to capture data by one shot, and high wavelength resolution. Among the methods discussed above, 2D spectroscopy with a fiber bundle can obtain quantitative and fine spectral information in a one-shot. As a result, we chose the first type category with fiber bundles for 2D spectroscopy and developed a method for a new 3D imaging technique with high spatial resolution and collection efficiency. In this study, we developed a setup using single-mode fiber (SMF) or multi-mode fiber (MMF) bundles for 2D spectroscopy and achieve the one-shot 3D imaging using 2D spectral imaging. The spatial resolution is greatly improved by designing fiber bundles with 190 SMFs and 55 MMFs. Then, we demonstrated that one-shot 3D imaging with an OFC and a 2D spectrometer with fiber bundles achieved high precision, high dynamic range, and one-shot measurement simultaneously.

2. Experiments

Figure 1 shows the experimental setup, which includes an image of the fiber bundle, experimental configuration, spectra, and pulse waveforms of the OFC. An OFC with a pulse width of 65 fs, center wavelength of 1.55 µm, average power of ∼100 mW, and repetition frequency of 51.06 MHz was generated by a lab-built mode-locked Er-doped fiber laser through an optical amplifier with an Er-doped fiber. At a gate time of 1 s, the OFC’s repetition frequency (f_rep) and carrier-envelope offset frequency (f_CEO) were stabilized to the microwave frequency standard with an Allan deviation of ∼10⁻¹². The OFC was first divided into the reference and probe pulses using a beam splitter before being used to construct the spectral interferometer for the proposed method. A probe pulse width is extended to 5.7 ps with highly chirped by passing through a 3.8-m long SMF (Fig. 1(a)).

Following irradiation through a beam expander, the probe pulse was focused on the measurement target. The reflected probe pulse was overlapped with the chirp-free reference pulse following a delay stage to form an interference fringe. The overlapping beams were then introduced into the input side of the fiber bundle with 2D arrangement. The 2D image was converted to a 1D arrangement by the fiber bundle at the output face. Here the probe beam was focused to ensure a high collection efficiency, whereas the reference beam was not because the optical power was sufficient. The 1D arranged image was then introduced to the imaging spectrometer (iHR320, HORIBA, Japan), which comprised a reflection-type grating with a groove number of 600 mm⁻¹, and captured with a 640 × 512 pixels InGaAs camera (C12741-03, Hamamatsu Photonics, Japan). The wavelength axis (i.e., depth information) was shown in the captured image's horizontal direction, while the transverse spatial information was shown in the image's vertical direction. Millions of pulses from the 51 MHz repetition pulse train were averaged because the camera's exposure time was 4.6 ms. Using the delay stage, the optical path of the interferometer was adjusted to overlap the adjacent pulses in a mode-locked pulse train so that they interfered with each other. Note that without a mechanical stage, a target object located at any position can be measured by scanning the repetition frequency of the OFC to make the initial alignment [18]. After the initial alignment, there was no need for delay scanning for shape measurement, allowing for one-shot 3D imaging.

In our experiments, we used two types of fiber bundles, SMF and MMF. The schematic of the SMF fiber bundle, which consisted of 190 SMFs (Core diameter: 9 µm, Clad diameter: 125 µm, Numerical Aperture (NA): 0.14, Length: 1.5 m), is shown in Fig. 1(b). In the case of the MMF fiber bundle, the structure is the same as the SMF fiber bundle, but it contains 55 MMFs (Core diameter: 200 µm, Clad diameter: 245 µm, NA: 0.22, Length: 1 m). The fiber bundle has two distinct ends. At one end, 190 SMFs were arranged in two dimensions (in a circle with a diameter of 2 mm) and 190 SMFs were arranged in a line at the other end. The fiber bundle functions as a 2D-to-1D image converter. Because the image was introduced in the fiber bundle after interference in our study, precise length control was not needed to re-arrange the interference image to the 1D line. By introducing the 2D spectral image to the fiber bundle, each SMF sampled each position of the image as one pixel, thus, we can capture the 3D image with 190 pixels using one-shot imaging (Fig. 1(b)). A focusing lens was then used to introduce the linearly rearranged image to the spectrometer. As a result, the spectral image with three-dimensional shape information was obtained with a single pulse.

