Compression-coding-based surface measurement using a digital micromirror device and heterodyne interferometry of an optical frequency comb

Guangyao Xu; Yue Wang; Yue Wang; Jiayang Chen; Shilin Xiong; Guanhao Wu

doi:10.1364/OE.432112

1. Introduction

An optical frequency comb (OFC) emits a series of ultrashort optical pulses with equal time intervals. In the frequency domain, the Fourier spectrum is composed of many optical longitudinal modes that are equidistantly spaced at the repetition frequency [1–4]. These optical longitudinal modes can be precisely phase locked to a frequency standard. Hence, the OFC can be regarded as an optical frequency ruler and applied extensively in diverse fields, including optical clocks [5,6], absolute distance measurement [7–9], refractive index measurement [10,11], angle measurement [12–14] and surface measurement [15–23]. In the field of high-precision surface measurement, the OFC can be used to perform profilometry with a higher accuracy and a larger range compared with the traditional light source. Hence, a series of measurement methods has been proposed, such as repetition frequency scanning [15,17], chirped pulses [16], spectral interferometry [24], mechanical scanning [25], and single-pixel-imaging (SPI) [18–20]. Among these methods, the SPI-combined OFC surface measurement method has garnered significant attention owing to its characteristics of lower cost, faster measurement speed, and no additional mechanical vibration, afforded by the use of a high-speed modulated digital micromirror device (DMD).

In the SPI surface measurement system, a series of coding masks for sampling and the corresponding reconstruction algorithm are essential. The sampling and reconstruction methods significantly affect the reconstruction speed and quality [18–20,26,27]. The first approach is based on compressive sensing (CS). By constructing a sparse measurement signal and solving the underdetermined equation sets, one can reconstruct the profile of an object [18,20,26]. This CS-based method can achieve compression measurements with a sampling rate lower than the Nyquist limit [20,28,29], i.e., the object’s profile is reconstructed completely at a sampling times less than the number of measuring points. However, the CS reconstruction process is time consuming because an iterative algorithm is required to solve the convex optimization problem. In addition, sparse measurement signals cannot be structured easily for complex surface profiles. The other approach is based on cyclic Hadamard transform imaging. By inverse matrix transformation, the object’s profile can be reconstructed rapidly without iteration [19,27]. This method affords high reconstruction efficiency, and the modulation matrix is binary, which is easy to use on high-speed spatial light modulators. However, the sampling times must be equal to the measuring point to reconstruct the object’s profile, hence, compression sampling cannot be realized.

To achieve effective compression sampling and rapid reconstruction, the SPI surface sampling and reconstruction method must be optimized. For SPI methods, alternative compression approaches have been proposed for real-time imaging with a tradeoff between image quality and real-time robustness, such as evolutionary compressive sensing [30–32] and optimized ordering of the Hadamard basis [33,34]. Because different bases contribute differently to image reconstruction, in these methods, more significant basis masks are projected and measured to reduce the sampling times. Moreover, for the abovementioned methods, the detected value of the single-pixel detector must satisfy the linear relationship with the measuring values at all measuring points. In the OFC-based surface measurement, the depth information is obtained by solving the phase of the longitudinal mode, and the phase of the signal detected by a single-pixel detector is typically nonlinear to the phase of each measuring point. Therefore, the nonlinear correlation between the phases of the detection signal and the measuring points must be solved to apply the optimized ordering of the Hadamard basis in the OFC surface measurement. Additionally, an appropriate ordering of the Hadamard basis must be constructed for surface measurements.

In this article, we propose a compression coding-based surface measurement method using SPI and OFC heterodyne interferometry. Using the fast Fourier transform (FFT) for the heterodyne interference signal (HIS) recorded by the single-pixel detector, we can obtain the real and imaginary parts at the beat frequency that exhibit a linear relationship with the real and imaginary parts of the HIS with all measuring points, respectively. Additionally, owing to the additional effect of the wavefront error on the measuring surface, it is difficult to effectively perform compression with a fixed ordering of Hadamard basis. To improve the sampling compression ratio, an adaptive optimized ordering of the Hadamard basis is applied to reconstruct the surface profile without a complex wavefront correction. Consequently, the measuring time of a three-step profile improves to several milliseconds with nanoscale precision.

