• Photonics Research
  • Vol. 9, Issue 5, 657 (2021)
Seongjin Bak1、†, Gyeong Hun Kim1、†, Hansol Jang, and Chang-Seok Kim*
Author Affiliations
  • Department of Cogno-Mechatronics Engineering, Pusan National University, Busan 46241, Republic of Korea
  • show less
    DOI: 10.1364/PRJ.416025 Cite this Article Set citation alerts
    Seongjin Bak, Gyeong Hun Kim, Hansol Jang, Chang-Seok Kim. Optical Vernier sampling using a dual-comb-swept laser to solve distance aliasing[J]. Photonics Research, 2021, 9(5): 657 Copy Citation Text show less

    Abstract

    Optical interferometry using comb-swept lasers has the advantage of efficiently reducing the acquisition bandwidth for high-speed and long-range detection. However, in general, the use of a comb-swept laser involves a critical limitation in that the absolute distance cannot be measured, and, thus, multiple layers cannot be distinguished when measuring each position. This is because of the distance ambiguity induced by optical aliasing, in which there is periodic repetition of the frequency of an interferometric signal owing to discrete spectral sweeping, which does not occur in conventional optical interferometry that uses a continuous swept laser. In this paper, we introduce an optical Vernier sampling method using a dual-comb-swept laser to measure the absolute distances in a multi-layer target. For this, we designed a new type of dual-comb-swept laser to include two different free spectral ranges (FSRs) in separated wavelength bands to provide a stable lasing condition. Using a principle similar to that of a Vernier caliper for length measurement, the two different FSRs can be used to recover a higher frequency of an optical interferometric signal to measure longer distances from different layers in a target. Using the dual-comb-swept laser in optical interferometry, we solved the optical aliasing issue and measured the absolute distances of three layers separated over 83 mm using a point-scanning imaging setup and the simultaneous absolute distance of the top surfaces separated over 45 mm using a full-field imaging setup at 14 and 8 times lower acquisition bandwidth than a conventional continuous swept laser that is based on optical interferometry.

    1. INTRODUCTION

    Optical interferometry is a well-established method for measuring distances with high accuracy and long range and in multiple layers of targets [13]. Currently, optical coherence tomography (OCT) is employed in biomedical imaging and industrial inspection applications [48]. Recently, the high-speed A-scan at a deep depth using a swept-source OCT (SS-OCT) system has become possible owing to the development of a continuous swept laser with a high sweep rate and long coherence length [915]. By employing a state-of-the-art continuous swept laser, a deeper imaging depth and faster imaging speed can be implemented. However, this will cause a considerable amount of burden on electric detectors with a limited electrical bandwidth [9,11,16].

    Recently, a solution was suggested for deeper and faster imaging without using a faster detector, i.e., using a comb-swept laser instead of a conventional continuous swept laser [5,9,11,1719]: D=T·c·f2B.

    Equation (1) [3,20] gives the relationship among the measured distance (D), time period of the swept-source (T), speed of light (c), frequency of the interferometric signal (f), and spectral bandwidth of the swept-source (B). Most of the factors in Eq. (1), except f, are fixed values during the measurement of varying distances under given measurement conditions. This implies that D can be proportionally changed by varying f, and, accordingly, the measurement of a large value of D must correspond to a large value of f. A faster continuous swept laser with a shorter time period, T, will induce a larger value of f for the same distance, D. Therefore, the longer coherence length and faster sweep rate of a continuous swept laser require a much faster detector and wider acquisition bandwidth in a conventional SS-OCT system. Therefore, comb-swept lasers have recently been investigated for their use in the SS-OCT system to capitalize on the advantage of reducing the electrical bandwidth of detecting devices. This feature of comb-swept lasers provides a periodic repetition of f as the target distance increases continuously [9,1719]. The periodic repetition has only limited values below the frequency limit of the interferometric signal, flim. As a result of the limited value of the measured f, D also has only periodic repetition and a limited value. This periodic repetition of the D is called distance aliasing and is induced from the discrete distribution of the spectrum over the wavelength domain of the comb-swept laser output [21].