In this study, we performed several one-shot 3D imaging using SMF and MMF fiber bundles. First, we tested the fundamental performance of obtaining depth information from the spectral interference fringe by varying the known delay, which yields a calibration curve for determining depth from the measured characteristic color. Second, we show and evaluate the surface profile measurement of a well-evaluated gauge block structure. Third, we used it on a large object with a high aspect ratio consisting of gauge blocks and a flat mirror. Here, we evaluated the method’s dynamic range using pulse-to-pulse interference. Finally, we used the MMF fiber bundle to measure the surface of the gauge blocks to maximize collection efficiency while minimizing the effect of speckle noise in spectral interference. The SMF and MMF fiber bundles’ transmitting efficiencies were also compared.

 

Fig. 1. (a) Experimental setup for one-shot 3D imaging. The above two plots in the red square show the spectra and autocorrelation traces of the OFC. The figure below is an overview of the experimental setup. Figure in blue dotted square shows a detail of an optical setup for a 2D spectrometer using the bundle fiber. LD: Laser diode, WDM: Wavelength division multiplexer, EDF: Erbium-doped fiber, PQH: Polarization controller with a polarizer, quarter-, and half-wave plate, ND: Neutral density filter, BS: Beam splitter (b) Structure of the SMF fiber bundle consisting of 190 SMFs. The left image shows the input side. The right image shows the spectral image captured by the imaging spectrometer. The spectral image including 190 spectral lines aligned to the vertical axis induced from the output side of the bundle fiber. The numbers, from 1^st to 190^th, next to the image indicate the corresponding fibers of the bundle. The wavelength range of the spectral image was approximately from 1440 to 1690 nm.

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3. Results

3.1 Obtaining the calibration curve

We evaluated the basic depth measurement performance before applying it to the surface and internal structure measurements of actual targets, by obtaining the calibration curve between the depth and color using a flat mirror measurement. We did this by changing the position of the delay stage in the reference beam path and measuring a series of spectral interference fringes. The depth information is then derived from a characteristic wavelength that gives the minimum fringe frequency of the chirped spectral interference, as described in Ref. [10]. The characteristic wavelength in the spectra changes as the delay between the probe and reference pulses changed. The calibration curve results from the relationship between the characteristic wavelength and the delay time or distance. Because the relationship reflects the chirp characteristic, obtaining such a curve when measuring the target is not required every time unless the chirp behavior changes.

Figure 2(a) shows the captured spectral interference image, including the chirped spectral interference fringe, in which horizontal and vertical directions correspond to the wavelength and space, respectively. Figure 2(a) displayed only 90 spectral lines out of 190 lines sampled with SMFs to show clear spectral lines. Each spectrum was slightly curved as a result of chromatic aberration in the focusing lens. We obtained an interference fringe image (Fig. 2(b)) after removing the constant background for the delay to enhance the fringe. The spectral interference fringe is visible in the enhanced image. Figure 2(c) shows the wavelength dependence of the interference signal at Y = 184 pixels in Fig. 2(b), which is a spectral line of 24 SMF in the bundle. Because of the chirped spectral interference, the fringe spectrum demonstrates a non-uniform pattern, with a characteristic wavelength region where the fringe span is broadest, that is, oscillates with the minimum fringe frequency (red dotted line in Fig. 2(c)). After calculating the convolution of the spectral signal in Fig. 2(c) [10], we get the peak shown in Fig. 2(d). The characteristic wavelength is represented by the position of the convolution signal’s peak, which is shown by the red circle in Fig. 2(d).

By adjusting the delay time between the chirped and chirp-free pulses, the fringe image pattern moves horizontally, that is, along the wavelength axis. Figure 2(e) shows the characteristic wavelengths obtained by convolution with varying delays at Y = 184 pixels. The red circle in Fig. 2(e) shows the characteristic wavelength in Fig. 2(d). Following a polynomial approximation analysis, we obtained the calibration curve between the depth and color, which can be used in the actual depth measurement based on the color in our method (blue line in Fig. 2(e)). In a delay range of 700 µm, the fitting residual’s uncertainty was 0.54 µm.