2. System configuration

Figure 1 shows the SPI-based OFC heterodyne interference system, which illustrates the process by which the SPI scheme of the optimized ordering of Hadamard basis is incorporated into the OFC heterodyne interference scheme. The light source is a mode-locked Er-doped fiber laser with a central wavelength of 1.56 µm and a spectral bandwidth of 53 nm. The repetition frequency f_r is ∼77.8 MHz, which is phase locked to a rubidium frequency standard (SIM940, Stanford Research Systems, USA, accuracy, 5 × 10⁻¹¹; instability, 2 × 10⁻¹¹ at 1 s) by controlling the laser cavity length using a piezoelectric transducer and a temperature-controlled box. The pulse train from the light source is segregated into two paths by a 50:50 beam splitter and served as reference and measurement pulses, respectively. A beam expander, which is composed of a micro-objective, pinhole, and collimating lens, is inserted in the measurement path to expand the measurement beams before they reached the measurement surface. The beam expander in the reference path has the same amplification parameters with the measurement path, and an acousto–optic modulator (AOM) is inserted into the reference path to shift the optical frequency prior to beam expansion. To construct a heterodyne interference signal with frequency f_b, the shift frequency (f_s) by the AOM is set to f_s= f_r + f_b in our system, furthermore, the f_s is also referenced to the rubidium frequency standard to ensure the stability of the heterodyne frequency. The measurement beam carrying information pertaining to the target surface is superimposed with the reference beam and results in heterodyne interference at BS₂. The heterodyne interference beam is segregated into two beams by BS₃. One beam is detected by a photodetector (PD₂) after lens focusing, and the other beam, modulated by a DMD (DLP6500, Texas Instruments), is focused to another photodetector (PD₁). The DMD’s reflecting surface contained more than 2 million micromirrors with a side length of 7.56 µm, and each micromirror has two operating states with the highest modulation frequency over 9.5 kHz corresponding to different reflection angles. The reflected light corresponding to one of the operating states is reflected to PD₁, whereas the other reflected light will not be received in our system. The single-pixel photodetectors (PDA20CS2) with a bandwidth of 11 MHz and an active area of 3.14 mm² (Ø2.0 mm) ensures that the high-frequency heterodyne signals are adequately sampled and fully received. Both heterodyne interference signals recorded by PD₁ and PD₂ are sampled synchronously using a data acquisition card. We measure a compensate flat surface in advance to eliminate wavefront error caused by different beam expansions and optimize the ordering of Hadamard basis. To improve the longitudinal measurement range of the surface measurement system, two band-pass filters with different central wavelengths are alternatively inserted into the system to construct the synthetic wavelength [8,17,35]. The inset in Fig. 1 illustrates the filtered spectra.

Fig. 1. Schematic illustration of surface measurement system based on OFC and DMD. Blue and orange paths stand for measurement and reference arms, respectively. OFC (optical frequency comb), BS (beam splitter), BPF (band-pass filter), AOM (acoustic optical modulator), DMD (digital micromirror device), PD (photodetector), DAQ (data acquisition).

Download Full Size | PDF

3. Principles of compression-coding-based surface measurement

Figure 2 shows the compression-coding-based surface measurement processing using the SPI of the optimized ordering of Hadamard basis and the OFC heterodyne interferometry. Owing to the equal intervals and the linearly dependent stable phases of the OFC longitudinal modes [36], the HIS containing the information of the sample surface can be expressed as shown in Eq. (1) before being modulated by the DMD.

(1)$${I_i} = {A_i}\cos (2\mathrm{\pi }{f_\textrm{b}}t + {\varphi _{\textrm{c,}i}}),$$

where I_i is the heterodyne interference signal corresponding to the i^th measuring point; A_i, f_b, and φ_c,i are the amplitude, beat frequency, and phase of the heterodyne interference signal, respectively. The phase of the heterodyne interference signal is equivalent to that of the effective central wavelength of the optical spectrum based on the previous study [37].

Fig. 2. Compression-coding-based surface measurement processing for obtaining HIS-reconstructed phase. CM (coding mask), PD (photodetector), FFT (fast Fourier Transform).

