Multicascade-linked synthetic wavelength digital holography using an optical-comb-referenced frequency synthesizer

Digital holography (DH) is a promising method for non-contact surface topography because the reconstructed phase image can visualize the nanometer unevenness in a sample. However, the axial range of this method is limited to the range of the optical wavelength due to the phase wrapping ambiguity. Although the use of two different wavelengths of light and the resulting synthetic wavelength, i.e., synthetic wavelength DH, can expand the axial range up to a few tens of microns, this method is still insufficient for practical applications. In this article, a tunable external cavity laser diode phase-locked to an optical frequency comb, namely, an optical-comb-referenced frequency synthesizer, is effectively used for multiple synthetic wavelengths within the range of 32 um to 1.20 m. A multiple cascade link of the phase images among an optical wavelength (= 1.520 um) and 5 different synthetic wavelengths (= 32.39 um, 99.98 um, 400.0 um, 1003 um, and 4021 um) enables the shape measurement of a reflective millimeter-sized stepped surface with the axial resolution of 34 nm. The axial dynamic range, defined as the ratio of the maximum axial range (= 0.60 m) to the axial resolution (= 34 nm), achieves 1.7*10^8, which is much larger than that of previous synthetic wavelength DH. Such a wide axial dynamic range capability will further expand the application field of DH for large objects with meter dimensions.


Introduction
Digital holography (DH) [1][2][3][4] has attracted attention as a three-dimensional (3D) imaging tool with nanometer axial resolution for biomedical imaging [5,6] and industrial inspection [7][8][9]. In DH, an interference fringe, formed by interfering diffracted light from an object with a reference light, is acquired using a digital imaging sensor such as a charge-coupled device (CCD) or a complementary metal-oxide semiconductor (CMOS) camera. Then, the amplitude and phase images of the object light can be reconstructed by a diffraction calculation of the acquired interference image with a computer. DH is featured by phase imaging, digital focusing, real-time imaging, and quantitative analysis. In particular, phase imaging enables nanometer axial resolution in the 3D shape measurement of transparent or reflective objects. When DH is performed using single-wavelength continuous wave (CW) laser light, the maximum axial range is limited within a half wavelength (l/2) for reflective objects or a full wavelength (l) for transparent objects due to the phase wrapping ambiguity. Although phase unwrapping processes can expand the maximum axial range [10], their adoption has been limited to a smooth shaped profile.
To extend the maximum axial range over l/2 without the need for a phase unwrapping process, synthetic wavelength DH, referred to as SW-DH, has been proposed [11][12][13][14][15]. In this method, DH is performed at two different wavelengths (l1, l2), and then the synthetic wavelength L between them [= (l1l2)/|l2-l1|] is used to increase the maximum axial range up to L/2, which is larger than l/2. However, since stable CW lasers operate at several discrete wavelengths of 532 nm, 612 nm, 633 nm, or 780 nm, the available L was limited to several microns. Even though three different wavelength lights were used for SW-DH, the maximum L value remained at approximately a few tens of microns [12,16]. One possible method to further increase L is the use of a tunable CW laser [17,18]. Although the synthetic wavelength was extended from 3.2 to 40 mm, its wavelength fluctuation and/or mode hopping behavior hinders us from using two slightly different wavelengths of light for the generation of further longer synthetic wavelengths [18]. Furthermore, the axial precision was remained at 35 µm due to no cascade link between phase images with synthetic wavelengths and an optical wavelength. If multiple synthetic wavelengths with high stability and accuracy can be arbitrarily generated in the micrometer to meter range from a single light source and their phase images are cascade linked with each other, the maximum axial range will be significantly greater than the millimeter scale while maintaining the nanometer-scale axial resolution. It is anticipated that the resulting wide axial dynamic range DH will find many applications in the -4 -precise profilometry of large objects.
Widely and finely tunable CW light with high stability and accuracy in wavelength or optical frequency can be obtained by an optical-comb-referenced frequency synthesizer (OFS), which is a tunable external cavity laser diode (ECLD) phase-locked to an optical frequency comb (OFC) [19][20][21]. The OFC is composed of a series of optical frequency modes regularly spaced by a repetition frequency frep with a carrier-envelope offset frequency fceo. The OFC can be used as an optical frequency ruler by phase locking both frep and fceo to a frequency standard.
By further phase locking the ECLD to one optical frequency mode of the OFC, the narrow linewidth, high stability, and high accuracy in the OFC are transferred to the ECLD, enabling the determination of the optical frequency based on the frequency standard. Furthermore, the optical frequency of the OFS can be tuned by switching the OFC mode phase-locked by the ECLD or by changing frep while maintaining the phase locking of the ECLD to the OFC mode. Such an OFS has been used for high-precision broadband spectroscopy in the nearinfrared [19] and terahertz [20,21] regions. However, there have been no attempts to generate multiple synthetic wavelengths within the micrometer to millimeter range in SW-DH.
In this article, we demonstrate multiple synthetic wavelength DH (MSW-DH) using an OFS. The OFS is used for the acquisition of multiple phase images at five different synthetic wavelengths and an optical wavelength. The resulting series of phase images were coherently cascade linked for surface profilometry of a millimeter-stepped structure with nanometer axial resolution. Figure 1 shows the schematic diagram of the OFC and OFS. A vast number of OFC modes (freq. = nm) can be used as optical frequency markers in a broad spectral range based on the following equation where m is the mode number of the OFC. The parameter nm is given with an uncertainty of a frequency standard by determining m with an optical wavemeter while phase locking fceo and frep to the frequency standard. Next, we consider that the ECLD is phase-locked to the m-th OFC mode, namely, the OFS. The optical frequency of OFS nofs is given by