The spectral and interference images obtained with a single MMF are shown in Figs. 2(f) and (g). It displays a spectral fringe pattern similar to the image obtained by the SMF bundle. As a result, the results show that the imaging system can capture interference images with both fiber types, SMF and MMF. Figure 2(h) shows the characteristic wavelength's dependence on delay. The calibration curve was also obtained using a polynomial approximation (Fig. 2(h)), and the uncertainty was 0.35 µm over a 650 µm delay range. Thus, the proposed method can measure sub-µm-level structure with SMF and MMF fiber bundles. Target 3D shape information can be measured in a single shot of ultrashort pulses because each probe pulse can record full 3D information using a fiber bundle. It should be noted that the slopes of the calibration curves in Figs. 2 (e) and (h) differ. This is because the captured spectral range was changed in both cases. The measurement range was discussed in Section 4.1.

3.2 Demonstration of the profile measurement

Additionally, we used the technique for the surface profile measurement of the gauge blocks to evaluate the method quantitatively. The target consisted of optically contacted three-gauge blocks on a glass plate, which provide two steps with the height of 480 and 980 µm with nanometer-level uncertainty.

The target is shown schematically in Fig. 3(a). Here, an SMF fiber bundle is used to analyze the spectral interference fringe to determine the surface profile of the target. Figure 3(b) shows the obtained 3D image of the target. To obtain the calibration curve, 100 spectral images were obtained in this experiment by varying the delay. The X and Y axes represent the position of the fiber bundle's input side, which has a diameter of 2 mm. Averaging 100 images yielded the Z position at each pixel, i.e., each SMF. The Z value was plotted relative to the surface of step A. Each area, A, B, and C, was captured with 48, 55, and 87 SMFs, respectively, in Fig. 3(b). The SMF fiber bundle captured the structure of the target. The obtained step heights of A to B and A to C were 478.9 ± 3.7 µm and 981.7 ± 3.0 µm, which were obtained from the average and the standard deviation of 55 and 87 points, that is pixels, respectively. The nominal values of the step heights from A to B and A to C were 480 µm and 980 µm, respectively. The standard deviations of the mean values at steps A to B and A to C were 0.5 µm and 0.3 µm, respectively, calculated by dividing the standard deviation by the root of the number of measurements, 55 and 87. This result shows that the system succeeds in measuring the mm-level range structure with sub-µm uncertainty.

 

Fig. 2. (a) Captured spectral interference image where the delay position was 600 µm using the SMF fiber bundle. The wavelength range was approximately from 1520 to 1620 nm. (b) Enhanced image of captured spectral interference image. This image was obtained by removing the background. (c) Interference fringe spectrum at Y = 184 pixels in (b) which shows a characteristic wavelength that gives the minimum fringe frequency. (d) Convolution of the interference spectrum of (c). The peak of this convolution signal shows the characteristic wavelength. (e) Delay dependence of the characteristic wavelength calculated with convolution analysis. Red line: measured data, Blue line: polynomial fitted line which can be used as the calibration curve. The red circle shows the position of the peak in (d). (f) Single MMF spectral interference image. The wavelength range was approximately from 1505 to 1625 nm. (g) Enhanced image of single MMF spectral interference image. (h) Center position of spectral fringe using the same analysis at (e).

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Fig. 3. (a) Target of three-gauge blocks. (b) 3D image of the gauge block target. Blue, green, and red show measurement points at A, B, and C, respectively. The origin of the Z value is the mean of measurement points at A. Four blue points outside of area A are crosstalk between the fibers due to the diffraction of the probe light at the edge of the gauge block.

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3.3 Large object measurement