Download Full Size | PDF

The coding masks (CMs) loaded and displayed on the DMD are based on an N × N Hadamard matrix H. Every CM is adjusted from a one-dimensional to two-dimensional n × n distribution, where N = n × n. Moreover, the CM unit corresponding to each measuring point is composed of many micromirrors with the same operating state within the unit areas. The binary elements of the CM correspond to states reflecting or not reflecting light to PD₁, respectively. After being modulated successively by the DMD, the HIS is expressed as

(2)$${Y^j} = \sum\limits_{i = 1}^\textrm{N} {w_i^j{I_i}} ,w_i^j \in \{{0,1} \},$$

where ${Y^j}$ is the HIS detected by PD₁ in the j^th measurement; “1” and “0” of w correspond to passing and blocking light, respectively. To reduce the effect of air disturbance, part of the HIS is detected at PD₂ as the reference HIS. Based on the Fourier transform of the HIS and the characteristics of the H matrix, the real parts of the HIS based on the reference signals for all CMs are expressed as

(3)$$\left[ {\begin{array}{{c}} {\begin{array}{{c}} {\textrm{Re} (Y_{{f_\textrm{b}}}^1)}\\ \vdots \\ {\textrm{Re} (Y_{{f_\textrm{b}}}^j)}\\ \vdots \\ {\textrm{Re} (Y_{{f_\textrm{b}}}^N)} \end{array}} \end{array}} \right] = \textrm{H} \times \left[ {\begin{array}{{c}} {\begin{array}{{c}} {\textrm{Re} ({I_1})}\\ \vdots \\ {\textrm{Re} ({I_i})}\\ \vdots \\ {\textrm{Re} ({I_N})} \end{array}} \end{array}} \right],$$

where $\textrm{Re} (Y_{{f_\textrm{b}}}^j)$ is the real part of the modulated HIS at frequency f_b in the j^th measurement, $\textrm{Re} ({I_i})$ is the real part of the HIS at frequency f_b for the i^th measuring point, and H is the Hadamard matrix. The specific derivation process is presented in the Appendix. Similarly, the imaginary parts of the HIS can be obtained as follows:

(4)$$\left[ {\begin{array}{{c}} {\begin{array}{{c}} {\textrm{Re} (Y_{{f_\textrm{b}}}^1)}\\ \vdots \\ {\textrm{Re} (Y_{{f_\textrm{b}}}^j)}\\ \vdots \\ {\textrm{Re} (Y_{{f_\textrm{b}}}^N)} \end{array}} \end{array}} \right] = \textrm{H} \times \left[ {\begin{array}{{c}} {\begin{array}{{c}} {\textrm{Re} ({I_1})}\\ \vdots \\ {\textrm{Re} ({I_i})}\\ \vdots \\ {\textrm{Re} ({I_N})} \end{array}} \end{array}} \right],$$

where ${\mathop{\rm Im}\nolimits} (Y_{{f_\textrm{b}}}^j)$ is the imaginary part of the modulated HIS at frequency f_b in the j^th measurement, and ${\mathop{\rm Im}\nolimits} ({I_i})$ is the imaginary part of the HIS at frequency fb for the i^th measuring point.

Because the Hadamard matrix is a symmetric orthogonal matrix (H = H^T + H⁻¹), the surface profile can be reconstructed perfectly by the inverse matrix transform when all bases are used for the measurement [19,38]. To reduce the measurement time and realize compression measurements, the real and imaginary parts of the measuring points can still be restored using partial bases. However, the reconstruction quality is related to the importance of partial measurement bases [30,32]. Therefore, the ordering of the measurement bases significantly affects the reconstruction efficiency when using partial bases for measurement [33]. In our system, the measurement results contain both the sample surface profile information and the wavefront error between the reference and measurement arms. The wavefront error can be determined by measuring the standard compensating plate, and the measurement bases are sorted in the order of priority for the error reconstruction. For some simple surfaces, the necessary bases are typically contained within the bases for wavefront error reconstruction. Therefore, rapid measurements can be realized effectively by reconstructing the wavefront error rapidly. To evaluate the importance of the measurement bases for reconstructing the wavefront error, all the measurement bases are used to measure the compensate plane and obtain the real and imaginary parts of the modulated HIS. The ordering factor can be calculated as follows:

(5)$${O_j} = |{\textrm{Re} (Y_{{f_\textrm{b}}}^j)} |+ |{{\mathop{\textrm {Im}}\nolimits} (Y_{{f_\textrm{b}}}^j)} |,$$

where $\textrm{Re} (Y_{{f_\textrm{b}}}^j)$ and ${\mathop{\rm Im}\nolimits} (Y_{{f_\textrm{b}}}^j)$ are the real and imaginary parts of the modulated HIS at the j^th CM, respectively. Subsequently, we optimize the ordering of the Hadamard basis using the magnitude of their corresponding ordering factors. Based on the optimized ordering of the Hadamard basis, the approximate real and imaginary parts of the measuring points can be reconstructed as

