Terahertz synthetic aperture in-line holography with intensity correction and sparsity autofocusing reconstruction

Zeyu Li; Ruijiao Zou; Weipeng Kong; Xuemin Wang; Qinghua Deng; Qiang Yan; Yu Qin; Weidong Wu; Xun Zhou

doi:doi:10.1364/PRJ.7.001391

Photonics Research, 2019, 7 (12): 12001391, Published Online: Nov. 14, 2019

Terahertz synthetic aperture in-line holography with intensity correction and sparsity autofocusing reconstruction Download： 810次

论文大纲

Zeyu Li ^1,2,3Ruijiao Zou ^1,2Weipeng Kong ^1,2Xuemin Wang ^1,2Qinghua Deng ^1,2Qiang Yan ^1,2Yu Qin ^1,2Weidong Wu ^1,2Xun Zhou ^1,2,*

Author Affiliations

¹ Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China

² Microsystem & Terahertz Research Center, China Academy of Engineering Physics, Chengdu 610200, China

³ e-mail: lzy.robin@foxmail.com

Abstract

We demonstrate high-resolution and high-quality terahertz (THz) in-line digital holography based on the synthetic aperture method. The setup is built on a self-developed THz quantum cascade laser, and a lateral resolution better than 70 μm (

～ λ

) is achieved at 4.3 THz. To correct intensity differences between sub-holograms before aperture stitching, a practical algorithm with global optimization is proposed. To address the twin-image problem for in-line holography, a sparsity-based phase retrieval algorithm is applied to perform the high-quality reconstruction. Furthermore, a new autofocusing criterion termed “reconstruction objective function” is introduced to obtain the best in-focus reconstruction distance, so the autofocusing procedure and the reconstruction are unified within the same framework. Both simulation and experiment prove its accuracy and robustness. Note that all the methods proposed here can be applied to other wavebands as well. We demonstrate the success of this THz synthetic aperture in-line holography on biological and semiconductor samples, showing its potential applications in bioimaging and materials analysis.

1 INTRODUCTION

Terahertz (THz) imaging, benefiting from THz radiation’s capabilities of non-ionizing and penetration of non-conducting materials, serves as a cutting-edge non-destructive evaluation technology [1]. The amplitude image indicates the absorption properties, while the phase image reveals the refractive and thickness information, thus simultaneously determining that the amplitude and phase distributions of the wave front are highly desirable for applications ranging from bioimaging to material characterization [2,3]. THz time-domain spectroscopy (THz-TDS) performs a direct measurement of the electric field and estimates both the amplitude and phase information of a sample [4]. Although single-pixel technology can be utilized to accelerate imaging [5,6], it is still time consuming. Recently, ptychography has been demonstrated to be very promising in the THz domain, but it still requires scanning the sample to record multiple overlapped diffraction patterns [7 9" target="_self" style="display: inline;">–9]. Another competitive phase-contrast imaging technique is digital holography, which is an essential full-field, lensless imaging approach without scanning the sample. Over the past decade, THz digital holography in off-axis [10 –15" target="_self" style="display: inline;">–15] and in-line [15 –21" target="_self" style="display: inline;">–21] configurations, based on THz quantum cascade lasers (THz-QCL) [12,13,18,20] or optically pumped THz gas lasers [10,11,14 –17" target="_self" style="display: inline;">–17,19,21], has been explored. Real-time THz holography has been demonstrated [12–14] with frame rates even up to 50 Hz [14]. The holographic processing algorithms, such as sub-pixel sampling [19], extrapolation [2], autofocusing [15], iterative phase retrieval [15,19], and compressive sensing [17,21], have been proposed or applied to enhance the resolution or reconstruction quality.

The biggest advantage of off-axis holography is that it can quantitatively reconstruct the complex amplitude wave front free of twin images, at the expense of space-bandwidth product (SBP) due to the requirement for spectral filtering. As a comparison, the in-line scheme can fully utilize the SBP of the detector without spectral filtering and does not need an additional reference beam, enabling a more compact optical configuration with a larger numerical aperture (NA), both of which result in a better resolution. Moreover, a compact lensless structure with fewer optical elements can make better use of THz energy, improving the imaging signal-to-noise ratio. However, it suffers from the twin-image artifacts caused by the lack of phase information in the recorded hologram. Fortunately, phase retrieval can be used to overcome this problem. According to the Abbe criterion and the Nyquist sampling theorem, the resolution of in-line holography is limited by the wavelength, recording distance, detector size, and pixel pitch, where the recording distance and detector size affect the NA, while the pixel pitch affects the sampling [19]. With the emergence of high-resolution THz detector arrays such as the THz imager ( $320 \times 240$ pixels) with a pitch of 23.5 μm from the NEC Corporation (Japan), the THz camera ( $384 \times 288$ pixels) with a pitch of 35 μm from the INO Corporation (Canada), and the THz detector array ( $640 \times 512$ pixels) with a pitch of 17 μm from the IRay Technology Co., Ltd. (China) [21], sampling is no longer a problem, but the detector size is still insufficient, becoming a bottleneck of THz holographic resolution. Synthetic aperture methods were proposed to improve the NA by combining multiple holograms at different spatial positions [11,18] or under multiple tilted illumination beams [22], which are equivalent to enlarging the size of the detector. The application of this technique in THz in-line holography has been reported with an enhanced resolution of 125 μm ( $1.29 λ$ ) at 3.1 THz, but the intensity calibration of the sub-holograms is not optimal [18].