Optical-comb-referenced frequency synthesizer (OFS)
where fbeat is the beat frequency between the m-th OFC mode and ECLD light. If fbeat is phase-locked to the frequency standard, then the frequency uncertainty in the OFC is transferred to the ECLD, while the ECLD maintains its inherent characteristics such as wide wavelength tunability and moderate power usage. In other words, the OFS is traceable to the frequency standard via the OFC. More importantly from the viewpoint of MSW-DH, nofs can be discretely tuned at a step of frep within the whole tunable range of the ECLD or the spectral range of the OFC by switching the m value in Eq. (2). Furthermore, fofs can be tuned more precisely or more continuously by changing frep while maintaining the phase locking of the ECLD to the OFC mode. Here, we use the frep-step discrete tuning of the OFS for the generation of multiple synthetic wavelengths with a wide dynamic range.

Multicascade link of the synthetic wavelengths and optical wavelength
Principle of operation in SW-DH using two different wavelengths (l1, l2) is given in detail elsewhere [11][12][13][14][15]. The synthetic wavelength L is given by In the phase-image-based shape measurement in a reflection configuration, the spatial height distribution H(x, y) of a sample is given by where Nl (x, y) and fl(x, y) are the spatial distributions of interference-fringe orders (integer) and phase values at l1 or l2 and NL(x, y) and fL(x, y) are those at L, respectively. The function fL(x, y) is calculated by taking the difference of fl(x, y) between l1 and l2. We here assumed that phase wrapping did not occur in fL(x, y). While fl(x, y) and fL(x, y) can be measured within the phase range of 0 to 2π rad, Nl(x, y) cannot be directly determined due to the phase wrapping ambiguity. If fL(x, y) is used to determine Nl(x, y) and later H(x,y) by the cascade link between l and L, the maximum axial range is expanded to L/2 (typically, ~ 10l) whereas the axial resolution of approximately l/100 ~ l/1000 is maintained. The resulting axial dynamic range is expanded to 10 3 ~ 10 4 .
However, if fL(x, y) also undergoes phase wrapping, then H(x,y) cannot be determined. The limited synthetic wavelengths (typically < a few tens of microns) by available CW lasers hinder the increase in the axial dynamic range.
The OFS can generate multiple synthetic wavelengths with different orders of magnitude (= L1<L2<••••<Ln-1<Ln) as multiple cascades, as shown in Fig. 2. When such a series of multiple synthetic wavelengths were used for MSW-DH together with l, H(x, y) is given by where NLi(x, y) and fLi(x, y) are the spatial distributions of interference-fringe orders and phase values at Li, respectively. If each NLi(x, y) value can be determined one after another from the longest synthetic wavelength Ln to the shortest synthetic wavelength L1, then one can finally determine Nl(x, y) without errors. For example, in Eq. (5), the no-wrapping phase image fLn(x, y) obtained at the longest synthetic wavelength Ln is used to calculate HLn(x, y), where HLn(x, y) is H(x, y) determined by Ln. Then, HLn(x, y) is used to determine NLn-1(x, y).
Subsequently, the determined NLn-1(x, y) and the measured fLn-1(x, y) are used to determine HLn-1(x, y) more precisely. NLi(x, y) is given by By repeating a similar procedure from the longest to the shortest synthetic wavelength to the optical wavelength, Nl(x, y) can be determined correctly. By using such a multicascade link from Ln to l, both the maximum axial range of Ln/2 and the axial resolution of l/100 ~ l/1000 can be achieved at the same time. The resulting axial dynamic range becomes several orders of magnitude larger than the previous single SW-DH.