(6)$$\left[ {\begin{array}{{c}} {\begin{array}{{c}} {\textrm{Re} ({{\hat{I}}_1})}\\ \vdots \\ {\textrm{Re} ({{\hat{I}}_i})}\\ \vdots \\ {\textrm{Re} ({{\hat{I}}_N})} \end{array}} \end{array}} \right]\textrm{ = H}_\textrm{P}^\textrm{T} \times \left[ {\begin{array}{{c}} {\begin{array}{{c}} {\textrm{Re} (Y_{{f_\textrm{b}}}^1)}\\ \vdots \\ {\textrm{Re} (Y_{{f_\textrm{b}}}^j)}\\ \vdots \\ {\textrm{Re} (Y_{{f_\textrm{b}}}^M)} \end{array}} \end{array}} \right],$$

(7)$$\left[ {\begin{array}{{c}} {\begin{array}{{c}} {{\mathop{\rm Im}\nolimits} ({{\hat{I}}_1})}\\ \vdots \\ {{\mathop{\rm Im}\nolimits} ({{\hat{I}}_i})}\\ \vdots \\ {{\mathop{\rm Im}\nolimits} ({{\hat{I}}_N})} \end{array}} \end{array}} \right]\textrm{ = H}_\textrm{P}^\textrm{T} \times \left[ {\begin{array}{{c}} {\begin{array}{{c}} {{\mathop{\rm Im}\nolimits} (Y_{{f_\textrm{b}}}^1)}\\ \vdots \\ {{\mathop{\rm Im}\nolimits} (Y_{{f_\textrm{b}}}^j)}\\ \vdots \\ {{\mathop{\rm Im}\nolimits} (Y_{{f_\textrm{b}}}^M)} \end{array}} \end{array}} \right],$$

where H_p is the partial Hadamard matrix comprising the former M bases, and ${\hat{I}_i}$ is the approximate value of I_i using partial measuring bases (M < N). Therefore, the approximate phase can be obtained as follows:

(8)$${\hat{\varphi }_i} = \arctan \left\{ {\frac{{{\mathop{\rm Im}\nolimits} ({{\hat{I}}_i})}}{{\textrm{Re} ({{\hat{I}}_i})}}} \right\},$$

Subsequently, the distance from the i^th measuring point can be expressed as

(9)$${\hat{d}_i} = \frac{{{{\hat{\varphi }}_i}}}{{2\mathrm{\pi }}} \times \frac{{{\lambda _\textrm{c}}}}{2}.$$

To expand the measuring longitudinal range, two filters with different central wavelengths are used to form the synthetic wavelength alternately. When the precision of the synthetic wavelength measurement is less than a quarter of a single wavelength, the distance can be precisely determined using a single wavelength [17].

4. Experiments and results

To evaluate the performance of the optimized ordering of the Hadamard basis, we derived an object’s profile using this compression-coding-based surface measurement, with N = 256 and n = 16, and the size of the unit pixel of the CM is 378 µm × 378 µm, which is equal to the spatial resolution due to a parallel light path in the equal arm interferometer. Figure 3(a) shows a photograph of a three-step sample comprising three gauge blocks of 1.23, 1.24 and 1.26 mm thickness respectively with different heights of 10, 20, and 30 µm. After measuring the compensated surface and optimizing the ordering of the measurement basis using the procedure described above, we measured the three-step sample in area II (6.048 mm × 6.048 mm) with a blue square box, as shown in Fig. 1(a). We used the root mean square (RMS) of the error between the compression measurement and the all-bases measurement to evaluate the performance of the compression measurement; it is calculated as

(10)$${d_{rms}} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {{{|{d_i^\textrm{P} - d_i^\textrm{A}} |}^2}} } ,$$

where $d_i^\textrm{P}$ is the measurement result from the i^th measuring point constructed via compression measurement with partial bases, and $d_i^\textrm{A}$ is reconstructed via an all-bases measurement. Figure 3(b) shows the relationship between the sampling rate and the RMS for the three-step measurement. Here, the sampling rate refers to the ratio of the number of bases used to all measuring bases and it stands for the ratio of the sampling time using part measurement bases to the one of uncompressed measurement, that is, the smaller the sampling rate, the shorter the sampling time. The RMS decreased as the sampling rate increased, particularly at the beginning, and reached 25 nm at a sampling rate of 20%, thereby validating the compression. To reconstruct the surface precisely, two filters with different central wavelengths of 1535 and 1559 nm were alternately inserted into the optical path to form a synthetic wavelength exceeding 80 µm. The standard deviations of the synthetic wavelength for 100 measurements are shown in Fig. 3(c) at a sampling rate of 20%. The precision of the synthetic wavelength measurement was less than a quarter of a single wavelength, except for individual points at the edge, and a more accurate reconstructed surface was obtained from the single-wavelength measurement results. Figure 3(d) shows the standard deviations of the single wavelength for 100 measurements at a 20% sampling rate, and the standard deviations of most measurement points were less than 5 nm. The DMD binary modulation rate was approximately 9.5 kHz in our system, and the measurement time was less than 6 ms at a 20% sampling rate. It is noteworthy that the measuring speed improved significantly without an iterative reconstruction process.