In this work, we demonstrate high-resolution and high-quality THz synthetic aperture in-line holography. The setup is based on a self-developed THz-QCL [23,24] with lateral resolution better than 70 μm ( $\sim λ$ ) at 4.3 THz. A global optimization algorithm is proposed to correct the sub-holograms’ intensities, a new autofocusing criterion based on “reconstruction objective function” is introduced to find the best reconstruction distance, and the sparsity-based phase retrieval algorithm [21] is applied to reconstruct the complex amplitude distribution of the sample without twin-image artifacts. The entire reconstruction process can be done automatically without manual intervention. Experiments are carried out to verify the proposed methods, and the potential THz biological imaging and materials analysis applications are illustrated.

The rest of this paper is structured as follows. Section 2 presents the experimental setup of THz synthetic aperture in-line holography. Section 3 describes the principle of the proposed intensity correction algorithm. In section 4, the sparsity-based autofocusing reconstruction method, including the reconstruction algorithm and the corresponding autofocusing criterion, is described. Section 5 shows the experimental results and discussions. The final section makes a conclusion.

2 EXPERIMENTAL SETUP

A schematic diagram of the experimental setup for THz synthetic aperture in-line holography is depicted in Fig. 1. The laser source is a THz-QCL developed by ourselves [23,24]. The working frequency in the experiments was 4.3 THz corresponding to a wavelength of 69.7 μm. The THz beam was shaped and collimated by a Si lens. The detector is an uncooled micro-bolometer array (IRV-T0830, NEC) with $320 \times 240$ pixels on a pitch of 23.5 μm; thus, the total size of the detector is $7.52 mm \times 5.64 mm$ . It was mounted on a two-axis motorized translation stage with micrometer accuracy and was placed downstream of the sample. To achieve high-NA imaging, the sample was placed as close as possible to the detector array. The wave scattered by the sample forms the object wave, and the unscattered part of the illumination wave forms the reference wave. They interfere to form an in-line hologram captured by the detector array. Note that when the wavelength is 69.7 μm and the pixel pitch is 23.5 μm, the Nyquist sampling can be satisfied, in which case the lateral resolution is limited only by the detection NA.

Schematic layout of the experimental setup. A Si lens was used to collimate the output THz beam. The wave scattered by the sample forms the object wave, and the unscattered part of the illumination forms the reference wave. The interference pattern recorded by the detector array is called the in-line hologram. The object wave in dark blue represents the low-frequency components, and the object wave in light blue denotes the high-frequency parts. By moving the detector, multiple sub-aperture holograms can be recorded and combined to be a synthetic aperture hologram for the resolution enhancement.

Fig. 1. Schematic layout of the experimental setup. A Si lens was used to collimate the output THz beam. The wave scattered by the sample forms the object wave, and the unscattered part of the illumination forms the reference wave. The interference pattern recorded by the detector array is called the in-line hologram. The object wave in dark blue represents the low-frequency components, and the object wave in light blue denotes the high-frequency parts. By moving the detector, multiple sub-aperture holograms can be recorded and combined to be a synthetic aperture hologram for the resolution enhancement.

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To increase the lateral resolution, the detection NA was increased by the synthetic aperture method. As shown in Fig. 1, the object waves colored by dark blue and light blue represent the low-frequency and high-frequency components, respectively. Therefore, by simply moving the detector, the detection NA can be expanded to collect a high-frequency object wave with a larger diffraction angle. In the experiments, the detector array was translated for a $3 \times 3$ raster scanning in the recording plane so that multiple sub-holograms at different spatial locations were recorded sequentially. After the recording of the sub-holograms, the sub-backgrounds were also recorded in the absence of the sample, and then nine normalized sub-holograms were obtained to suppress the non-uniform illumination by dividing the hologram patterns by the corresponding background patterns. The translation step was set to 5 mm to effectively cover the THz beam aperture while ensuring that the adjacent frames were overlapped. These overlapped parts are necessary for image registration and intensity correction.

3 APERTURE SYNTHESIS WITH INTENSITY CORRECTION

Aperture synthesis is performed on the normalized sub-holograms. Although it seems straightforward, accurate registration and intensity correction must be taken in the image stitching to keep both the fringes and the intensity continuous. For the former, the shift between the adjacent sub-holograms is determined by cross-correlation [25]. The registration in THz is not as challenging as that in the visible band, because the micrometer translation accuracy is high enough to make the translation error usually negligible relative to the 23.5 μm pixel size. For the latter, due to various factors such as the fluctuation of the THz-QCL and the non-uniform response of the detector, the captured intensity value is changing during data acquisition. If not corrected, the background of the synthetic aperture hologram is not uniform, resulting in degradation of the reconstruction.