We can use this technique to capture a large object of a high aspect ratio target using pulse-to-pulse interference. To demonstrate this advantage, we measured a large target consisting of a step structure made of two gauge blocks and a flat mirror spaced 3 m apart. The high aspect ratio target is shown in Fig. 4(a). The step height of the gauge blocks C and D was 980 µm. We then used a handmade SMF fiber bundle with four SMFs. The reflected probe pulse from A, B, C, and D was captured by each fiber in Fig. 4(a). The 3D image of the high aspect ratio target is shown in Fig. 4(b). For quantitative evaluation, the calibration curve was derived from 10 spectral images with varying delay. In this demonstration, first, we experimented to capture the points of A, B, and C with one shot. Because of the pulse-to-pulse interference, all the probe pulses from the three positions can interfere with the same reference pulse, thus a captured one-shot image contains the spectral interference fringes from all the points of interest, simultaneously. In this case, the pulse-to-pulse separation could be precisely calculated using the comb's repetition frequency. As a result, one-shot imaging of a 3 m object was demonstrated. Following that, we measured the points A, B, and D by one shot after adjusting the small delay caused by the chirped pulse duration limitation, which was 700 µm, as determined by the chirp value of the pulses. Because the chirp value can be extended by simply inserting longer SMF in the probe arm in the setup, we can upgrade to capture all 4 steps simultaneously. Because the comb provided highly stable pulse-to-pulse separation, these two measurement results could be precisely linked and are shown in Fig. 4. In either case, we were able to show 3D imaging of a large object by one shot, with the distances from A to C and D precisely measured as 2.932363 m and 2.933343 m, respectively. The left plot in Fig. 4(b) shows an overview of the experimentally obtained target structure, with the Z-axis in meters. The two plots on the right side of Fig. 4(b) show zoomed views of each surface whose Z-axis was plotted in µm. Here, the origin of the Z-axis was set by the mean of the surface of step D. Points A and B, which are on the surface of the same flat mirror placed 3 m away from the gauge blocks, were obtained as they have a difference of 1 ± 3 µm. The result reflects the measurement's uncertainty because these two points were on the same mirror. The distance between points C and D on the surface of the gauge blocks was determined to be 980 ± 2 µm, which agreed well with the nominal value of 980 µm, although the two points were obtained by two independent measurements.

As a result, the developed system is shown to be capable of measuring the 3D profile of 3 m targets with micrometer accuracy. Therefore, we could demonstrate the dynamic range of the system, that is, range/uncertainty was at least 6 digits, which corresponds to the relative uncertainty (Δz) to the measurement range (z), i.e., Δz/z = 3 µm/3 m ∼ 10⁻⁶.

 

Fig. 4. (a) High aspect ratio target. We placed the two gauge blocks, C and D, and 3 m apart A and B were on the same mirror. (b) The obtained 3D plot of the target. The left plot shows the target with a meter scale on the Z-axis. The two right plots show a detailed view of the target with a micrometer scale on the Z-axis. Here, the position at D was defined as the origin on the Z-axis. The mean values of the distances from point A to C and A to D were 2.932363 m and 2.933343 m, respectively.

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3.4 3D imaging using the MMF fiber bundle

As demonstrated, we could successfully measure 3D shape by one shot with the fiber bundle. However, with SMFs, the light collection efficiency is insufficient in some applications. Thus, we demonstrated and evaluated 3D imaging using an MMF fiber bundle. Because MMFs have a larger core radius than SMFs, their coupling efficiency is higher. However, due to the multi-mode nature of the beam, speckle noise may cause spectral interference. This is especially important when using the method on a target with low reflectance, such as a rough surface object. Here, we developed the method to remove the speckle noise for analyzing the spectral interference of MMFs.

Here, we measured the surface of the gauge blocks, which was the same target as in Section 3.2(Fig. 3(a)). Figure 5(a) shows the raw spectral interference image, whose horizontal and vertical axes correspond to wavelength and space, respectively. The image in Fig. 5(a) was noisy due to speckle noise in MMF, as opposed to Fig. 2(a), which was obtained using the SMF fiber bundle. Figure 5(b) shows the interference image after the background has been removed. As a result, the spectral interference pattern can be seen in Fig. 5(b). This result suggests that by removing the speckle noise, an MMF fiber bundle can be used. To remove the speckle noise, we slightly lose the focus of the spectral image corresponding to each MMF on the image sensor so that it illuminates approximately 5 pixels width on the vertical axis. The speckle noise was smoothed and the visibility of the spectral interference signal improved by averaging the spectral interference image along the vertical axis. Following averaging, we used the convolution method to analyze the spectral interference fringe and obtained the 3D structure of the target shown in Fig. 3(a). Section 4.3 will discuss the effect of averaging in greater detail.

Figure 5(c) shows the 3D structure of the gauge block target with 54 points of 55 MMFs. One of the 55 spectral lines in the captured spectral image could not be measured due to insufficient optical power. The measured results allowed the surface profile of the three-gauge blocks to be reconstructed. By evaluating the measured results, the mean and the standard deviation of the Z values at step A, B, and C were 0.0 ± 4.2 µm (N=23), 481.3 ± 2.9 µm (N=14) and 984.4 ± 2.9 µm (N=17), respectively. Here N is the number of sampled points, the origin of the Z-axis was the mean of the Z of step A. The standard deviations of the means of the number N data were 0.9, 0.8, and 0.7 µm, respectively. The uncertainty reached the sub-micrometer level, just as it did with the SMF fiber bundle. As a result, we proved the MMF fiber bundle's functionality.