Fig. 3. (a) Photograph of three-step surface; red-square and blue-square boxes represent measurement areas by WLI and compression measurement, respectively; (b) RMS error between compression and all-bases measurements; red point indicates standard deviation of 100 measurements at 20% sampling rate; (c) results obtained using synthetic wavelength method; (d) results obtained using single wavelength at 1559 nm.

Download Full Size | PDF

Next, we conducted a comparative experiment using a commercial white light interferometer (WLI, Bruker, Contour GT-IM) with a step height accuracy of 0.75% to validate the height accuracy. The measurement results are shown in Fig. 4(a) for an area measuring 1.3 mm × 0.9 mm, which was limited by the field of view of WLI, and the step heights corresponding to the two reference lines were 20.049 and 29.962 µm, separately. Figure 4(b) shows the results reconstructed via the OFC-based compression measurement at a 20% sampling rate. Considering the reference lines at the same positions, the step heights were 29.990 µm (cross-section A) and 20.132 µm (cross-section B), which are consistent with the results of WLI. It was verified that this method can achieve nanoscale precision in measuring the step height.

Fig. 4. Comparison of reconstruction results between commercial WLI and compression-coding-based OFC surface measurement at 20% sampling rate. Results obtained from (a) WLI and (b) compression-coding-based surface measurement of present method.

Download Full Size | PDF

The experiments conducted show that a basal number of measurement bases is required to reconstruct the measuring surfaces effectively. Different surface morphologies require different number of the measurement bases. To validate the effect of surface complexity on the reconstruction quality, we conducted a comparative experiment for different surfaces which are parts of the MEMS device having a step-shaped platform and a teardrop-shaped groove, separately. The nominal height of both parts was approximately 10 µm. The reconstructed results at a 20% sampling rate for the step-shaped platform and teardrop-shaped groove are shown in Figs. 5(a) and 5(b), respectively, which are consistent with the actual morphology of the objects. Figure 5(c) shows the RMS error between the compression and all-bases measurements for the measuring objects, where the blue and red lines correspond to the results of the step-shaped and teardrop-shaped MEMS devices, respectively. The RMS error curve of the teardrop-shaped surface decreased more slowly than that of the step-shaped surface. This shows that the reconstruction of a more complex surface was less efficient than that of a simple surface. The experimental results prove that the proposed compression coding surface measurement is effective for different surfaces. Nevertheless, a higher sampling rate is preferred for complex surface measurements. Hence, our compression measurement method is more suitable for sparse surfaces, and the compression efficiency is limited by the complexity of the surfaces.

Fig. 5. Comparison of reconstruction results at 20% sampling rate between (a) step-shaped and (b) teardrop-shaped MEMS surfaces; (c) comparison of RMS error as a function of sampling rate.

Download Full Size | PDF

5. Discussion and conclusion

The proposed scheme of optimized ordering of the Hadamard matrix was verified to achieve high-precision single-pixel compression-coding-based surface measurements. Compared with CS technology, the reconstruction processing is more efficient without requiring iterations, which reduces the reconstruction time at a permissible loss of the reconstruction quality. In optimizing the ordering of the Hadamard basis, some efficient ordering methods include the Walsh–Hadamard ordering [33], which is suitable for recovering images of low spatial frequency because the bases for low spatial frequency are always used preferentially. However, owing to the wavefront error in the surface measurement, the measuring surface is always a complex surface, even if the sample is simple. Our adaptive optimization method orders the measurement bases based on the surface error and can effectively compress the sampling rate without strict wavefront alignment; Furthermore, the f_ceo can be free-running. Because the experimental upset is based on an equal arm interferometer and the overlapping pulses from the measurement and reference arms have the same pulse number. Therefore, the phase difference between the measurement and reference pulses doesn’t change with the f_ceo. However, the measurement speed of this method is limited by the modulation rate of DMD and the measurement speed will increase as the development of the DMD modulation rate. In addition, the spatial resolution is limited by the light intensity. By using a stronger intense light source, less micro-mirrors are needed to be used as an element and the spatial resolution is expected to enable a high lateral resolution up to the limitation of a micro-mirror size.

