Phase imaging without a phase wrapping ambiguity is required for wide-axial-range 3D imaging in the fields of surface topography measurement and biomedical imaging. Although multicascade-linked synthetic-wavelength digital holography (MCL-SW-DH) using an optical frequency synthesizer (OFS) is a promising method to meet this requirement, the slow switching of multiple optical wavelengths in the OFS prevents rapid imaging. In the work described in this article, a line-by-line spectral-shaped electro-optics-modulator-based optical frequency comb (EOM-OFC) is used as a light source in MCL-SW-DH to achieve rapid image acquisition. While MCL-SW-DH enables surface topography measurement with millimeter-order axial range and micrometer-order axial resolution, the line-by-line spectral-shaped EOM-OFC extracts a single narrow-linewidth OFC mode from the 10 GHz-spacing EOM-OFC at a center wavelength of 1545 nm within a spectral range of 30 nm at an interval of 500 ms. The effectiveness of the proposed MCL-SW-DH was highlighted by performing surface topography measurement with four step differences of sub-millimeter to millimeter size with an axial uncertainty of 2.08 µm in the image acquisition time of several seconds. The proposed MCL-SW-DH will be a powerful tool for 3D imaging with a wide axial range and high axial resolution.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Digital holography (DH) [1–4] is a promising method for reconstructing both amplitude and phase images of a sample from interference patterns, that is, a hologram, between an object beam and a reference beam using an interferometric setup. The advantages of DH include phase-image-based three-dimensional (3D) imaging, digital focusing, real-time measurement, and quantitative analysis. Therefore, DH has attracted attention as an imaging modality in the field of biomedical imaging [5,6] and surface topography measurement [7–9]. However, when single-wavelength continuous-wave (CW) light is used for reflection-type DH, the maximum axial range (MAR) is limited within the range of half the wavelength (λ/2) due to phase wrapping ambiguity. To avoid this problem, a phase unwrapping process can be used ; however, its use is limited to smooth surfaces with wavelength-order unevenness. Another method to extend the MAR greater than a phase wrapping period of λ/2 is to use synthetic-wavelength DH (SW-DH). For example, DH is performed at two different wavelengths (λ1, λ2), and then a synthetic wavelength Λ between them (= λ1λ2/|λ2-λ1|) is used to extend the MAR in the phase image [11–15]. Unfortunately, as the wavelength selectivity of stable CW lasers is limited to several discrete wavelengths (for example, 532 nm, 612 nm, 633 nm, or 780 nm), the MAR is still limited to within a few micrometers. Even in SW-DH with three different wavelengths, the MAR remains around a few tens of micrometers .
To further extend the MAR, multicascade-linked synthetic-wavelength DH (MCL-SW-DH)  was demonstrated by using an optical-comb-referenced frequency synthesizer (OFS) [18–20]. Since an optical frequency comb (OFC) can be used as a precise optical frequency ruler traceable to a microwave or RF frequency standard, one can discretely tune the optical frequency of the OFS in steps of the repetition frequency frep of the OFC over a broad spectral range by phase-locking a wavelength-tunable single-mode CW laser to one optical frequency mode of the OFC. Using an OFS enables us to perform multiple SW-DHs with different orders of synthetic wavelengths and to cascade-link them to each other. In other words, MCL-MS-DH can extend the MAR up to a few millimeters while maintaining an axial resolution of a few tens of nanometers. However, the need for sequential phase-locking operations at multiple optical wavelengths in an OFS (which typically takes a few minutes for each wavelength) prevents rapid imaging.