    Thus, solving this distance aliasing has been a challenging issue when conducting optical interferometry using comb-swept lasers. Recently, a dual-reference method was suggested for measuring the absolute distance in the first aliasing order, but the dual-reference method cannot measure the higher order of absolute distance after the first order [18]. Another method for resolving the absolute depth was suggested using a degenerated frequency comb that resolves an absolute depth of approximately 10 mm, but this degenerated frequency comb method is expected to be difficult to adapt in a dispersive medium or longer-range measurement because the method detects a chirped point spread function (PSF) as increasing the aliasing order [22]. In this paper, we introduce a dual-comb-swept laser that includes two different free spectral ranges (FSRs) for the incorporation of the optical Vernier sampling method [16]. When conducting length measurement using a Vernier caliper, two different scales of the ruler are used. Using these two scales, one can measure the length with accuracy much higher than that obtained using a normal one [23]. We applied this simple principle of the Vernier caliper to our optical interferometry system consisting of a dual-comb-swept laser. Similar to the length measurement using a Vernier caliper, our novel method incorporates the optical Vernier sampling method and uses two different FSRs from a dual-comb-swept laser, which corresponds to different scales of the two rulers in a Vernier caliper. In this proposed optical Vernier sampling method, we can recover a much higher frequency of the original distance by combining two lower frequencies of each aliased interferometric signal [16].

    2. PRINCIPLE AND EXPERIMENTAL SETUP

    A. Principle of the Optical Vernier Sampling Method

    Principle of the optical Vernier sampling method for solving distance aliasing. (a) A dual-comb-swept laser using comb A and comb B shows two different FSR values, namely, FSRa and FSRb, respectively. (b) Optical interferometer to measure the absolute distance of the target. BS, beam splitter; RM, reference mirror. (c) Actual interferometric signal using a continuous swept laser with a frequency of f0 (dashed line), and sampled interferometric signals using a proposed dual-comb-swept laser using comb A to indicate the aliasing frequency of fa (blue line) and using comb B to indicate the aliasing frequency of fb (red line), respectively. (d) Absolute distance recovery algorithm by measuring two aliasing frequencies of fa and fb together.

    Figure 1.Principle of the optical Vernier sampling method for solving distance aliasing. (a) A dual-comb-swept laser using comb A and comb B shows two different FSR values, namely, FSRa and FSRb, respectively. (b) Optical interferometer to measure the absolute distance of the target. BS, beam splitter; RM, reference mirror. (c) Actual interferometric signal using a continuous swept laser with a frequency of f0 (dashed line), and sampled interferometric signals using a proposed dual-comb-swept laser using comb A to indicate the aliasing frequency of fa (blue line) and using comb B to indicate the aliasing frequency of fb (red line), respectively. (d) Absolute distance recovery algorithm by measuring two aliasing frequencies of fa and fb together.

    In comparison, for the same d0, but for the light from the dual-comb-swept laser using combs A and B, the discrete spectral sweeping of the laser output generates an f with only limited values below certain levels fa,lim and fb,lim, respectively, which are determined by FSRa and FSRb and represented by the black dashed lines in Fig. 1(c). The blue and red solid curves plotted as a function of time are the measured aliased interferometric signals with combs A and B, respectively. The aliased interferometric signal caused different intervals of sampling points of each comb A and comb B. The sampling intervals τA and τB correspond to the FSRs of comb A and comb B, respectively. The FFT spectra of both interferometric signals correspond to the f using comb A of fa and using comb B of fb. As d0 varies continuously, the aliased frequencies of fa and fb periodically increase and decrease below fa,lim and fb,lim for the light from the dual-comb-swept laser using combs A and B, respectively.

    As shown in Fig. 1(d), the FFT peak values of fa and fb are processed using the absolute distance recovery algorithm, which includes the relationship between the actual frequency of the continuous swept laser and two aliased frequencies measured using the dual-comb-swept laser, as shown by the blue and red solid lines. When FSRa is smaller than FSRb for a dense comb A and sparse comb B, the frequency limit of the interferometric signal using comb A, fa,lim, is higher than the frequency limit of the interferometric signal using comb B, fb,lim, as indicated by the black dashed lines. When comb A is applied, the aliased frequency using comb A, fa, corresponds to the different possible positions of the actual frequency, which are represented by the half-filled blue circles in the figure. However, when comb B is applied, the aliased frequency using comb B, fb, also corresponds to different actual frequencies, which are represented by the half-filled red circles in the figure. Thus, using only a single measurement of the aliased frequency of fa or fb, it is impossible to find f0 and d0.