In summary, we demonstrate a compression-coding-based surface measurement method using SPI and OFC heterodyne interferometry. By optimizing the ordering of the Hadamard basis, the phase of the heterodyne interference signals can be reconstructed rapidly through the reconstruction of real and imaginary parts. A group of synthetic wavelengths of the OFC is used to extend the measurement depth, and a three-step surface with heights of 10, 20, and 30 µm is measured in 6 ms at a sampling rate of 20% with a precision of 5 nm. The experimental results showed that the surface profile can be reconstructed at a sampling rate lower than the Nyquist limit with an acceptable loss of reconstruction quality, and that the reconstruction is fast because no iteration is required. In addition, the measurement bases are contained in a binary Hadamard matrix that is generated rapidly by the Hadamard transform, thereby avoiding the storage of many matrices and easing their use in a DMD with a fast binary modulation rate. Our proposed method based on the OFC and SPI is feasible for rapid surface measurements, particularly for objects with a simple surface topography.

Appendix

A. Construction of real parts of modulated signals

Figure 2 shows the compression-coding-based surface measurement procedures using SPI with the optimized ordering of the Hadamard bases and the OFC heterodyne interference. When a series of CMs is displayed on the DMD, the initial phase of the modulation HIS is affected by the unstable modulation intervals. To reduce the effects of air disturbance and determine the effective initial phase, a portion of the HIS is assigned to the PD₂ and sampled synchronously with PD₁ as the reference HIS. Both the amplitude and phase of the modulated and reference HIS at frequency f_b can be obtained via FFT, and the relative complex value of the effective signal can be expressed as

(11)$$Y_{{f_\textrm{b}}}^j = A_\textrm{m}^j({f_\textrm{b}})\textrm{exp} [{\varphi _j}({f_\textrm{b}})i],$$

where $A_\textrm{m}^j({f_\textrm{b}})$ is the amplitude of the modulated HIS, and ${\varphi _j}({f_\textrm{b}})$ is the phase difference between the reference and modulated HISs. The real part of the complex value corresponding to each CM can be expressed as

(12)$$\left[ {\begin{array}{{c}} {\begin{array}{{c}} {{{\textrm{Re} }^ + }(Y_{{f_\textrm{b}}}^1)}\\ \vdots \\ {{{\textrm{Re} }^ + }(Y_{{f_\textrm{b}}}^j)}\\ \vdots \\ {{{\textrm{Re} }^ + }(Y_{{f_\textrm{b}}}^N)} \end{array}} \end{array}} \right]\textrm{ = }{\textrm{H}^\textrm{ + }} \times \left[ {\begin{array}{{c}} {\begin{array}{{c}} {\textrm{Re} ({I_1})}\\ \vdots \\ {\textrm{Re} ({I_i})}\\ \vdots \\ {\textrm{Re} ({I_N})} \end{array}} \end{array}} \right],$$

where H⁺ is the matrix constructed from the Hadamard matrix H with all elements “−1” replaced by “0.” Here, the matrix H satisfies

(13)$$\textrm{H = }{\textrm{H}^\textrm{T}}\textrm{ = }{\textrm{H}^{ - 1}},$$

where H^T is the transpose of matrix H, and H⁻¹ is the inverse of matrix H. In addition, all the elements in the first row of H are “1”; hence, the relationship between H and H+ can be expressed as

(14)$$\textrm{H} = \textrm{2} \cdot {\textrm{H}^ + }\textrm{ - ones( }N\textrm{, }N\textrm{),}$$

and the real parts of the HIS at frequency fb with modulation by H can be expressed as

(15)$$\left[ {\begin{array}{{c}} {\begin{array}{{c}} {\textrm{Re} (Y_{{f_\textrm{b}}}^1)}\\ \vdots \\ {\textrm{Re} (Y_{{f_\textrm{b}}}^j)}\\ \vdots \\ {\textrm{Re} (Y_{{f_\textrm{b}}}^N)} \end{array}} \end{array}} \right]\textrm{ = }\left[ {\begin{array}{{c}} {\begin{array}{{c}} {{{\textrm{Re} }^ + }(Y_{{f_\textrm{b}}}^1)}\\ \vdots \\ {2{{\textrm{Re} }^ + }(Y_{{f_\textrm{b}}}^j)\textrm{ - }{{\textrm{Re} }^ + }(Y_{{f_\textrm{b}}}^1)}\\ \vdots \\ {2{{\textrm{Re} }^ + }(Y_{{f_\textrm{b}}}^N)\textrm{ - }{{\textrm{Re} }^ + }(Y_{{f_\textrm{b}}}^1)} \end{array}} \end{array}} \right] = \textrm{H} \times \left[ {\begin{array}{{c}} {\begin{array}{{c}} {\textrm{Re} ({I_1})}\\ \vdots \\ {\textrm{Re} ({I_i})}\\ \vdots \\ {\textrm{Re} ({I_N})} \end{array}} \end{array}} \right],$$