If a single optical frequency mode can be directly and rapidly extracted from a mode-resolved OFC spectrum, the phase-locking operation of the CW laser to the OFC mode is not necessary any more. A line-by-line spectral shaping of OFC [21–23] enables us to extract a single OFC mode and hence will be further used for a simplified OFS in MCL-SW-DH. In this article, MCL-SW-DH is proposed based on the line-by-line spectral shaping of an electro-optic-modulator-based OFC (EOM-OFC). An arbitrary mode was selected from a 10 GHz-spacing EOM-OFC [24,25] by a combination of a two-dimensional (2D) spatial disperser [23,26–28] with a reflection-type spatial light modulator (SLM). Then, this line-by-line spectral-shaped OFC is applied to rapid MCL-SW-DH. A series of phase images was successfully cascade-linked with different synthetic wavelengths covering an MAR of a few tens of millimeters for a sample object.
2. Principle of operation
2.1 Line-by-line spectral shaping of OFC
As the details of a line-by-line spectral-shaping OFC are given elsewhere [21–23], the brief principle of operation is given here. Figure 1 shows a schematic drawing of the line-by-line spectral shaping OFC. The optical frequency νm of each mode in the OFC is given by
2.2 Multicascade linking of synthetic wavelengths and optical wavelength
Figure 2 shows the principle of operation of MCL-SW-DH . The line-by-line spectral-shaped OFC enables us to discretely select narrow-linewidth, single-mode light within the spectral range of the OFC. This makes it possible to generate multiple synthetic wavelengths with different orders of magnitude (= Λ1 < Λ2 < •••• < Λn-1 < Λn) as multiple cascades. In Fig. 2, the green arrows indicate different synthetic wavelengths; the right and left ends of the arrows correspond to one half of Λi and the phase resolution (typically, Λi/100), respectively. Such a series of multiple synthetic wavelengths can be used for MCL-SW-DH together with an optical wavelength λ (see the blue arrow in Fig. 2). In phase-image-based shape measurement in a reflection configuration, the spatial distribution of height, H(x, y), in a sample is given by
By repeating a similar procedure from the longest to the shortest synthetic wavelength to the optical wavelength, Nλ(x, y) can be determined correctly. By using such a multicascade link from Λn to λ, both the maximum axial range of Λn/2 and the minimum axial resolution of λ/100 can be achieved at the same time. The resulting axial dynamic range becomes several orders of magnitude larger than the previous single SW-DH.
3. Experimental setup
For the line-by-line spectral shaping, it is essential to spatially develop OFC modes as a discrete 2D spectrograph on the SLM. Since the spectral resolving power of a usual 2D spatial disperser is on the order of GHz, one has to use an OFC with GHz frequency spacing. To this end, an EOM-OFC [24,25] was used, as shown in Fig. 3(a). A 10 GHz-spacing EOM-OFC generator was composed of a 1550 nm CW external-cavity laser diode (ECLD; Redfern Integrated Optics, Inc., PLANEX, FWHM < 2.0 kHz) and a dual-drive Mach Zehnder modulator (MZM; Sumitomo-Osaka Cement Co., Ltd., electrooptic response of 20 GHz). After the first amplification using a semiconductor optical amplifier (SOA), the chirp compensation using a standard polarization maintaining fiber (PMF; length = 1000 m), and the second amplification by a two-stage erbium-doped fiber amplifier system (EDFA1 and EDFA2) with a maximum output power of 30 dBm, an optical spectrum of the 10-GHz-spacing EOM-OFC was broadened to be covered over a wavelength range of a few tens of nanometers by a highly nonlinear dispersion-shifted fiber (HNL-DSF; zero dispersion wavelength of 1559.37 nm, length = 118.82 m).