 

Fig. 5. (a) Captured spectral interference image using the MMF fiber bundle. Horizontal and vertical axes correspond with wavelength and space position. In this case, each fiber was placed on the space axis. (b) Spectral interference image after removing the background. (c) The 3D plot of the gauge block target. Blue, dark green and light green show steps A, B, and C. At each step, number of measurement points were 23 points at A, 14 points at B, and 17 points at C.

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3.5 Transmitting efficiency using the single-mode and multi-mode fiber bundles

Finally, to compare the transmitting efficiency in the case using SMF and MMF fiber bundles, we measured the power collecting efficiency. The experimental setup for determining transmitting power is shown in Fig. 6. The SMF fiber bundle used in the previous experiment was the same as in this one. For comparison, the MMF fiber bundle used here had the same number of fibers as the SMF fiber bundle, 190 MMFs. MMFs were grouped in a 3 mm circle. After combining the probe and reference beams in the experiment, we irradiated the fiber bundles. For simplicity of the experiment, the laser beam was focused only on the center part of the area consisting of 50 fibers of each fiber bundle using a focusing lens (f = 300 mm (Fig. 6)). After passing through the fiber bundle, we measured the output power on the output side of each fiber bundle.

 

Fig. 6. Experimental setup for evaluating the transmitting efficiency of a MMF bundle

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Table 1 shows a comparison of the properties of SMF and MMF bundles. We irradiated the SMF and MMF bundles with laser beams with diameters of 1.01 and 1.50 mm, respectively, to ensure that the same number of fibers were irradiated. Given the difference in core and clad sizes between SMF and MMF, the effective area, or the ratio of core to the clad area, was 0.518% for SMF and 66.6% for MMF, respectively. Thus, the transmitting power of MMF was 128 times larger than that of SMF based only on the size of each fiber, i.e., cross-section.

Table 1. Properties of the SMF and MMF bundles

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The results of the experiment are shown in Table 2. We irradiated the beam with nearly the same power. However, the output power ratio of the two fiber bundles differed significantly from that shown in Table 1. As a result, the MMF fiber bundle had a 299-fold higher transmitting efficiency than the SMF fiber bundle (Table 2). The reason will be discussed in Section 4.4.

Table 2. Measured power using the SMF and MMF bundles

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4. Discussions

4.1 Uncertainty and range of the depth measurement

Here, we discussed the depth resolution of the proposed method. In the experiment to measure the 3D profile of the gauge blocks, the uncertainty of the depth measurement was 0.3 µm. In the measurement, we calculated the convolution to obtain the depth information from the characteristic wavelength that appeared in the captured spectral interference images, and the wavelength was then converted to the depth using the calibration curve. We measured a depth range of 700 µm with an image range of 300 pixels to obtain wavelength information to obtain the calibration curve (Fig. 2(e)). A depth of 1-µm corresponds to 0.43 pixels, that is, 0.43 pixel/µm. Therefore, the depth uncertainty obtained in the experiment, 0.3 µm, corresponds to 0.13 pixels. On the other hand, the spectral fringe analysis method using a convolution should be able to identify the position of the characteristic wavelength with 0.5 pixels. The measured data in Fig. 2(e) shows a step-like structure with 0.5-pixel resolutions, which corresponded to the fringe analysis resolution. As a result of this measurement result, we succeeded to obtain a 3D profile with less uncertainty than a minimum step of the spectral fringe analysis. Because the standard deviation of the residual between the measured 700 data and the fitted calibration curve was used to estimate the uncertainty, we were able to resolve the position at the sub-pixel level with statistical analysis. Using a high-resolution image sensor can help to improve depth resolution without averaging, according to experiments. Furthermore, because the spectral interference fringe is obtained in this method (Fig. 2(b)), the optical phase information could be used to deduce more precise distance information. Because the analysis method used in this study is based on the envelope analysis [10], further implementation of the analysis technique using the phase information can provide the distance information to realize the uncertainty less than an optical wavelength with nm level. Such absolute distance measurement using the coherent link between the envelope and phase information is well known in the comb distance measurements technique [6,19].