Experimental setup of MSW-DH
We used an off-axis Michelson-type interferometer for MSW-DH, as shown in Fig. 3 Fig.   3(b) shows a schematic drawing of a sample with a stepped surface.

Angular spectrum method for the wavefront reconstruction of DH
We used an angular spectrum method (ASM) [22,23] for the wavefront propagation calculation. The angular spectrum A(kx, ky; z) is given by the Fourier transform of the optical field E(x, y; z). Therefore, the angular spectrum at the propagation distance z = 0 is given by Next, the angular spectrum after propagation is specified as the product of A0(kx, ky) and a phase term for the propagation distance as follows Finally, the optical field after propagation is obtained by the inverse Fourier formation of A(kx, ky; z) as follows The ASM has three advantages in wavefront reconstruction: First, the digital filtering in the spatial frequency domain eliminates the zero-order diffraction light and the conjugate first-order diffraction light images, both of which are unnecessary for the wavefront reconstruction. The spatial filtering of unnecessary components leads to the improved quality of reconstructed amplitude and phase images of the object. Second, the pixel size of the reconstructed image is the same as that of the obtained digital hologram. Third, the ASM has no limitation for the reconstruction distance z because the optical field is treated as a plane wave.

Basic performance of the OFS
We first estimated the frequency fluctuation of nofs. Since fceo, frep, and fbeat in Eq. (2) were phase-locked to the rubidium frequency standard, their frequency instability should be identical to that of the frequency standard.
To evaluate the frequency instability of the rubidium frequency standard, we prepared two independent rubidium frequency standards with equivalent performance and used them for the frequency signal generation and the frequency signal measurement with an RF frequency counter (53220A, Keysight Technologies). The resulting frequency instability, calculated by Allan deviation, is shown in Fig. 4(a). From this frequency instability, we estimated the frequency fluctuation of fceo, frep, and fbeat with respect to gate time, as shown in Fig. 4(b). We  The present OFS can generate single-mode CW light at a step of frep (= 250 MHz) or 1.9 pm within the wavelength range of 1520 nm ~ 1595 nm, that can be used for two wavelength lights with a wavelength difference of 1.9 pm to 75 nm. Fig. 4(c) shows a relation between the wavelength difference ∆l (= l2 -l1, l1 = 1520 nm) and the corresponding L [see Eq. (3)], indicating that the available L ranges from 32 µm to 1.20 m.
The maximum L of 1.20 m is three orders of magnitude larger than that in the previous study [18]

-12 -
We evaluate the basic performance of the phase imaging by measuring a gauge block (164042, Mitutoyo, Kawasaki, Japan, thickness = 1 mm ± 0.45 µm, surface roughness = 21.6 nm) as a sample. We first evaluated spatial phase noise. We reconstructed the phase images of the sample at a reconstruction distance, z, of 134.2 mm using the ASM. Figures 5(a), 5(b), 5(c), 5(d), 5(e), and 5(f) show a series of phase images at l, L1, L2, L3, L4, and L5, respectively. Each phase image shows the spatial distribution of the phase value at the corresponding wavelength. We here defined the standard deviation of the spatial phase distribution fl(x, y) or fLi(x, y) as a spatial phase noise, limiting the precision of the surface unevenness measurement. Figure 5(g) shows a comparison of the spatial phase noise among the optical wavelength l and synthetic wavelengths L1 ~ L5. Each phase image of Li has a phase noise around 0.02 rad, which corresponds to a phase resolving power of around 1/314. The reason for the large phase noise at l is mainly due to the surface roughness of the gauge block or the mirror in the referenced arm. By decreasing the wavelength from L5 to l, the precision of the surface unevenness measurement improved from 6.59 µm to 0.01 µm, as shown in Fig. 5(h).
We next evaluated temporal phase noise. The temporal phase noise was obtained by calculating the standard deviation of the phase values at the same pixel in 100 repetitive phase images with the same wavelength. The temporal phase noise depends on the robustness of the optical systems to an environmental disturbance, such as air turbulence or mechanical vibration, and hence is a critical factor to determine the uncertainty in the height measurement based on the phase image. Figures 6(a), 6(b), 6(c), 6(d), 6(e), and 6(f) show the spatial distribution of the temporal phase noise at l, L1, L2, L3, L4, and L5, respectively. The similar distribution of temporal phase noise was confirmed in each phase image. Figure 6(g) compares the mean of the temporal phase noise among the optical wavelength l and synthetic wavelengths L1 ~ L5. The temporal phase noise of around 0.1 rad, which corresponds to the phase resolving power of 1/63, respectively, was confirmed at each wavelength; this value is larger than the spatial phase noise due to the phase noise in the time domain. Figure 6     Dependence of (g) temporal phase noise and (h) height uncertainty on wavelength.