Here we present a global optimization algorithm, considering not only the intensity of a pixel but also its location, to correct the sub-holograms’ intensity before stitching. As illustrated in Fig. 2, for a sub-aperture tile, there are four possible overlapped regions marked in Fig. 2(a), and for convenience, a number $i$ ( $i = 1, \dots, 9$ ), as well as an intensity correction factor $β_{i}$ ( $i = 1, \dots, 9$ ), is assigned to each tile of a synthetic aperture hologram shown in Fig. 2(b). One of these sub-holograms should be set as a reference tile whose intensity is regarded as a baseline. In our work, the central one numbered 5 is designated as the reference tile, so its intensity correction factor $β_{5}$ is 1. Ideally, the intensity of an overlapped part should be consistent between the adjacent tiles. Therefore, every overlapped part can be associated with a cost function based on the least-squares method. For example, the cost function for the overlapped part of tile 1 and tile 2 is $E_{1, 2} = \sum {(β_{1} T_{1, 2} - β_{2} T_{2, 4})}^{2},$ (1)where $E_{i, j}$ denotes the cost function for the overlapped part of tile $i$ ( $i = 1, \dots, 9$ ) and tile $j$ ( $j = 1, \dots, 9$ ), and $T_{i, j}$ denotes a matrix for the overlapped region $j$ ( $j = 1, \dots, 4$ ) of the tile $i$ ( $i = 1, \dots, 9$ ). When taking four neighbors of a tile into consideration, the global cost function for all the overlapped parts is given as $E = E_{1, 2} + E_{2, 3} + E_{1, 4} + E_{2, 5} + E_{3, 6} + E_{4, 5} + E_{5, 6} + E_{4, 7} + E_{5, 8} + E_{6, 9} + E_{7, 8} + E_{8, 9} .$ (2)

Schematic diagram of the aperture synthesis. (a) Four potential overlapped regions within a sub-hologram. (b) Synthetic aperture hologram with 3×3 tiles numbered by 1–9.

Fig. 2. Schematic diagram of the aperture synthesis. (a) Four potential overlapped regions within a sub-hologram. (b) Synthetic aperture hologram with $3 \times 3$ tiles numbered by 1–9.

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Expand the items related to tile 5, and then Eq. (2) can be expressed as follows: $E = E_{1} + E_{2} + E_{3},$ (3) $E_{1} = E_{1, 2} + E_{2, 3} + E_{1, 4} + E_{3, 6} + E_{4, 7} + E_{6, 9} + E_{7, 8} + E_{8, 9} + β_{2}^{2} \sum T_{2, 3}^{2} + β_{4}^{2} \sum T_{4, 2}^{2} + β_{6}^{2} \sum T_{6, 4}^{2} + β_{8}^{2} \sum T_{8, 1}^{2},$ (4) $E_{2} = - 2 β_{2} \sum (T_{2, 3} ⊙ T_{5, 1}) - 2 β_{4} \sum (T_{4, 2} ⊙ T_{5, 4}) - 2 β_{6} \sum (T_{6, 4} ⊙ T_{5, 2}) - 2 β_{8} \sum (T_{8, 1} ⊙ T_{5, 3}),$ (5) $E_{3} = \sum T_{5, 1}^{2} + \sum T_{5, 2}^{2} + \sum T_{5, 3}^{2} + \sum T_{5, 4}^{2},$ (6)where $E_{1}$ is a quadratic term, $E_{2}$ is a linear term, $E_{3}$ is a constant term, and $⊙$ denotes the Hadamard product, which is an elementwise operation for matrices. Equation (3) can be simplified to the following matrix form: $E = β^{T} A β - 2 β^{T} B + E_{3},$ (7) $A = [\begin{matrix} A_{11} & A_{12} & 0 & A_{14} & 0 & 0 & 0 & 0 \\ A_{21} & A_{22} & A_{23} & 0 & 0 & 0 & 0 & 0 \\ 0 & A_{32} & A_{33} & 0 & A_{35} & 0 & 0 & 0 \\ A_{41} & 0 & 0 & A_{44} & 0 & A_{46} & 0 & 0 \\ 0 & 0 & A_{53} & 0 & A_{55} & 0 & 0 & A_{58} \\ 0 & 0 & 0 & A_{64} & 0 & A_{66} & A_{67} & 0 \\ 0 & 0 & 0 & 0 & 0 & A_{76} & A_{77} & A_{78} \\ 0 & 0 & 0 & 0 & A_{85} & 0 & A_{87} & A_{88} \end{matrix}],$ (8)where $β = {[β_{1}, β_{2}, β_{3}, β_{4}, β_{6}, β_{7}, β_{8}, β_{9}]}^{T}$ , $B = {[0, \sum (T_{2, 3} ⊙ T_{5, 1}), 0, \sum (T_{4, 2} ⊙ T_{5, 4}), \sum (T_{6, 4} ⊙ T_{5, 2}), 0, \sum (T_{8, 1} ⊙ T_{5, 3}), 0]}^{T}$ , $A$ is a symmetric matrix, and each element $A_{i j}$ ( $i, j = 1, \dots, 8$ ) corresponds to the coefficient of $β [i] β [j]$ . Note that $β [i]$ is the $i$ th element of $β$ , which may not equal $β_{i}$ . For instance, $β [1] = β_{1}$ , $β [5] = β_{6}$ , and the first row of $A$ is determined from the coefficients of $β [1] β [j]$ ( $j = 1, \dots, 8$ ) so that $A_{11} = \sum (T_{1, 2}^{2} + T_{1, 3}^{2}$ ), $A_{12} = - \sum (T_{1, 2} ⊙ T_{2, 4})$ , and $A_{14} = - \sum (T_{1, 3} ⊙ T_{4, 1})$ .