Following that, we compared the wavelength resolution of the captured spectral interference image with the imaging spectrometer's performance. In the experiment, 512 pixels were used to capture a spectrum with a bandwidth of 100 nm (Fig. 2(a)). As a result, one pixel resolved a spectrum of about 0.2 nm. However, the imaging spectrometer has a wavelength resolution of 0.06 nm. Therefore, the experimental wavelength resolution was limited by the resolution of the camera pixel. The wavelength resolution can be improved using a high-resolution camera, and thus the uncertainty of the depth measurement can be further reduced.

Using pulse-to-pulse interference, the measurement range can be easily and significantly extended. The proposed method achieved an extensive dynamic range of 6 digits (Section 3.3). Here, we used the interference between 0^th and 1^st pulses. Using the interference between 0^th and N^th pulses, we can capture the 3D images of larger targets. Scanning the repetition frequency of the OFC yielded the number of pulses for any arbitrary object as far away as km-level [18].

In addition, we discussed the measurement range's dead zone. The pulse-to-pulse separation of the used Er-comb, f_rep ∼ 50 MHz, was approximately 6 m. However, the measurement range of the one pulse was approximately 1 mm, which corresponded to the chirped pulse duration. As a result, there is a large dead zone, which is the delay time where probe and reference pulses cannot interfere. This problem can be solved by using a high repetition rate OFC and a highly chirped pulse. A high repetition rate comb, f_rep ∼ 1 GHz, had been developed using a small cavity Yb-comb [20] or a Fabry–Pérot cavity [21]. Considering the time interval of 1 GHz is 1 ns, if we generated a ∼1 ns chirped pulse, which is easily obtained by using a pulse stretcher with a grating pair or fiber Bragg grating, the pulse train can cover arbitrary positions in the time domain, which means that the pulses can interfere at any time. Such dead zone-free measurement has been demonstrated elsewhere [22].

The preceding discussion assumed that the light source's spectrum was fixed. Optical nonlinear phenomena such as fiber, amplifier, or supercontinuum generation in a high nonlinear fiber can be used to broaden the spectrum of the optical frequency comb. Even when the chirp value of the pulse remains constant, the measurement range broadens without sacrificing depth resolution for the same spectrometer wavelength resolution. In this way, the measurement performances of the proposed method can be tailored using chirp value, optical source spectrum, and spectrometer, to match the application needs.

4.2 Transverse spatial resolution using 2D spectroscopy with the fiber bundle

The transverse spatial resolution of one-shot 3D imaging was determined using an optics diffraction limit [10]. We used a 2D spectrometer made up of an imaging spectrometer and a fiber bundle in this study. The light from the object is focused on the image plane of the target by imaging lenses. The image was also detected at the image plane by the fiber bundle. As a result, the spatial resolution of the sampled image cannot exceed the spatial resolution at the image plane and is limited by the imaging optical setup's diffraction limit. Furthermore, when the imaging lens's diffraction is adequate, i.e., when the NA of the imaging lens's system is large enough, the NA of the fiber determines the spatial resolution limit because the fiber allows beams matched to the NA of the fiber to be coupled. We used SMF and MMF with NAs were 0.14 and 0.22, respectively, and the center wavelength of the OFC was ∼1.5 µm. Therefore, the diffraction limits calculated from the fibers’ NA and wavelength were 5.3 µm and 3.4 µm for the SMF and MMF, respectively. These values can be improved using fiber with a large NA.

The number of fibers in the bundle that serve as image pixels, however, limits the number of image positions sampled. A fiber bundle's pixels are typically less than those of image sensors. As a result, by using more fibers, an image's resolution can be increased. However, because of the large clad area, the image obtained is sparse because the pixel density of the image using conventional SMF or MMF is low. This issue can be solved using a thin clad fiber.

4.3 Effect and reduction of speckle noise

A coherent light introduced in an MMF can cause speckle noise because the higher transverse modes can be coupled and interfere with each other. We must remove the speckle noise to get a clear image of the spectral interference. A mode scrambler is commonly used to reduce speckle noise [23]. However, to avoid mechanical vibration while capturing the interference signal and achieve one-shot imaging, we must develop other techniques for removing speckle noise from the captured spectral images.