    However, when the measurement results of both fa and fb are used together, it is possible to determine fr and dr. Two different baselines of scales A and B can be used along with the two aliased frequencies. On these two different scales, a matching position between the two half-filled blue and red circles at a specific actual frequency position can be found. This matching position is the recovered frequency, which is represented by the completely filled blue–red circle. The recovered frequency, fr, has a one-to-one correspondence with the pair of fa and fb with different orders for each aliasing order oa and ob, such that fr=(fa;oa,fb;ob),where oa and ob represent the aliasing order using combs A and B, respectively. Thus, the recovered frequency, fr, can be used to reconstruct the recovered distance, dr using Eq. (1). This implies that the recovered distance, dr, can also be represented with a bijective functional form, including the pair da and db with a different order for each aliasing order, oa and ob: dr=(da;oa,db;ob),where da and db represent the aliased distance using combs A and B, respectively. Furthermore, each aliasing order of oa and ob represents data compression in the electrical bandwidth. A more detailed description [represented by Eq. (A1)] is provided in Appendix A.

    B. Experimental Setup

    (a) Setup of the dual-comb-swept laser. The black and red lines represent the fiber and free-space part, respectively. SOA, semiconductor optical amplifier; PC, polarization controller; COL, collimator; ISO, isolator; DM, dichroic mirror; OC, optical coupler; FFP-TF, fiber Fabry–Perot tunable filter; BOA, boosting optical amplifier; TEC, thermo-electric cooler; AFG, arbitrary function generator; TEC CTRL, TEC controller. (b) Optical spectrum of the peak hold mode obtained from the dual-comb-swept laser. Enlarged views of the spectra obtained using (c) comb A in the wavelength range of 775.5 to 776 nm and (d) comb B in the wavelength range of 805.5 to 806 nm.

    Figure 2.(a) Setup of the dual-comb-swept laser. The black and red lines represent the fiber and free-space part, respectively. SOA, semiconductor optical amplifier; PC, polarization controller; COL, collimator; ISO, isolator; DM, dichroic mirror; OC, optical coupler; FFP-TF, fiber Fabry–Perot tunable filter; BOA, boosting optical amplifier; TEC, thermo-electric cooler; AFG, arbitrary function generator; TEC CTRL, TEC controller. (b) Optical spectrum of the peak hold mode obtained from the dual-comb-swept laser. Enlarged views of the spectra obtained using (c) comb A in the wavelength range of 775.5 to 776 nm and (d) comb B in the wavelength range of 805.5 to 806 nm.

    Equation (4) shows the comb spacing relationship among an FSR of a comb source (Δf), the speed of light (c), the refractive index of the etalon (ne), and the thickness of the etalon (le) [11]. Because the designed thicknesses of etalons A and B are 4.3 and 3.8 mm, respectively, the designed FSRs of combs A and B sources are 47.5 pm (23.7 GHz) and 58.1 pm (26.8 GHz). The linewidths of etalons A and B are 2.13 pm and 2.41 pm, respectively, which correspond to 1  GHz. Therefore, as scanning the transmission window of FFP-TF, it is sufficient to generate a narrow comb-swept lasing output without the gain competition. Figure 2(b) shows the peak hold mode spectral data obtained from a dual-comb-swept laser, in which the spectra for both comb sources have been presented for bands A (a shorter wavelength band from 768.8 to 787.2 nm) and B (a longer wavelength band from 791.1 to 812.5 nm). Of note, the laser spectrum from 787.2 to 791.1 nm was not clearly generated because of the low lasing efficiency resulting from the limited sectioning between the free-space reflection and transmission in the DM. Figures 2(c) and 2(d) show the enlarged views of the comb A spectrum in the wavelength range from 775.5 to 776.0 nm and the comb B spectrum from 805.5 to 806.0 nm, respectively. The measured FSRs from the average of combs A and B sources are 47.2 and 58.1 pm, respectively, which are similar to the designed values. The high stability of the dual-comb-swept laser can be guaranteed because the lasing of both comb sources is independently generated in the spectrally separated wavelength bands of A and B, respectively: Δd=c2·Δf=ne·le.

    Equation (5) provides the numerical relation of the distance aliasing period (Δd) obtained from comb-swept laser [9,11]. Factors ne and le represent the refractive index and thickness of the etalon, respectively [19]. As illustrated in Figs. 1(a) and 1(c) above, compared to the sparse comb spacing (a larger value of Δf) of the comb B source, the dense comb spacing (a smaller value of Δf) of the comb A source can obtain more sampling points and higher frequency terms from the actual f, f0. This implies that the frequency limit of the interferometric signal, fa,lim, is higher than fb,lim, and, thus, the aliasing distance period, Δda,lim, is longer than Δdb,lim. Using Eq. (5), the theoretical aliasing distance periods of combs A and B were calculated to be 6.350 and 5.600 mm, respectively. These values were utilized in the numerical simulation of the optical Vernier sampling.