The goal is to find the values of all the intensity correction factors $β$ that minimize the cost function as follows: $\arg \min_{β} E = β^{T} A β - 2 β^{T} B + E_{3} .$ (9)It is easy to prove that $E_{1}$ , or $β^{T} A β$ , is always greater than 0 unless $β = 0$ , which means that $A$ is a positive definite matrix and is invertible. There exists a global optimal solution to the Eq. (9), which can be found such that $\frac{\partial E}{\partial β} = 2 A β - 2 B = 0,$ (10) $β = A^{- 1} B .$ (11)The optimal solution for $4 \times 4$ tiles or other dimensions can be obtained by the same principle. When establishing the optimization objective function, theoretically any tile can be set as the reference, but the central one may be a better choice because it contains the main information of the hologram. Besides, eight neighbors of a tile can also be adopted instead of four neighbors.

4 SPARSITY-BASED AUTOFOCUSING RECONSTRUCTION

4.1 A. Sparse Phase Retrieval

Due to the absence of phase information during hologram recording, the twin-image-free reconstruction for in-line holography can be considered as a phase retrieval problem. In recent years, with the introduction of compressed sensing into in-line holography, this ill-conditioned inverse problem can be solved by optimization methods with sparsity regularization [21,26,27]. Here, the sparsity-based phase retrieval algorithm proposed in our previous work [21] is used to do the reconstruction. The optimization objective function can be expressed as $\arg \min_{X, W, μ} L (X, W, μ) = \frac{1}{2} {‖ H ⊙ W - T (d) * (X + μ 1) ‖}_{2}^{2} + τ {‖ X ‖}_{1} s.t. P_{s} (X) = X, | W | = 1,$ (12)where $H$ is the amplitude of the normalized synthetic aperture hologram; $W \in C^{m \times n}$ corresponds to the missing phase distribution; $T (d)$ is the point spread function of diffraction propagation over a recording distance $d$ ; $X \in C^{m \times n}$ and $μ \in C$ represent the wave front of the object and the uniform background at the object plane, respectively; $τ$ is a hyperparameter that weights the relative contribution of the $L_{1}$ norm term; * denotes the 2D convolution operation, and $P_{s}$ denotes the projection operator for object constraint, which is the positive absorption constraint [28] in this work. To calculate diffraction propagation, the angular spectrum method is adopted for its validity at small propagation distances.

The alternating minimization method is used to solve Eq. (12) iteratively by minimizing the subproblem corresponding to each variable sequentially. The phase of illumination at the object plane is assumed to be zero, and then the closed-form solutions are as follows: $μ^{k + 1} = \arg \min_{μ} L (X^{k}, W^{k}, μ) = \frac{1}{m n} ⟨ | T (- d) * (H ⊙ W^{k}) - X^{k} |, 1 ⟩,$ (13) $W^{k + 1} = \arg \min_{W} L (X^{k}, W, μ^{k + 1}) = \frac{T (d) * (X^{k} + μ^{k + 1} 1)}{| T (d) * (X^{k} + μ^{k + 1} 1) |},$ (14) $X^{k + 1} = \arg \min_{X} L (X, W^{k + 1}, μ^{k + 1}) = P_{s} {{SFT}_{τ} [T (- d) * (H ⊙ W^{k + 1}) - μ^{k + 1} 1]},$ (15)where $⟨ ., . ⟩$ denotes the inner product operator and ${SFT}_{τ}$ denotes the complex soft-thresholding operator [29], given by ${SFT}_{τ} (Z) [i, j] = {\begin{cases} (| Z [i, j] | - τ) \frac{Z [i, j]}{| Z [i, j] |} & | Z [i, j] | > τ \\ 0 & | Z [i, j] | \leq τ \end{cases},$ (16)and the initial hyperparameter $τ$ is estimated from the result of the first iteration, given by $τ = \frac{1}{m n} ⟨ | T (- d) * (H ⊙ W^{1}) - μ^{1} 1 |, 1 ⟩ .$ (17)Equations (13)–(15) are performed sequentially in each iteration. This iterative process is actually a combination of the conventional iterative phase retrieval [28] and the sparsity projection operation, which is illustrated in Fig. 3. Usually, the result converges with 10–20 iterations.

Iterative sparse reconstruction scheme. SA_HN is the normalized synthetic aperture hologram, which is composed of the normalized sub-holograms with intensity correction.

Fig. 3. Iterative sparse reconstruction scheme. $SA_H_{N}$ is the normalized synthetic aperture hologram, which is composed of the normalized sub-holograms with intensity correction.

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4.2 B. Autofocusing Algorithm

Numerical reconstruction of digital holography relies on a precise a priori knowledge about the recording distance. Basically, most autofocusing strategies depend on a criterion function, which is evaluated from a set of reconstructed images at different reconstruction distances. The distance corresponding to the maximum or minimum of this criterion is considered as the in-focus distance for that given hologram. Various autofocusing criteria have been demonstrated for different holographic applications in the past decade, such as variance [30], self-entropy [31], and sparsity [32]. The comprehensive overview and comparison of different measures can be learned from Ref. [32] and Ref. [33]. These metrics, however, are rarely specific to Gabor in-line holography, where the twin-image artifacts without phase retrieval make autofocusing challenging, especially for the case of a short recording distance. Zhang et al. proposed two edge sparsity criteria that perform well for their lens-free in-line holography [32], but they still regard autofocusing and phase retrieval as two independent steps; the reconstruction used for autofocusing is not the same as the one obtained by phase retrieval. This means that the estimated best in-focus distance may not be accurate because of the twin-image disturbance.