where $\textrm{Re} (Y_{{f_\textrm{b}}}^j)$ is the real part of the HIS at frequency f_b for the j^th measurement, and $\textrm{Re} ({I_i})$ is the real part of the relative HIS at frequency f_b of the i^th measuring point.

Funding

Tsinghua University; Beijing Science and Technology Planning Project (Z191100007419011); National Natural Science Foundation of China (51835007).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation and Applications (Springer US, 2005).

2. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002). [CrossRef]

3. T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, “Accurate measurement of large optical frequency differences with a mode-locked laser,” Opt. Lett. 24(13), 881–883 (1999). [CrossRef]

4. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000). [CrossRef]

5. M. Takamoto, F. L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435(7040), 321–324 (2005). [CrossRef]

6. B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10(−18) level,” Nature 506(7486), 71–75 (2014). [CrossRef]

7. H. Y. Zhang, H. Y. Wei, X. J. Wu, H. L. Yang, and Y. Li, “Absolute distance measurement by dual-comb nonlinear asynchronous optical sampling,” Opt. Express 22(6), 6597–6604 (2014). [CrossRef]

8. Z. Zhu, G. Xu, K. Ni, Q. Zhou, and G. Wu, “Synthetic-wavelength-based dual-comb interferometry for fast and precise absolute distance measurement,” Opt. Express 26(5), 5747–5757 (2018). [CrossRef]

9. K. N. Joo and S. W. Kim, “Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser,” Opt. Express 14(13), 5954–5960 (2006). [CrossRef]

10. G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Rep. 3(1), 1894 (2013). [CrossRef]

11. P. Balling, P. Mašika, P. Křen, and M. Doležal, “Length and refractive index measurement by Fourier transform interferometry and frequency comb spectroscopy,” Meas. Sci. Technol. 23(9), 094001 (2012). [CrossRef]

12. S. Zhou, V. Le, S. Xiong, Y. Yang, K. Ni, Q. Zhou, and G. Wu, “Dual-comb spectroscopy resolved three-degree-of-freedom sensing,” Photonics Res. 9(2), 243–251 (2021). [CrossRef]

13. X. Liang, J. Lin, L. Yang, T. Wu, Y. Liu, and J. Zhu, “Simultaneous Measurement of Absolute Distance and Angle Based on Dispersive Interferometry,” IEEE Photonics Technol. Lett. 32(8), 449–452 (2020). [CrossRef]

14. Y.-L. Chen, Y. Shimizu, J. Tamada, Y. Kudo, S. Madokoro, K. Nakamura, and W. Gao, “Optical frequency domain angle measurement in a femtosecond laser autocollimator,” Opt. Express 25(14), 16725–16738 (2017). [CrossRef]

15. W.-D. Joo, J. Park, S. Kim, S. Kim, Y. Kim, S.-W. Kim, and Y.-J. Kim, “Phase shifting interferometry for large-sized surface measurements by sweeping the repetition rate of femtosecond light pulses,” Int. J. Precis. Eng. Manuf. 14(2), 241–246 (2013). [CrossRef]

16. T. Kato, M. Uchida, Y. Tanaka, and K. Minoshima, “High-resolution 3D imaging method using chirped optical frequency combs based on convolution analysis of the spectral interference fringe,” OSA Continuum 3(1), 20–30 (2020). [CrossRef]

17. Y. Wang, G. Xu, S. Xiong, and G. Wu, “Large-field step-structure surface measurement using a femtosecond laser,” Opt. Express 28(15), 22946–22961 (2020). [CrossRef]

18. Q. D. Pham and Y. Hayasaki, “Combining phase images measured in the radio frequency and the optical frequency ranges,” Opt. Lett. 42(11), 2062–2065 (2017). [CrossRef]