    3. RESULTS

    By combining the two aliased frequencies with a dual-comb-swept laser, our novel optical Vernier sampling was verified by conducting the following tests. (i) PSF was measured, and the coherence length of the dual-comb-swept laser was determined. (ii) The recovered distance by the aliased distance using combs A and B was compared with that obtained from the numerical simulation by employing Eq. (5), and the analysis of accuracy and stability was performed. (iii) A multi-layer target was measured using the point-scanning imaging setup, and the three-dimensional (3D) full-field imaging setup was verified to confirm the usefulness of the optical Vernier sampling method based on the dual-comb-swept laser.

    A. Measurement of the PSF of the Dual-Comb-Swept Laser

    PSF measurement was performed to demonstrate the optical aliasing characteristics and evaluate the improvement in the coherence length of the comb-swept laser compared to the conventional continuous swept laser.

    PSF measurements at every 0.4 mm interval using (a) comb A for the 1st and 12th orders and (c) comb B for the 1st and 11th orders. (b) and (d) show the collected first 0.4 mm positions of each forward aliased distance using combs A and B sources, respectively, for the 1st to 13th orders.

    Figure 3.PSF measurements at every 0.4 mm interval using (a) comb A for the 1st and 12th orders and (c) comb B for the 1st and 11th orders. (b) and (d) show the collected first 0.4 mm positions of each forward aliased distance using combs A and B sources, respectively, for the 1st to 13th orders.

    In a general SS-OCT system, the coherence length of the swept-source is defined to have a 6  dB roll-off-length axial position for the continuously decreasing peak values of the FFT intensity starting from the zero-length difference position of the FFT intensity distribution [10]. The coherence length can be newly defined for the comb-swept laser because the aliasing with unexpected back-bounce of the FFT peak intensity is induced by increasing the absolute distance due to the aliasing of the interference subsampling from discrete spectral sweeping. Thus, the coherence length of the comb-swept laser is newly defined by the 6  dB roll-off-length position for decreasing peak values of the FFT intensity at the same aliased distance at every aliasing order. Figures 3(b) and 3(d) show the collection of the first 0.4 mm positions of each forward aliased distance for the 1st to 13th orders of the PSF measurement of combs A and B sources, respectively. In Fig. 3(b), the approximate 6  dB roll-off position corresponds to the 12th order in the comb A source. Similarly, in Fig. 3(d), the approximate 6  dB roll-off point is on the 11th order in the comb B source. Thus, the coherence lengths of combs A and B can be measured to be 140 mm and 112  mm, respectively. The measured coherence lengths of comb A and comb B can be converted to the linewidth of comb A and comb B, which are 1.38  pm and 1.84  pm, respectively [24].

    The axial resolution of combs A and B can be represented from the full width at half-maximum (FWHM) of the PSF [24]. Based on the PSF measurement at the absolute distance of 0.4 mm, the measured FWHMs of combs A and B [represented by the arrows in Figs. 3(b) and 3(d)] were 0.03408  mm and 0.03143  mm, respectively. So, we estimated the overall resolution of recovered distance will be 0.03408  mm, which is the lower resolution value from combs A and B. Figures 3(a) and 3(b) show the detailed PSF measurement performed at every 0.4 mm interval of absolute distance. The aliasing orders of comb A are the 1st and 12th, and the aliasing orders of comb B are the 1st and 11th. This distance aliasing phenomenon, which depends on the absolute distance, is the main feature of comb-swept lasers employed in optical interferometry. The advantages of comb-swept lasers include an enhanced coherence length of over 100 mm owing to the narrower linewidth achieved when using an etalon and the reduced electrical bandwidth owing to the lower optical frequency for detection [1719]. However, their main disadvantage has been that an absolute distance cannot be measured, and multiple layers cannot be distinguished when measuring each position because of the distance aliasing. Thus, on the basis of the results obtained in this study, we can conclude that distance aliasing can be solved by the optical Vernier sampling method according to theoretical and experimental investigations.

    B. Recovering the Absolute Distance Using the Optical Vernier Sampling Method Compared with Numerical Simulations

    Comparison of the recovered distance measurement with a numerical simulation.

    Figure 4.Comparison of the recovered distance measurement with a numerical simulation.

    As shown in Eq. (3), the recovered distance, dr, is represented by a bijective functional form that includes the pair da and db with different aliasing orders, namely, oa and ob. However, in the practical recovery algorithm that incorporates the optical Vernier sampling method, only the aliased distances, da and db, were sufficient for recovering dr. Because almost all of the ordered pairs of (da, db) indicate a single dr, within the coherence length of a given dual-comb-swept laser source, we can ignore few exceptional cases of blind spots where the same ordered pair (da, db) indicates multiple recovered distances, dr. This is described in detail in Appendix B. Furthermore, the information regarding oa and ob can be consequently obtained from the confirmed value of the recovered distance, dr.