Figure 3(b) shows the experimental setup of the line-by-line spectral shaping of the EOM-OFC and the following MCL-SW-DH system. The output beam from the EOM-OFC was fed into a 2D spatial disperser after being formed into horizontal linear polarization by a polarization controller (PC) and subjected to beam expansion (beam diameter = 12 mm) by a beam expander (BE). The 2D spatial disperser was composed of a cylindrical lens (CL), a virtually imaged phased array (VIPA; Light Machinery, Inc., OP-6721-1686-8, free spectral range FSRVIPA = 60 GHz, finesse = 100), relay lenses (RLs), a diffraction grating (Spectrogon AB, GR50-1210, groove density = 1200 grooves/mm), and a lens (L). The 2D spatial disperser developed the EOM-OFC modes in 2D space, forming the 2D spectrograph of the EOM-OFC modes on a reflection-type liquid-crystal-on-silicon SLM (SLM; Santec Corp., SLM100-04, number of pixels = 1440 × 1050, response time = 100 ms). The 2D spectrograph of the EOM-OFC was spatially modulated as an on-and-off binary pattern by a combination of the SLM and a polarization beam splitter (PBS). A Peltier-cooled infrared CCD camera (CCD1; Allied Vision Technol. GmbH, Goldeye P-008, 320 × 256 pixels, exposure time = 10 ms, digital output resolution = 14 bit) was used to monitor the 2D spectrograph of the EOM-OFC on SLM with another lens. After propagating through the same optics in the reverse direction, the line-by-line spectral-shaped EOM-OFC beam was fed into an off-axis Michelson interferometer for MCL-SW-DH by a beam splitter cube (BS1, reflection = 50%, transmittance = 50%). In the interferometer, the object beam was reflected by another beam splitter cube (BS2, reflection = 50%, transmittance = 50%), reflected at a sample, and then passed through BS2. The reference beam passed through BS2, was reflected at a mirror, and was then reflected by BS2. Both beams were incident on another Peltier-cooled infrared CCD camera (CCD2; Allied Vision Technol. GmbH, Goldeye P-008, 320 × 256 pixels, exposure time = 10 ms, digital output resolution = 14 bit) at an off-axis angle of 0.07° to generate interference patterns with a fringe spacing of 70 µm, namely, the digital hologram. Finally, an angular spectrum method (ASM) [29,30] was used for the wavefront propagation calculation. The details of the wavefront propagation calculation are given in a previous paper .
4.1 Basic performance of line-by-line spectral-shaped EOM-OFC
First, the basic performance of the EOM-OFC was evaluated. Figures 4(a) and 4(b) show linear and logarithmic optical spectra of the EOM-OFC (total power = 500 mW) measured by an optical spectrum analyzer (Yokogawa Test & Measurement Corp., Tokyo, Japan, AQ-6315A, wavelength range = 350–1750 nm, wavelength resolution = 0.05–10 nm). The optical spectrum had a center wavelength of 1554 nm and covered a wavelength range of 1540 nm to 1570 nm, corresponding to a center optical frequency of 192.9 THz and an optical frequency range of 191.0 THz to 194.7 THz. Multiple spikes of the spectrum reflect a series of discrete, equally spaced frequency lines of EOM-OFC. As the frequency spacing of the EOM-OFC was set to 10 GHz, the optical spectrum included 370 comb modes within the spectral range. This EOM-OFC generated multiple synthetic wavelengths within the range of 86 µm to 30 mm.
Next, the basic performance of the line-by-line spectral-shaped EOM-OFC was evaluated. Figures 5(a) and 5(b) respectively show a whole image and an expanded image of a 2D spectrograph of the EOM-OFC on the SLM acquired by CCD1, indicating comb modes as 2D discrete points. Due to different spectral resolving powers in the VIPA and grating, the resulting 2D spectrograph was spatially developed as a zigzag line in the SLM plane [see blue dashed line in Fig. 5(b)]. The horizontal and vertical dimensions of each spot were limited by the spectral resolving power of the VIPA ( = 600 MHz, corresponding to 0.0048 nm in wavelength). On the other hand, their spacing was equal to FSRVIPA ( = 60 GHz, corresponding to 0.48 nm in wavelength) for horizontally adjacent spots and frep ( = 10 GHz, corresponding to 0.08 nm in wavelength) for vertically adjacent spots. By using another tunable ECLD (OptoComb, Inc., Tokyo, Japan, LT-5001N, tuning range = 1520–1595 nm, mean power = 30 mW) and an optical wavelength meter (Yokogawa Test & Measurement Corp., Tokyo, Japan, AQ6151, wavelength range = 1270–1650 nm, wavelength accuracy = ±0.3 ppm), the wavelengths of the comb modes were calibrated as shown in Fig. 5(c).