19. K. Shibuya, T. Minamikawa, Y. Mizutani, H. Yamamoto, K. Minoshima, T. Yasui, and T. Iwata, “Scan-less hyperspectral dual-comb single-pixel-imaging in both amplitude and phase,” Opt. Express 25(18), 21947–21957 (2017). [CrossRef]

20. Q. D. Pham and Y. Hayasaki, “Optical frequency comb profilometry with a compressive sensing-based single-pixel camera composed of digital micromirror devices and a two-frequency method for meter-order depth measurements,” J. Micro/Nanolith. MEMS MOEMS 14(4), 041305 (2015). [CrossRef]

21. D. Kim, Y. Lu, J. Park, B. Kim, L. Yan, L. Yu, K.-N. Joo, and S.-W. Kim, “Rigorous single pulse imaging for ultrafast interferometric observation,” Opt. Express 27(14), 19758–19767 (2019). [CrossRef]

22. K. Koerner, G. Pedrini, I. Alexeenko, T. Steinmetz, R. Holzwarth, and W. Osten, “Short temporal coherence digital holography with a femtosecond frequency comb laser for multi-level optical sectioning,” Opt. Express 20(7), 7237–7242 (2012). [CrossRef]

23. J. S. Oh and S. W. Kim, “Femtosecond laser pulses for surface-profile metrology,” Opt. Lett. 30(19), 2650–2652 (2005). [CrossRef]

24. H. Zhao, Z. Zhang, X. Xu, H. Zhang, J. Zhai, and H. Wu, “Three-Dimensional Imaging by Frequency-Comb Spectral Interferometry,” Sensors 20(6), 1743 (2020). [CrossRef]

25. Y. Na, C.-G. Jeon, C. Ahn, M. Hyun, D. Kwon, J. Shin, and J. Kim, “Ultrafast, sub-nanometre-precision and multifunctional time-of-flight detection,” Nat. Photonics 14(6), 355–360 (2020). [CrossRef]

26. Q. D. Pham and Y. Hayasaki, “Optical frequency comb profilometry using a single-pixel camera composed of digital micromirror devices,” Appl. Opt. 54(1), A39–44 (2015). [CrossRef]

27. W. K. Pratt, J. Kane, and H. C. Andrews, “Hadamard transform image coding,” Proc. IEEE 57(1), 58–68 (1969). [CrossRef]

28. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008). [CrossRef]

29. Q. D. Pham and Y. Hayasaki, “Optical frequency comb interference profilometry using compressive sensing,” Opt. Express 21(16), 19003–19011 (2013). [CrossRef]

30. M. J. Sun, L. T. Meng, M. P. Edgar, M. J. Padgett, and N. Radwell, “A Russian Dolls ordering of the Hadamard basis for compressive single-pixel imaging,” Sci. Rep. 7(1), 3464 (2017). [CrossRef]

31. M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015). [CrossRef]

32. M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7(1), 12010 (2016). [CrossRef]

33. M. Li, L. Yan, R. Yang, and Y. Liu, “Fast single-pixel imaging based on optimized reordering Hadamard basis,” Acta Phys. Sin. 68(6), 064202 (2019). [CrossRef]

34. M. F. Li, Y. R. Zhang, X. F. Liu, X. R. Yao, K. H. Luo, H. Fan, and L. A. Wu, “A double-threshold technique for fast time-correspondence imaging,” Appl. Phys. Lett. 103(21), 211119 (2013). [CrossRef]

35. J. Lee, S. Han, K. Lee, E. Bae, S. Kim, S. Lee, S. W. Kim, and Y. J. Kim, “Absolute distance measurement by dual-comb interferometry with adjustable synthetic wavelength,” Meas. Sci. Technol. 24, 045201 (2013). [CrossRef]

36. Y.-S. Jang and S.-W. Kim, “Distance Measurements Using Mode-Locked Lasers: A Review,” Nanomanuf. Metrol. 1(3), 131–147 (2018). [CrossRef]

37. S. Xiong, Y. Wang, Y. Cai, J. Liu, J. Liu, and G. Wu, “Calculating the Effective Center Wavelength for Heterodyne Interferometry of an Optical Frequency Comb,” Appl. Sci. 8(12), 2465 (2018). [CrossRef]

38. M. Harwit, Hadamard transform optics (Elsevier Science, 2012).

Compression-coding-based surface measurement using a digital micromirror device and heterodyne interferometry of an optical frequency comb

Abstract

1. Introduction

2. System configuration

3. Principles of compression-coding-based surface measurement

4. Experiments and results

5. Discussion and conclusion

Appendix

A. Construction of real parts of modulated signals

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (15)

Optics Express