    According to the experimental results shown in Fig. 3, the coherence length of the dual-comb-swept laser is more than 100 mm, but the maximum moving range of the translation stage in our laboratory is less than 11 mm. Therefore, the range of the absolute distance measurement was separated into a 0–11 mm region and a 98–109 mm region, and two different translation stages with a separation distance of 100 mm were used. The aliased distances using combs A and B sources are shown by the red filled circles in Fig. 4. Clearly, the ordered pair of the two aliased distances (da, db) is sufficient to recover dr by one-to-one correspondence for a long range of over 100 mm.

    We also compared the recovered distance measured in the discrete interval of the absolute distance with the corresponding values obtained from our numerical simulation. The simulation was performed by combining the theoretical aliasing periods, and the results are represented by the blue solid line in Fig. 4. According to Fig. 4, the recovered distances and numerical simulation according to absolute distance have a similar tendency, and, thus, the proposed method for solving distance aliasing by employing the optical Vernier sampling method works properly over a distance of 100 mm to the target.

    Table 1 shows the means and standard deviations (SDs) of distance measurement at four different absolute distances with each 100 data sets. The measurement values (mean±SD) of dr were 1.015  mm±0.39  μm, 4.009mm±0.67  μm, 7.035  mm±0.34  μm, and 9.997  mm±0.71  μm, which confirmed that the optical Vernier sampling method has reliable accuracy and stability.

    Result of Distance Measurement at Four Different Absolute Distances for 100 Times of Repetition (Unit: mm)

     Absolute Distance (d0)
    14710
    MeanSDMeanSDMeanSDMeanSD
    da1.0120.000642.3430.001090.6790.000632.6750.00144
    db1.0170.000301.5880.000521.4410.000451.1640.00019
    dr1.0150.000394.0090.000677.0350.000349.9970.00071

    C. Multi-Layer Target Measurement Using a Point-Scanning Imaging Setup

    Multi-layer target measurement using a point-scanning setup. (a) Schematic of the point-scanning setup. OC, optical coupler; CIR, circulator; DL, delay line; BD, balanced detector. (b) Image of a multi-layer target. (c) Result of the multi-layer target measurement with refractive index compensation. (d) Cross-sectional view of the result at 15.68 mm along the Y axis (32nd pixel of the Y axis).

    Figure 5.Multi-layer target measurement using a point-scanning setup. (a) Schematic of the point-scanning setup. OC, optical coupler; CIR, circulator; DL, delay line; BD, balanced detector. (b) Image of a multi-layer target. (c) Result of the multi-layer target measurement with refractive index compensation. (d) Cross-sectional view of the result at 15.68 mm along the Y axis (32nd pixel of the Y axis).

    Figure 5(d) shows a cross-sectional view of the result at a transverse position of 15.68 mm on the Y axis (32nd pixel of the Y axis) and shows the difference between layers. The average physical distances of layers 3, 2, and 1 were 57.352, 47.441, and 5.950 mm, respectively. The difference in the physical distance between layers 3 and 2 was 9.911 mm, and the physical distance difference was 10 mm. The corresponding physical distance difference between layers 2 and 1 was 41.491 mm, and the physical distance difference was 41.5 mm. The results shown in Fig. 5 indicate that our solution for distance aliasing using the optical Vernier sampling method was successful for the absolute distance measurement of a multi-layer target with a long-distance difference of over 83 mm (corresponding to 166 mm of optical-path-length difference).

    D. 3D Target Measurement Using a Full-Field Imaging Setup

    3D target measurement using a full-field imaging setup. (a) Schematic of the setup. L, lens; BS, beam splitter; RM, reference mirror. (b) Configuration of the 3D targets. Results showing the aliased distance using (c) comb A and (d) comb B sources. (e) Recovered distance data using combs A and B sources based on the optical Vernier sampling method.

    Figure 6.3D target measurement using a full-field imaging setup. (a) Schematic of the setup. L, lens; BS, beam splitter; RM, reference mirror. (b) Configuration of the 3D targets. Results showing the aliased distance using (c) comb A and (d) comb B sources. (e) Recovered distance data using combs A and B sources based on the optical Vernier sampling method.