Five comb modes with different wavelengths ( = 1540.732 nm, 1540.810 nm, 1542.635 nm, 1557.218 nm, and 1568.625 nm) were then extracted at intervals of 500 ms. The corresponding optical spectra are shown in Figs. 6(a), 6(b), 6(c), 6(d), and 6(e), indicating narrow-linewidth, single-mode light with well-rejected neighboring modes. These five comb modes were used to generate four synthetic wavelengths with different orders of magnitude (Λ1 = 86.646 µm, Λ2 = 145.53 µm, Λ3 = 1,249.0 µm, and Λ4 = 30,436 µm) together with a single optical wavelength (λ = 1540.732 nm) for the following MCL-SW-DH.
4.2 Phase noise of MCL-SW-DH
To evaluate the phase noise of MCL-SW-DH, a plane gold mirror (front surface flatness = 63.3 nm) was used as a sample. Phase images at five different optical wavelengths ( = 1540.732 nm, 1540.810 nm, 1542.635 nm, 1557.218 nm, and 1568.625 nm, see Fig. 6) were respectively obtained from their corresponding holograms by the ASM-based wavefront propagation calculation. Then, we calculated the phase image at four different synthetic wavelengths (Λ1 = 86.646 µm, Λ2 = 145.53 µm, Λ3 = 1,249.0 µm, and Λ4 = 30,436 µm) from those five optical wavelengths. Figure 7 shows phase images of the sample at Λ4, Λ3, Λ2, Λ1, and λ ( = 1540.732 nm): (a) ϕΛ4(x, y), (b) ϕΛ3(x, y), (c) ϕΛ2(x, y), (d) ϕΛ1(x, y), and (e) ϕλ(x, y), respectively (pixel size = 191 by 140 pixels). The standard deviation of ϕΛn(x, y) or ϕλ(x, y) is corresponding to spatial phase noise, which limited the axial resolution of the surface unevenness measurement. The resulting standard deviation of ϕi(x, y) was calculated to be 0.009 rad in ϕΛ4(x, y), 0.010 rad in ϕΛ3(x, y), 0.067 rad in ϕΛ2(x, y), 0.087 rad in ϕΛ1(x, y), and 0.233 rad in ϕλ(x, y), as shown in Fig. 7(f). For comparison, the blue line in Fig. 7(f) shows a spatial phase distribution expected from the mirror flatness. By comparing them, the spatial phase noise in ϕΛ4(x, y), ϕΛ3(x, y), and ϕλ(x, y) is in reasonable agreement with the mirror flatness.
The temporal phase noise depends on the robustness of the optical system to environmental disturbances, such as air turbulence or mechanical vibration, and hence is a critical factor in determining the uncertainty in the height determined by the phase image. Therefore, the temporal phase noise was evaluated too. We obtained 50 phase images at Λ4, Λ3, Λ2, Λ1, and λ, respectively, and then calculated a standard deviation of 50 phase values at the same pixel as temporal phase noise. Figure 8 shows the temporal phase noise in (a) ϕΛ4(x, y), (b) ϕΛ3(x, y), (c) ϕΛ2(x, y), (d) ϕΛ1(x, y), and (e) ϕλ(x, y), respectively (pixel size = 191 by 140 pixels). The spatial means and standard deviations of the temporal phase noise were 0.044 ± 0.001 rad in ϕΛ4(x, y), 0.058 ± 0.012 rad in ϕΛ3(x, y), 0.081 ± 0.025 rad in ϕΛ2(x, y), 0.082 ± 0.020 rad in ϕΛ1(x, y), and 0.072 ± 0.006 rad in ϕλ(x, y), respectively, as shown in Fig. 8(f).