Here, different from previous work, we propose a new criterion, termed the “reconstruction objective function” (ROF), making autofocusing and phase retrieval unified within the same method as one step. Both the fidelity and the sparsity of the reconstruction are taken into account to give a more accurate in-focus position. The idea is sensible and straightforward that the objective function in Eq. (12) is employed as the autofocusing criterion function. For each reconstruction distance d, the iterative sparse reconstruction algorithm is performed, and then the value of the objective function is calculated. The distance that corresponds to the minimum is determined as the best in-focus distance.

Numerical simulation was carried out to verify the feasibility of the proposed ROF criterion. The pixel pitch and wavelength were set to 23.5 μm and 69.7 μm, respectively, as in the experiment. Three types of objects were simulated: type A, a complex amplitude object; type B, a pure amplitude object; and type C, a pure phase object, shown in Fig. 4. The illumination at object plane was assumed to be 1, the iteration number was set to 20 for the sparse phase retrieval algorithm, and the step for $d$ was set to 0.01 mm. To show the superior accuracy and robustness of our ROF criterion, the state-of-the-art metrics, “Gini of gradient” (GoG) and “Tamura of gradient” (ToG) [32] were selected for comparison. All the curves were normalized to [0,1].

Simulated three types of objects with a THz pattern. (a) Type A: complex amplitude object. (b) Type B: pure amplitude object. (c) Type C: pure phase object.

Fig. 4. Simulated three types of objects with a THz pattern. (a) Type A: complex amplitude object. (b) Type B: pure amplitude object. (c) Type C: pure phase object.

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Figures 5(a)–5(c) illustrate the autofocusing curves for the complex amplitude object (type A), the pure amplitude object (type B), and the pure phase object (type C), respectively, with the recording distance of 8 mm. The correct position is marked by black dashed lines for clarity. For the object of type A, ROF and ToG succeed in finding the correct distance, while the error for GoG is 0.02 mm. For the object of type B, ROF also reaches its minimum at the actual distance, while both GoG and ToG fail with the errors of 0.01 mm and 0.13 mm, respectively. For the object of type C, ToG gets the accurate position, while the errors of both ROF and GoG are 0.01 mm.

Autofocusing curves (left column) and their zoomed-in local counterparts (right column) for (a) the complex amplitude object (type A), (b) the pure amplitude object (type B), and (c) the pure phase object (type C). The black dashed lines show the correct position.

Fig. 5. Autofocusing curves (left column) and their zoomed-in local counterparts (right column) for (a) the complex amplitude object (type A), (b) the pure amplitude object (type B), and (c) the pure phase object (type C). The black dashed lines show the correct position.

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To further compare these three criteria, simulations were performed with different recording distances. The search range for all the simulations was [ $d - 0.2$ , $d + 0.2$ ]. As shown in Table 1, both ROF and GoG perform very well and stably, with average errors of 0.0038 mm and 0.0071 mm, respectively. Although the errors are both within the depth of focus, ROF is superior in focusing accuracy, which is reflected in the fact that for the complex amplitude object (type A) and pure amplitude object (type B), ROF is perfectly able to find the exact recording distances, marked with light green; for the pure phase object (type C), their performances are comparable. Interestingly, ToG is very suitable for a pure phase object (type C) without any focusing error; however, it is not stable for a pure amplitude object (type B). In seven tests, two errors happen between 0.05 and 0.1 mm indicated in yellow, two errors are greater than or equal to 0.1 indicated in red, and there is another test marked with blue and an asterisk that fails to even obtain a global maximum near the best position, whose focusing curves are shown in Fig. 6. To quantify the performance of the three focusing criteria, we define the average focusing error as a figure of merit for focusing accuracy. According to the data in Table 1, the average focusing errors for ROF, GoG, and ToG are 0.0038, 0.0071, and 0.0205, respectively. If we consider the value of ROF to be 1, then the corresponding values of GoG and ToG are 1.87 and 5.39, respectively.

Table 1. Absolute Errors (mm) of Three Criteria on Different Object Types with Various Recording Distances

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Fig. 6. Autofocusing curves for the pure amplitude object (type B) with $d = 6 mm$ .

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5 Experimental Results

First, a dragonfly forewing shown in Fig. 7(d) was used to verify the proposed intensity correction algorithm and the sparsity-based autofocusing reconstruction method. Figure 7(a) illustrates the forewing’s nine normalized sub-holograms with intensity correction, and Fig. 7(b) presents the corresponding seamless synthetic aperture hologram whose intensities are uniform and continuous. As a comparison, Fig. 7(c) shows the synthetic aperture hologram without intensity correction, where the boundary and intensity difference between different tiles can be clearly observed. The effect of non-uniform intensity on reconstruction can be seen in Figs. 7(e) and 7(f), which are amplitude distributions reconstructed from synthetic aperture holograms with and without intensity correction, respectively, by our sparsity phase retrieval algorithm. As shown in Fig. 7(f), without intensity correction for the sub-holograms, the object part marked by the white dotted circle is missing, and the background part marked by the blue dotted circle is disturbed by an unwanted pattern. All the reconstruction results are scaled to [0,1] based on the range of reconstruction values for improving the display contrast.