    Figure 6(a) shows a schematic of the full-field imaging setup. Light from the dual-comb-swept laser passed through lens 1 (L1) and was collimated as an area beam. This beam entered a BS and was split into two paths. One path was toward the 3D target, whereas the other was toward the RM. The light reflected from the RM, and the backscattered light from the 3D target produced an optical interference signal at the BS, which further passed through L2 and L3 and was detected by a specific pixel of the CMOS camera. We used 500  ×500   pixels of the CMOS camera.

    An image showing the side view (left) and configuration (right) of the 3D target is presented in Fig. 6(b). The 3D target was assembled with multiple 2-mm-high magnetic blocks. Figures 6(c) and 6(d) show the results of the aliased distance using only comb A or B. The black-and-white scale bars in the figures indicate the aliased distance. Because of distance aliasing, each aliased distance using combs A and B could not measure the recovered distance of the 3D target. The maximum difference in the distance, corresponding to the results shown in Figs. 6(c) and 6(d), was limited to values below the PMR or Δd/2, which were 3.175 and 2.800 mm, respectively. Thus, we could use these two pieces of data from the optical Vernier sampling method to reconstruct the recovered distance and real shape of the target. Figure 6(e) shows the solving of distance aliasing that was successfully recovered by combining the results of combs A and B. The color scale bar indicates a range of 31–47 mm because the 3D target was intentionally located with an offset distance of 31  mm to demonstrate the merit of a long coherence length.

    According to Eq. (A1), each surface of recovered distance can be represented as follows: Surface:  45.294  mm=(0.844  mm;8even,0.494  mm;9even),Surface:  43.271  mm=(1.179  mm;7odd,1.529  mm;8odd),Surface:  41.290  mm=(3.160  mm;7odd,2.090  mm;8even),Surface:  39.334  mm=(1.234  mm;7even,0.134  mm;8even),Surface:  37.260  mm=(0.840  mm;6odd,1.940  mm;7odd),Surface:  35.259  mm=(2.841  mm;6odd,1.659  mm;7even),Surface:  33.370  mm=(1.620  mm;6even,0.230  mm;6odd).

    Recovered distance and SD obtained in the 3D target measurement using the full-field setup.

    Figure 7.Recovered distance and SD obtained in the 3D target measurement using the full-field setup.

    4. CONCLUSION

    In this work, we demonstrated the dual-comb-swept laser for solving distance aliasing using the novel optical Vernier sampling method for the first time, to the best of our knowledge. Similar to the principle of using a Vernier caliper for measuring the length of an object, the optical Vernier sampling method also uses two different frequencies of interferometric signals from a dual-comb-swept laser, which correspond to the different intervals of the two rulers in a Vernier caliper.

    According to two different designed FSRs of combs A and B of 47.5 pm (23.7 GHz) and 58.1 pm (26.8 GHz), the corresponding PMR values of 3.175 and 2.800 mm, respectively, were determined. The combination of the two aliased distances below these PMR limits can be used to obtain the recovered distance by solving the distance aliasing using the optical Vernier sampling method. We also measured the recovered distances of three layers separated over 83 mm using the point-scanning imaging setup. The simultaneous absolute distance of the top surfaces separated over 45 mm is also measured using the full-field imaging setup. The key advantage of optical Vernier sampling is data compression in the long range. In the multi-layer target, a recovered distance of 83 mm has data compression rates of 14 times using comb A and 15 times using comb B. In the 3D target, a recovered distance of 45 mm has data compression rates of 8 times using comb A and 9 times using comb B.

    On the basis of these results, we expect to expand the application of solving distance aliasing by employing the novel optical Vernier sampling method in systems, such as the highly efficient SS-OCT imaging system, that exhibit an enhanced coherence length for deeper imaging and reduced subsampling frequency for a decreased electrical bandwidth.

    APPENDIX A: DETAILED DESCRIPTION OF THE DISTANCE EQUATION

    From Eq.?(3), the recovered distance, dr, in the optical Vernier sampling method can be represented in detail as a bijective function as follows: dr=(da;oa(even or odd),db;ob(even or odd)),where dr, da, oa(even or odd), db, and ob(even or odd) represent the recovered distance, aliased distance using comb A, aliasing order using comb A, aliased distance using comb B, and aliasing order using comb B. Thus, it is possible to show that the recovered distance consists of the abovementioned four measured factors for combs A and B together using a single correspondence expression, as given by Eq.?(A1). Figure?8 shows the schematic depiction of the use of the recovery algorithm for obtaining the recovered distance, dr. The main difference between Figs.?1 and 8 is the axis conversion from the frequency scale to the distance scale using Eq.?(1). Each aliasing order of comb A and comb B is divided into even and odd orders, respectively. Each even order has a forward correlation, whereas each odd order has a backward correlation between the actual and aliased distances for combs A and B. To numerically display a bijective function of dr, we used a well-known definition of the greatest integer function (floor function), as expressed by ?x?=max{nZ:nx},where ?x? of a real number x denotes the greatest integer less than or equal to x. For example, ?0.1?=0, ?9.966?=9, ?3.5?=3, and ??π?=?4.