4.3 3D imaging of a stepped object
Finally, shape measurement of a sample having a stepped surface to highlight the wide dynamic range of the axial dimension was demonstrated. The sample was composed of four steel gauge blocks with different thicknesses (Grade K, Mitutoyo, Kawasaki, Japan; thickness = 11.0000 ± 0.0003 mm, 2.0000 ± 0.0002 mm, 1.1100 ± 0.0002 mm, and 1.0000 ± 0.0002 mm). Figure 9 shows (a) an optical photograph and (b) a schematic drawing of the sample. Step differences of 10.0000 ± 0.0004 mm, 9.0000 ± 0.0004 mm, 0.8900 ± 0.0003 mm, and 0.1100 ± 0.0003 mm were used as a sample. MCL-SW-DH with four synthetic wavelengths (Λ1, Λ2, Λ3, and Λ4) and one optical wavelength (λ) was performed. We acquired 100 images of hologram at five different optical wavelengths (total acquisition time = 0.01 × 100 × 5 = 5 s) and accumulated them. We also acquired images of three backgrounds (only object beam, only reference beam, and no beams). From those images, we calculate the phase image. Based on Eqs. (2) and (3), we obtained the spatial distribution of height, H(x, y). Figure 9 compares H(x, y) with respect to the number of cascade links: (c) HΛ4(x, y) with no cascade links, (d) HΛ3(x, y) with one cascade link between Λ4 and Λ3, (e) HΛ2(x, y) with two cascade links among Λ4, Λ3, and Λ2, (f) HΛ1(x, y) with three cascade links among Λ4, Λ3, Λ2, and Λ1, and (g) Hλ(x, y) with full cascade links among Λ4, Λ3, Λ2, Λ1, and λ. The black color in these images indicates a null region due to the low signal-to-noise ratio at the boundary.
Figure 10 shows the mean and standard deviation of (a) 10.0000-mm step difference, (b) 9.0000-mm step difference, (c) 0.8900-mm step difference, and (d) 0.1100-mm step difference. According to the increased number of cascade links, the uncertainty of the measured values of the step difference was significantly improved. Figure 10(e) and Visualization 1 show the 3D shape of the stepped-surface sample. The four step differences were determined to be 10.000966 ± 0.000359 mm, 9.003529 ± 0.000375 mm, 0.889270 ± 0.000408 mm, and 0.108167 ± 0.000372 mm. Table 1 summarizes a comparison of step difference between the measured value and the specification value for four step differences. When an axial uncertainty is defined as a mean of a root mean squared error (RMSE) between the measured value and the specification value for those four step differences, the uncertainty was achieved to 2.08 µm.
In Fig. 10(e) and Visualization 1, while the four stepped surfaces were clearly visualized, some errors in the surface profile were confirmed as discrete unevenness in each step surface. This unevenness was caused by error of the determined NΛ3(x, y) in the first cascade link between Λ4 and Λ3. Figure 11(a) shows a comparison of the height profile between HΛ4(x, y) along a red line in Fig. 9(c) and HΛ3(x, y) along a red line in Fig. 9(d), in which the HΛ4(x, y) value fluctuated around a threshold to change NΛ3(x, y). This results in the discrete unevenness in HΛ3(x, y). Such discrete unevenness is likely to occur when a ratio of HΛ4(x, y) to NΛ3(x, y) is large. Similar error was occurred in the final cascade link between Λ1 and λ. Figure 11(b) shows a comparison of the height profile between HΛ1(x, y) along a red line in Fig. 9(f) and Hλ(x, y) along a red line in Fig. 9(g). The error of NΛ4(x, y) and Nλ(x, y) depends on a relation between synthetic wavelength and temporal phase noise. To link the large difference of wavelength between them, the temporal phase noise has to be sufficiently low. The error in Nλ(x, y) is mainly due to the spatial unevenness of temporal phase noise (see Fig. 8).