Synthetic aperture hologram with intensity correction for a dragonfly forewing. (a) Nine normalized sub-holograms with intensity correction. (b) Synthetic aperture hologram composed of (a). (c) Synthetic aperture hologram without intensity correction. (d) Optical image of the dragonfly forewing sample. (e) Amplitude distribution reconstructed from (b) with 20 iterations. (f) Amplitude distribution reconstructed from (c) with 20 iterations. The effect of non-uniform intensity on reconstruction can be seen from the parts marked by the white and blue dotted circles.

Fig. 7. Synthetic aperture hologram with intensity correction for a dragonfly forewing. (a) Nine normalized sub-holograms with intensity correction. (b) Synthetic aperture hologram composed of (a). (c) Synthetic aperture hologram without intensity correction. (d) Optical image of the dragonfly forewing sample. (e) Amplitude distribution reconstructed from (b) with 20 iterations. (f) Amplitude distribution reconstructed from (c) with 20 iterations. The effect of non-uniform intensity on reconstruction can be seen from the parts marked by the white and blue dotted circles.

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Based on the synthetic aperture hologram in Fig. 7(b), the sparsity-based autofocusing reconstructions are shown in Fig. 8. Figure 8(a) shows the autofocusing curves, where the proposed ROF criterion reaches its minimum at $d = 14.55 mm$ , while the maximum positions of GoG and ToG happen at $d = 14.85 mm$ and $d = 14.93 mm$ , respectively. The corresponding complex amplitude reconstructions are shown in Figs. 8(b)–8(d). The reconstruction difference between Figs. 8(c) and 8(d) is not obvious, while the reconstruction in Fig. 8(b) shows more focused details based on visual judgment, especially in the regions marked by the dotted circles. As shown in Fig. 8(b), compared with the other two reconstructions, the veins and the cross vein in the white dotted circle are much clearer and sharper, the gap between the subcosta and the radius vein can be better resolved, and the structure in the blue dotted circle shows more details. In addition, the focusing curves of GoG and ToG have multiple local maxima, making the selection of focus position more hesitant, while the focusing curve of ROF has only one minimum without oscillations.

Sparsity-based autofocusing reconstruction for a dragonfly forewing with 20 iterations. (a) Autofocusing curves for three criteria. (b) Complex amplitude reconstruction at d=14.55 mm. (c) Complex amplitude reconstruction at d=14.85 mm. (d) Complex amplitude reconstruction at d=14.93 mm. The regions in (b) marked by the dotted circles show sharper and clearer details than their counterparts.

Fig. 8. Sparsity-based autofocusing reconstruction for a dragonfly forewing with 20 iterations. (a) Autofocusing curves for three criteria. (b) Complex amplitude reconstruction at $d = 14.55 mm$ . (c) Complex amplitude reconstruction at $d = 14.85 mm$ . (d) Complex amplitude reconstruction at $d = 14.93 mm$ . The regions in (b) marked by the dotted circles show sharper and clearer details than their counterparts.

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Next, to quantify the lateral resolution and show its enhancement for this synthetic aperture in-line holography, three resolution targets, all consisting of three aluminum strips on a silicon wafer, were used in the experiment. The widths of the strips, as well as the interval between two adjacent lines, for the three samples are $\sim 100$ , $\sim 80$ , and $\sim 70 μm$ , respectively, as shown in Fig. 9.

Optical images of the three samples for the resolution quantification. (a) 100 μm resolution target with a measured width and separation of ∼99 μm. (b) 80 μm resolution target with a measured width and separation of ∼77 μm. (c) 70 μm resolution target with a measured width and separation of ∼69 μm.

Fig. 9. Optical images of the three samples for the resolution quantification. (a) 100 μm resolution target with a measured width and separation of $\sim 99 μm$ . (b) 80 μm resolution target with a measured width and separation of $\sim 77 μm$ . (c) 70 μm resolution target with a measured width and separation of $\sim 69 μm$ .

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For the 100 μm resolution sample, the complex amplitude distributions reconstructed from the synthetic aperture hologram are shown in Figs. 10(a) and 10(b), and the reconstructions from only the center sub-hologram are given in Figs. 10(c) and 10(d). The intensity distributions along the black lines are displayed on the right side of each reconstruction. For better observation, only the local area around the pattern is presented. Obviously, the strips in Figs. 10(c) and 10(d) are completely unresolvable without aperture synthesis. In contrast, with the synthetic aperture method, those lines, as well as the corresponding peaks and valleys in the intensity curves, can be clearly distinguished in Figs. 10(a) and 10(b).

Reconstructions of the 100 μm resolution target with and without synthetic aperture. (a), (b) The amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations based on the synthetic aperture hologram. (c), (d) The amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations based on the center sub-hologram. On the right side of each reconstruction, the intensity distributions along the black lines are displayed.