    Schematic showing the use of the recovery algorithm to obtain the recovered distance.

    Figure 8.Schematic showing the use of the recovery algorithm to obtain the recovered distance.

    Each element in Eq.?(A1) can be defined using Eqs.?(A3) and (A4) where PMRa and PMRb represent PMR when using combs A and B, respectively: da=PMRa×(drPMRa??drPMRa?),oa(even)=?dr2×PMRa+1?,}when???drPMRa?=even number,da=PMRa?PMRa×(drPMRa??drPMRa?),oa(odd)=?dr2×PMRa+1?,}when???drPMRa?=odd number,db=PMRb×(drPMRb??drPMRb?),ob(even)=?dr2×PMRb+1?,}when???drPMRb?=even number,db=PMRb?PMRb×(drPMRb??drPMRb?),ob(odd)=?dr2×PMRb+1?,}when???drPMRb?=odd number.

    Using these two sets of equations, all of the recovered distances can be bijectively correlated to the factors in Eq.?(A1). In the practical recovery algorithm that incorporates the optical Vernier sampling method, only the aliased distances using combs A and B are sufficient for recovering the absolute distance. The order can be consequently obtained via the confirmed value of the recovered distance.

    Appendix B: Solutions for a Blind Spot

    Most of the recovered distances, dr, were matched by a one-to-one correspondence with the ordered pair of two aliased distances, (da, db), using combs A and B. However, a few exceptional cases still exist because of the folded backward correlation. For example, for an absolute distance movement below 10?mm, six positions of blind spots exist when multiple recovered distances, dr, correspond to the same combination pair of (da, db). In our practical measurement system, the blind spots can be easily solved using a specific algorithm for the reference movement, as shown in Fig.?9.

    Flowchart for the reference movement to solve the blind spot problem.

    Figure 9.Flowchart for the reference movement to solve the blind spot problem.

    Results of the numerical simulation employing the I/Q demodulation and frequency shifter.

    Figure 10.Results of the numerical simulation employing the I/Q demodulation and frequency shifter.

    References

    [1] D. Huang, F. Li, C. Shang, Z. Cheng, S. T. Chu, P. K. A. Wai. Frequency comb swept laser with a high-Q microring filter. Photon. Res., 8, 904-911(2020).

    [2] J. Jerwick, Y. Huang, Z. Dong, A. Slaudades, A. J. Brucker, C. Zhou. Wide-field ophthalmic space-division multiplexing optical coherence tomography. Photon. Res., 8, 539-547(2020).

    [3] X. Zhang, J. Pouls, M. C. Wu. Laser frequency sweep linearization by iterative learning pre-distortion for FMCW LiDAR. Opt. Express, 27, 9965-9974(2019).

    [4] H. D. Lee, M. Y. Jeong, C. S. Kim, J. G. Shin, B. H. Lee, T. J. Eom. Linearly wavenumber-swept active mode locking short-cavity fiber laser for in-vivo OCT imaging. IEEE J. Sel. Top. Quantum Electron., 20, 1101008(2014).

    [5] M. Siddiqui, B. J. Vakoc. Optical-domain subsampling for data efficient depth ranging in Fourier-domain optical coherence tomography. Opt. Express, 20, 17938-17951(2012).

    [6] J. Cao, P. Wang, Y. Zhang, G. Shi, B. Wu, S. Zhang, Y. Liu. Methods to improve the performance of the swept source at 10 μm based on a polygon scanner. Photon. Res., 5, 245-250(2017).

    [7] F. Hou, M. Zhang, Y. Zheng, L. Ding, X. Tang, Y. Liang. Detection of laser-induced bulk damage in optical crystals by swept-source optical coherence tomography. Opt. Express, 27, 3698-3709(2019).

    [8] R. Su, M. Kirillin, E. W. Chang, E. Sergeeva, S. H. Yun, L. Mattsson. Perspectives of mid-infrared optical coherence tomography for inspection and micrometrology of industrial ceramics. Opt. Express, 22, 15804-15819(2014).