It is important to adjust the synthetic wavelength by expanding the spectral bandwidth and flattening the spectral shape of the EOM-OFC as well as suppressing the temporal phase noise by increasing the power of each mode of the EOM-OFC. The EOM-OFC used in this work had a limited spectral bandwidth and an uneven spectral shape [see Figs. 4(a) and 4(b)]. The limited spectral bandwidth made it difficult to reduce the minimum synthetic wavelength of the EOM-OFC. On the other hand, the uneven spectral shape not only limited the selectivity of multiple synthetic wavelengths but also increased the temporal phase noise when generating a synthetic wavelength with weaker OFC modes. An EOM-OFC with a broad spectral bandwidth and even spectral shape is ideal for MCL-SW-DH. Fortunately, the state-of-the-art spectral broadening in nonlinear waveguides based on silicon-nitride enables us to expand the spectral bandwidth over 120 THz with an even spectral shape in a 10 GHz EOM-OFC . Work is in progress to apply such a broadband EOM-OFC to MCL-SW-DH.
We here make a comparison of the present method with the previous method based on OFC. In the present system, each mode has a power of 1.35 mW from a total power of 500 mW and a total number of 370 modes although the actual mode power is influenced by the spectral shape and the signal loss of the 2D spatial disperser. In usual, the power is a problem when using the single mode of OFC. There are several methods to generate different synthetic wavelengths with OFC. For example, four distributed-feedback (DFB) lasers phase-locked to different modes of the OFC have been used for multi-wavelength interferometry . Since DFB laser has a typical power of a few to a few tens mW, such system may benefit from high signal-to-noise ratio in the measurement. However, such light source just gets complicated. On the other hand, the present line-by-line spectral-shaped EOM-OFC has a potential to further simplify the source setup by using an arrayed-waveguide-grating-based spectral shaper. Also, there is room for further increase of the mode power by optical amplification.
Another OFC-based method for step surface measurement was demonstrated based on the low coherence interferometry . By linking between the envelope peak determination and the carrier phase determination, the precision of the height measurement was enhanced from 1.820 µm with the envelope peak determination to 7 nm with the carrier phase determination. Although this method uses the phase of single optical wavelength to improve the uncertainty in the same manner as the present method, there are some differences of uncertainty between those two methods. The reason for this difference will be in the optical phase instability of EOM-OFC seeded by the free-running ECLD in addition to the limited spectral range and the uneven spectral shape mentioned above. Evaluation of such influence will be a future work.
A line-by-line spectral-shaped EOM-OFC was applied for MCL-SW-DH to accelerate the multicascade-linking operation of multiple synthetic wavelengths and an optical wavelength. To generate different orders of multiple synthetic wavelengths, a single narrow-linewidth OFC mode was extracted from the 10 GHz-spacing EOM-OFC within a spectral range of 1540 nm to 1570 nm at an interval of 500 ms. Although the procedure of MCL-SW-DH is similar to that in the previous paper , rapid switching of multiple optical wavelengths in the line-by-line spectral-shaped EOM-OFC significantly enhances the speed of measurement (typically, 100 times faster). MCL-SW-DH was demonstrated with a combination of four synthetic wavelengths of 86 µm, 145 µm, 1249 µm, and 30440 µm and a single optical wavelength of 1540 nm for measuring the stepped surface of a sample. Four step differences of 10 mm, 9 mm, 0.89 mm, and 0.11 mm was determined with an axial uncertainty of 2.08 µm. Although there is room for improvement in the temporal phase noise and the optical spectrum to achieve sub-µm precision, the proposed method will be a powerful tool for 3D imaging with mm-order axial range and nm-order axial resolution.
Cabinet Office, Government of Japan (Subsidy for Reg. Univ. and Reg. Ind. Creation); Japan Society for the Promotion of Science (18K04981, 19H00871); Exploratory Research for Advanced Technology (JPMJER1304).
The authors declare no conflicts of interest.
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.
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