Fig. 10. Reconstructions of the 100 μm resolution target with and without synthetic aperture. (a), (b) The amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations based on the synthetic aperture hologram. (c), (d) The amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations based on the center sub-hologram. On the right side of each reconstruction, the intensity distributions along the black lines are displayed.

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For the samples of 80 μm and 70 μm, their complex amplitude reconstructions by our sparse phase retrieval algorithm are given in Figs. 11 and 12, respectively. Three lines of both 80 μm and 70 μm can be easily distinguished, indicating that the lateral resolution of this THz synthetic aperture in-line holography system is better than 70 μm ( $\sim λ$ ).

Reconstructions of the 80 μm resolution target: the amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations.

Fig. 11. Reconstructions of the 80 μm resolution target: the amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations.

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Reconstructions of the 70 μm resolution target: the amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations.

Fig. 12. Reconstructions of the 70 μm resolution target: the amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations.

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The intrinsic resolution $R$ of an in-line holography system, according to the Abbe criterion, is limited by NA as follows: $R = \frac{λ}{2 NA} = \frac{λ}{2 \sin θ_{\max}},$ (18)where $θ_{\max}$ denotes the maximum diffraction angle. In the work presented here, the diffraction limit, for the direct measurement with center sub-aperture, amounts to $\sim 170 μm$ , with the shortest recording distance of $\sim 13.5 mm$ we can make in the experiment. The synthetic aperture method increased the effective detection area from $7.52 mm \times 5.64 mm$ ( $320 \times 240 pixels$ ) to $17.484 mm \times 15.604 mm$ ( $744 \times 664 pixels$ ) and thus enhanced the resolution to $\sim 69 μm$ , which is in good agreement with the experimental results.

Finally, three other biological samples, a beetle’s leg, a cicada’s wing, and a spider, as well as a semiconductor sample, a silicon wafer, were imaged with the proposed methods. Figure 13 shows the complex amplitude reconstructions of the biological samples, and Fig. 14 gives the reconstruction of the silicon wafer where the internal non-uniformity can be observed, showing the potential applications of THz synthetic aperture in-line holography in bioimaging and materials analysis.

Reconstructions of three biological samples with the proposed methods. (a) Complex amplitude reconstructions of a beetle’s leg. (b) Complex amplitude reconstructions of a cicada’s wing. (c) Complex amplitude reconstructions of a spider.

Fig. 13. Reconstructions of three biological samples with the proposed methods. (a) Complex amplitude reconstructions of a beetle’s leg. (b) Complex amplitude reconstructions of a cicada’s wing. (c) Complex amplitude reconstructions of a spider.

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Fig. 14. Reconstructed (a) amplitude and (b) phase distributions of a silicon wafer, where the internal non-uniformity can be observed.

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6 DISCUSSION AND CONCLUSION

We have demonstrated a high-resolution and high-quality THz synthetic aperture in-line holography system built on a self-developed THz-QCL with lateral resolution better than 70 μm ( $\sim λ$ ) at 4.3 THz. To correct the intensity difference between sub-holograms, a global optimization algorithm was proposed with which the reconstruction quality improvement is shown by experiment. Intensity correction is necessary before aperture stitching since many reconstruction algorithms require a uniform background. To overcome the twin-image problem for in-line holography, a sparsity-based iterative phase retrieval algorithm was used to give high-quality reconstructions. Moreover, to obtain the best in-focus reconstruction distance, a new autofocusing criterion based on the “reconstruction objective function” was introduced into in-line holography for the first time, so the autofocusing procedure and the reconstruction are unified within the same framework. Both simulation and experiment showed its accuracy and robustness. Since iterative reconstruction is performed for each candidate distance during autofocusing, it may be a little time consuming depending on the number of iterations. According to Gabor, reliable reconstruction for in-line holography can be obtained if the object distribution in the illuminated field is sparse [34]. This is why sparsity can be exploited for reconstruction, and it also implies the limitation of sample size for in-line holography. For the extended samples, ptychography or multi-height [35] methods based on multiple diffraction patterns can be adopted. When the $L_{1}$ -based sparse model in this paper is used to perform the reconstruction, due to the soft-thresholding operator, the regions with weak object waves, such as the transparent membrane pieces between veins, may be eliminated along with the twin-image artifacts as shown in Figs. 8 and 13(b). Using constraints that are not directly imposed on the amplitude distribution, such as total variation, may be a way to alleviate this problem. The proposed autofocusing reconstruction model is based on the 2D object; when dealing with the 3D volume object, there will be multiple local minima related to the targets at different depths in the autofocusing curve. However, the diffraction patterns from out-of-focus objects will disturb the reconstruction at any given depth with the current 2D reconstruction model. Therefore, for 3D objects, the current model should be extended, and the corresponding autofocus criterion can be more optimized.

Overall, a complete solution for THz high-resolution complex amplitude imaging was presented. Note that the proposed approaches can be applied to other wavebands as well, such as visible light and X-ray band. We demonstrated its success on biological samples and a semiconductor silicon wafer, showing the potential applications of THz synthetic aperture in-line holography in bioimaging and materials analysis.