Figure 7 shows the spectrum and analyzed correlation signal of the spectral interference fringe image in Fig. 2(g). Figure 7(a) shows the series of spectra of the image in Fig. 2(g) at different spatial positions inside the same fiber core, where there are 9 different points along the Y-axis. Although each spectrum in Fig. 7(a) shows that the spectral interference fringe pattern was distorted due to speckle noise, the stable interference phase was still visible because each spectrum displays the same spectral interference fringe. This means that the speckle noise does not destroy the relative phase of the reference and probe pulses. Thus we discovered a method for reducing speckle noise while retaining interference information by spatial averaging as mentioned earlier. Figure 7(b) shows the averaged spectrum from Fig. 7(a) where the spectral interference fringe pattern is observed. Figures 7(c) and (d) show the correlation signal obtained from one spectrum in Fig. 7(a) and the averaged spectrum in Fig. 7(b). In Fig. 7(c), the convolution signal of a single spectrum has several spurious peaks near the center peak (red maker). The convolution signal in Fig. 7(d), on the other hand, has a distinct signal peak after averaging and no significant spuriousness. Because all of the areas for averaging are in the same core of the MMF fiber, such spatial averaging can be applied to one-shot measurements without degrading image resolution. In these results, we confirmed that removing speckle noise through spatial averaging is effective for one-shot 3D imaging using the MMF fiber bundle. This is useful for sampling a weak intensity image using an MMF bundle; it is especially useful for measuring rough surface targets.

 

Fig. 7. (a) Spectrum at each spatial point (Y = 230 ∼ 254 pixels in Fig. 2(g)) extracted from the spectral interference image in Fig. 2(g). (b) Averaging of 9 spectra in (a). (c) Correlation signal obtained from a single spectrum in (a). (d) Correlation signal obtained from the averaged spectrum (b).

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4.4 Collection efficiency of the SMF and MMF fiber bundles

The use of MMF was motivated by our desire to improve the collection efficiency of the interference image signal. It is critical when measuring a rough surface object with low reflectance. In Section 4.3, we demonstrated the effectiveness of the proposed one-shot 3D imaging method, the MMF fiber bundle. The collection efficiencies of SMF and MMF fiber bundles are discussed in this section.

Comparing the SMF and MMF bundles shown in Section 3.5, the MMF fiber bundle transmitted more power than expected from the cross-section of the fibers. The discrepancy was attributed to the fiber's NA because NA affects coupling efficiency. The NA for the MMF was greater than the NA for the SMF (Table 1). The NA of the imaging optical setup also has an impact on collection efficiency. In general, a lens-based imaging setup focuses object light on the image. Therefore, when the focal lens NA on the input side of the fiber bundle is matched to the fiber’s NA, the fiber can efficiently accept the input light intensity. Finally, because of the large cladding area and some buffer space between each fiber, the power of the focused image cannot be collected efficiently. This problem can be solved by using thin cladding fiber or multi-core fiber.

5. Conclusion

We demonstrated one-shot 3D imaging using an optical frequency comb by 2D spectroscopy with SMF and MMF fiber bundles. In this experiment, the uncertainty of the proposed method was assessed using a mirror and a gauge block target, and it was shown to reach a sub-µm level. In addition, we captured the high aspect ratio target, which was made up of a mirror and a gauge block 3 m apart, without any scanning. The proposed method achieved a wide dynamic range of 6 digits by one-shot with profile image. Finally, in the experiment with the MMF fiber bundle, we confirmed the capability for high light collection efficiency and established the method to reduce the speckle noise caused by MMF with one-shot.

From these results, we demonstrated that the proposed 3D imaging method achieved high precision, a wide dynamic range, and one-shot imaging simultaneously. In principle, the imaging method's depth and spatial resolution were independent, so two measurement performances could be designed independently, providing a practical advantage of the method. The depth resolution was equal to the precision of the interference measurement and can ultimately reach the nanometer level. The diffraction limit of the imaging optics setup and the NA of the fiber was used to calculate the transverse spatial resolution. The spatial resolution of the experimental setup can reach 3.4 µm when the effect of the fiber NA is considered. This one-shot 3D imaging method based on the OFC has broad scientific and industrial applications, including bio-engineering.