    [9] N. Lippok, M. Siddiqui, B. J. Vakoc, B. E. Bouma. Extended coherence length and depth ranging using a Fourier-domain mode-locked frequency comb and circular interferometric ranging. Phys. Rev. Appl., 11, 014018(2019).

    [10] H. D. Lee, G. H. Kim, J. G. Shin, B. Lee, C. S. Kim, T. J. Eom. Akinetic swept-source optical coherence tomography based on a pulse-modulated active mode locking fiber laser for human retinal imaging. Sci. Rep., 8, 17660(2018).

    [11] M. Siddiqui, A. S. Nam, S. Tozburun, N. Lippok, C. Blatter, B. J. Vakoc. High-speed optical coherence tomography by circular interferometric ranging. Nat. Photonics, 12, 111-116(2018).

    [12] T. S. Kim, J. Y. Joo, I. Shin, P. Shin, W. J. Kang, B. J. Vakoc, W. Y. Oh. 9.4 MHz A-line rate optical coherence tomography at 1300 nm using a wavelength-swept laser based on stretched-pulse active mode-locking. Sci. Rep., 10, 9328(2020).

    [13] Z. Wang, B. Potsaid, L. Chen, C. Doerr, H.-C. Lee, T. Nielson, V. Jayaraman, A. E. Cable, E. Swanson, J. G. Fujimoto. Cubic meter volume optical coherence tomography. Optica, 3, 1496-1503(2016).

    [14] D. Huang, F. Li, C. Shang, Z. Cheng, P. K. A. Wai. Reconfigurable time-stretched swept laser source with up to 100 MHz sweep rate, 100 nm bandwidth, and 100 mm OCT imaging range. Photon. Res., 8, 1360-1367(2020).

    [15] J. P. Kolb, W. Draxinger, J. Klee, T. Pfeiffer, M. Eibl, T. Klein, W. Wieser, R. Huber. Live video rate volumetric OCT imaging of the retina with multi-MHz A-scan rates. PLoS ONE, 14, e0213144(2019).

    [16] S. Bak, G. H. Kim, C. S. Kim. Dual-band wavelength-comb-swept laser to extend displacement measurement range. European Conference on Biomedical Optics, 11078_67(2019).

    [17] E. J. Jung, J.-S. Park, M. Y. Jeong, C.-S. Kim, T. J. Eom, B.-A. Yu, S. Gee, J. Lee, M. K. Kim. Spectrally-sampled OCT for sensitivity improvement from limited optical power. Opt. Express, 16, 17457-17467(2008).

    [18] S.-K. Chun, H. Jang, S.-W. Cho, N. S. Park, C.-S. Kim. Unfolding displacement measurement method for the aliasing interferometer signal of a wavelength-comb-swept laser. Opt. Express, 26, 5789-5799(2018).

    [19] N. S. Park, S. K. Chun, G. H. Han, C. S. Kim. Acousto-optic-based wavelength-comb-swept laser for extended displacement measurements. Sensors, 17, 140(2017).

    [20] G. Berkovic, E. Shafir. Optical methods for distance and displacement measurements. Adv. Opt. Photon., 4, 441-471(2012).

    [21] T.-H. Tsai, C. Zhou, D. C. Adler, J. G. Fujimoto. Frequency comb swept lasers. Opt. Express, 17, 21257-21270(2009).

    [22] N. Lippok, B. J. Vakoc. Resolving absolute depth in circular-ranging optical coherence tomography by using a degenerate frequency comb. Opt. Lett., 45, 371-374(2020).

    [23] A. D. Gomes, M. S. Ferreira, J. Bierlich, J. Kobelke, M. Rothhardt, H. Bartelt, O. Frazão. Optical harmonic Vernier effect: a new tool for high performance interferometric fibre sensors. Sensors, 19, 5431(2019).

    [24] W. Drexler, J. G. Fujimoto. Optical Coherence Tomography: Technology and Applications(2015).

    [25] S. H. Yun, G. J. Tearney, J. F. de Boer, B. E. Bouma. Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting. Opt. Express, 12, 4822-4828(2004).

    [26] M. Siddiqui, S. Tozburun, E. Z. Zhang, B. J. Vakoc. Compensation of spectral and RF errors in swept-source OCT for high extinction complex demodulation. Opt. Express, 23, 5508-5520(2015).

    Seongjin Bak, Gyeong Hun Kim, Hansol Jang, Chang-Seok Kim. Optical Vernier sampling using a dual-comb-swept laser to solve distance aliasing[J]. Photonics Research, 2021, 9(5): 657
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