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1 INTRODUCTION

2 EXPERIMENTAL SETUP

3 APERTURE SYNTHESIS WITH INTENSITY CORRECTION

4 SPARSITY-BASED AUTOFOCUSING RECONSTRUCTION

4.1 A. Sparse Phase Retrieval

4.2 B. Autofocusing Algorithm

5 Experimental Results

6 DISCUSSION AND CONCLUSION

Zeyu Li, Ruijiao Zou, Weipeng Kong, Xuemin Wang, Qinghua Deng, Qiang Yan, Yu Qin, Weidong Wu, Xun Zhou. Terahertz synthetic aperture in-line holography with intensity correction and sparsity autofocusing reconstruction[J]. Photonics Research, 2019, 7(12): 12001391.

Terahertz synthetic aperture in-line holography with intensity correction and sparsity autofocusing reconstruction Download： 810次

1 INTRODUCTION

2 EXPERIMENTAL SETUP

3 APERTURE SYNTHESIS WITH INTENSITY CORRECTION

Fig. 2. Schematic diagram of the aperture synthesis. (a) Four potential overlapped regions within a sub-hologram. (b) Synthetic aperture hologram with $3 \times 3$ tiles numbered by 1–9.

4 SPARSITY-BASED AUTOFOCUSING RECONSTRUCTION

4.1 A. Sparse Phase Retrieval

Fig. 3. Iterative sparse reconstruction scheme. $SA_H_{N}$ is the normalized synthetic aperture hologram, which is composed of the normalized sub-holograms with intensity correction.

4.2 B. Autofocusing Algorithm

Fig. 4. Simulated three types of objects with a THz pattern. (a) Type A: complex amplitude object. (b) Type B: pure amplitude object. (c) Type C: pure phase object.

Fig. 5. Autofocusing curves (left column) and their zoomed-in local counterparts (right column) for (a) the complex amplitude object (type A), (b) the pure amplitude object (type B), and (c) the pure phase object (type C). The black dashed lines show the correct position.

Table 1. Absolute Errors (mm) of Three Criteria on Different Object Types with Various Recording Distances

Fig. 6. Autofocusing curves for the pure amplitude object (type B) with $d = 6 mm$ .

5 Experimental Results

Fig. 11. Reconstructions of the 80 μm resolution target: the amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations.

Fig. 12. Reconstructions of the 70 μm resolution target: the amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations.

Fig. 13. Reconstructions of three biological samples with the proposed methods. (a) Complex amplitude reconstructions of a beetle’s leg. (b) Complex amplitude reconstructions of a cicada’s wing. (c) Complex amplitude reconstructions of a spider.

Fig. 14. Reconstructed (a) amplitude and (b) phase distributions of a silicon wafer, where the internal non-uniformity can be observed.

6 DISCUSSION AND CONCLUSION

Article Outline

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Terahertz synthetic aperture in-line holography with intensity correction and sparsity autofocusing reconstruction Download： 810次

1 INTRODUCTION

2 EXPERIMENTAL SETUP

3 APERTURE SYNTHESIS WITH INTENSITY CORRECTION

Fig. 2. Schematic diagram of the aperture synthesis. (a) Four potential overlapped regions within a sub-hologram. (b) Synthetic aperture hologram with 3×3 tiles numbered by 1–9.

4 SPARSITY-BASED AUTOFOCUSING RECONSTRUCTION

4.1 A. Sparse Phase Retrieval

Fig. 3. Iterative sparse reconstruction scheme. SA_HN is the normalized synthetic aperture hologram, which is composed of the normalized sub-holograms with intensity correction.

4.2 B. Autofocusing Algorithm

Fig. 4. Simulated three types of objects with a THz pattern. (a) Type A: complex amplitude object. (b) Type B: pure amplitude object. (c) Type C: pure phase object.

Fig. 5. Autofocusing curves (left column) and their zoomed-in local counterparts (right column) for (a) the complex amplitude object (type A), (b) the pure amplitude object (type B), and (c) the pure phase object (type C). The black dashed lines show the correct position.

Table 1. Absolute Errors (mm) of Three Criteria on Different Object Types with Various Recording Distances

Fig. 6. Autofocusing curves for the pure amplitude object (type B) with d=6 mm.

5 Experimental Results

Fig. 11. Reconstructions of the 80 μm resolution target: the amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations.

Fig. 12. Reconstructions of the 70 μm resolution target: the amplitude and phase-contrast distributions reconstructed by the sparse phase retrieval algorithm with 20 iterations.

Fig. 13. Reconstructions of three biological samples with the proposed methods. (a) Complex amplitude reconstructions of a beetle’s leg. (b) Complex amplitude reconstructions of a cicada’s wing. (c) Complex amplitude reconstructions of a spider.

Fig. 14. Reconstructed (a) amplitude and (b) phase distributions of a silicon wafer, where the internal non-uniformity can be observed.

6 DISCUSSION AND CONCLUSION

Article Outline

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Fig. 2. Schematic diagram of the aperture synthesis. (a) Four potential overlapped regions within a sub-hologram. (b) Synthetic aperture hologram with $3 \times 3$ tiles numbered by 1–9.

Fig. 3. Iterative sparse reconstruction scheme. $SA_H_{N}$ is the normalized synthetic aperture hologram, which is composed of the normalized sub-holograms with intensity correction.

Fig. 6. Autofocusing curves for the pure amplitude object (type B) with $d = 6 mm$ .