3D Shape measurement of the stepped surface
Finally, we performed a 3D shape measurement of a stepped surface based on the cascade-linked phase images. We attached the same gauge block (164042, Mitutoyo, Kawasaki, Japan, thickness = 1 mm ± 0.45 µm, surface roughness = 21.6 nm) to another equivalent gauge block for wringing, as shown in the inset of Fig. 3(b).
This stepped surface was used as a sample.  -16 -

Discussion
We first discuss a possibility of a further enhancement in the axial dynamic range in the phase-image-based 3D shape measurement. In the demonstration above, an axial resolution of 34 nm was achieved within the axial range of 2 mm by using cascade link between 5 different synthetic wavelengths and an optical wavelength.
However, the present OFS has a potential to increase the axial range up to 0.60 m. The resulting axial dynamic range was 1.7×10 8 . Such a wide axial dynamic range is a distinctive feature in MSW-DH. However, there is still room to further expand the axial range. From the viewpoint of increased synthetic wavelength, the more precise tuning of fofs can be achieved by changing frep while maintaining the phase locking of the ECLD to the OFC mode. In this case, the minimum step is limited by the linewidth of the OFS [typically, 10 kHz at a gate time of 0.1 s in Fig. 4(b)]. Two wavelengths of light with an optical frequency difference of 10 kHz results in the generation of several tens kilometers in L, which is four orders of magnitude larger than the present maximum L. On the other hand, from the viewpoint of decreased axial resolution, use of more robust optical system will further reduce the temporal phase noise down to l/1000. Such improvement for the increased synthetic wavelength and the decreased axial resolution will further increase the axial dynamic range.
We next discuss the possibility of real-time MSW-DH measurements. In this article, we applied five different synthetic wavelengths and one optical wavelength for MSW-DH. To this end, we acquired holograms at six different optical wavelengths and calculated the phase images at synthetic wavelengths. Those holograms were acquired in order while ECLD was phase-locked to different mode of the OFC; the total acquisition time for the multiple holograms was typically a few minutes. In this article, we performed fine cascade links among phase images with different wavelengths as a proof of concept; however, considering the temporal phase noise at Li and l, we can further reduce the number of cascade links while maintaining the same performance in MSW-DH, leading to the reduction of the total acquisition time. Furthermore, if the phase locking procedure of ECLD is excluded, the acquisition time of multiple holograms will be largely reduced. One possible method is use of a line-by-line pulse-shaping technique in the OFC [24]. In this case, a single OFC mode is arbitrarily and quickly extracted by the use of a spatial light modulator and then is directly used for the rapid generation of multiple synthetic wavelengths. If the selection of an arbitrary OFC mode is performed in synchronization with a frame timing in the infrared camera, all holograms required for MSW-DH will be acquired in real-time. Such an approach will enable us to achieve real-time MSW-DH.

Conclusion
We demonstrated wide axial dynamic range MSW-DH using an OFS. The OFS was effectively used for multiple synthetic wavelengths within the range of 32 µm to 1.20 m. The cascade link of phase images was demonstrated among a single optical wavelength and 5 synthetic wavelengths within the range from 1.5 µm to 4000 µm, and enabled us to achieve an axial resolution of 34 nm in a 3D shape measurement of a 1-mm-stepped surface with a precision of 25 nm. The . The wide axial dynamic range MSW-DH can be a powerful tool for industrial inspection of large objects.