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Abstract
We demonstrate terahertz (THz) lens-free in-line holography on a chip in order to achieve 40 μm spatial resolution corresponding to ~0.7λ with a numerical aperture of ~0.87. We believe that this is the first time that sub-wavelength resolution in THz holography and the 40 μm resolution were both far better than what was already reported. The setup is based on a self-developed high-power continuous wave THz laser at 5.24 THz (λ = 57.25 μm) and a high-resolution microbolometer detector array (640 × 512 pixels) with a pitch of 17 μm. This on-chip in-line holography, however, suffers from the twin-image artifacts which obfuscate the reconstruction. To address this problem, we propose an iterative optimization framework, where the conventional object constraint and the L1 sparsity constraint can be combined to efficiently reconstruct the complex amplitude distribution of the sample. Note that the proposed framework and the sparsity-based algorithm can be applied to holography in other wavebands without limitation of wavelength. We demonstrate the success of this sparsity-based on-chip holography by imaging biological samples (i.e., a dragonfly wing and a bauhinia leaf).
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Attribute to the unique properties of terahertz (THz) radiations, THz imaging is a cutting-edge nondestructive evaluation technique for applications including biomedicine, security, artwork, etc [1]. Due to the long wavelength of the THz wave, the imaging resolution is usually at the level of hundreds of micrometers or even millimeters, which has become an important factor limiting its application in many fields. Several efforts have been made to enhance the resolution to sub-wavelength scale, including near-field scanning [2], super-resolution focusing with dielectric metalens [3] or dielectric cube [4], aspheric or solid immersion lens [5,6], etc. Near-field scanning and super-resolution focusing need long time scanning, meanwhile, near-field probe requires a wavelength scale working distance. The lens-based scheme enables real-time imaging, but only intensity image can be obtained, which is not suitable for phase objects. Performing full-field, real-time, and complex amplitude THz imaging on a sub-wavelength scale remains challenging.
THz digital holography, a combination of THz technology and digital holography, is a competitive technique to address this problem. It is an essential full-field, real-time imaging approach, which can simultaneously reconstruct the amplitude and phase distributions of the sample. Benefiting from the compact and lens-free structure, it has a potential of sub-wavelength resolution with a large numerical aperture (NA). During the past several years, THz digital holography has received more and more attentions [7–15]. Rong et al. utilized an optically pumped THz laser (OPTL) and a pyroelectric array detector to perform THz Gabor in-line holographic measurements at 2.52 THz. An iterative self-extrapolation post-processing method was adopted to enhance lateral resolution and suppress twin-image. Based on this, the 35 μm width cross veins of a dragonfly hindwing were reconstructed [9] and the fibrosis in human hepatocellular carcinoma tissue was observed in the phase distribution [10], which demonstrated the potential of THz digital holography for biomedical applications. With the same detector and THz source, subpixel sampling method was applied to break through the limitation of the detector pixel pitch, and then imaging resolution of 150 μm (1.26λ) was achieved on a pitch of 100 μm [11]. To expand the NA of the detector array, the synthetic aperture method was implemented to improve lateral resolution [8,12,14]. By use of a high-resolution (640 × 480 pixels) bolometer detector array with a pitch of 25 μm, Zolliker et al. realized a lateral resolution of 200 μm (1.68λ) in off-axis reflection [8]. Huang et al. showed a resolution of 125 μm (1.29λ) at 3.1 THz in in-line configuration based on a THz quantum cascade laser (QCL) and an uncooled microbolometer THz detector [12]. Later, Deng et al. pushed the lateral resolution to a wavelength level of about 70 μm at 4.3 THz [14].
In this work, we demonstrate THz lens-free in-line holography on a chip with an NA of ~0.87 and a spatial resolution better than 0.7λ, which, to the best of our knowledge, is the first time to achieve sub-wavelength resolution in THz holography. Although the on-chip holographic structure has been widely used for microscopy over a large field-of-view in the visible range [16], there is still no report on the THz band. The setup here is based on a self-developed high-power continuous wave (CW) THz laser and a microbolometer detector array (640 × 512 pixels) with a pitch of 17 μm. The working frequency is at 5.24 THz, making the imaging resolution reach 40 μm corresponding to a wavelength of 57.25 μm, which may greatly promote THz imaging applications in biomedical, material characterization, and others.
This on-chip in-line holography, however, suffers from the twin-image artifacts due to the lack of phase information in the hologram [17]. The influence of the twin-image is inversely proportional to the recording distance, so for our case, the contrast between the real image and its twin is low at a small recording distance. The most common numerical approach for this ill-posed inverse problem is finite-support based iterative phase retrieval. It requires a mask to estimate the outline of the object, which brings new challenges. Latychevskaia et al. proposed a novel method employing positive absorption constraint instead of the object support [18]. The original algorithm is based on a normalized hologram under the assumption that the background is 1, which is not valid for samples on the substrate and for the case where the laser power is unstable. So a correction factor needs to be applied to the normalized hologram [10]. Recent years, compressive sensing has been introduced to in-line holography, where sparsity was explored to reconstruct the object via optimization procedures [19–25]. Denis et al. proposed a reconstruction model based on the L1 norm [19]. Brady et al enforced a sparsity on total variation based on an approximation model [20], which was then first applied to THz in-line holography by Li et al. [21]. But all of above were restricted to the real domain. Song et al. developed a two-constraint iterative scheme to find the optimal solution of the L0 norm which is typically NP-hard, so it might be very time-consuming for large images [22]. Rivenson made use of sparsity in multi-height phase recovery [23]. Lately, Zhang et al. proposed a physics-driven model on the total variation sparsity constraint, which was proved effective and powerful [25].
Here, we regard in-line holography reconstruction as a constrained minimization problem and propose an iterative optimization framework. Alternating minimization method is used to estimate the missing phase of the hologram, as well as to obtain the optimal solution of the object wavefront. Within this framework, the classical error reduction (ER) algorithm [26] can be well explained, and conventional object constraints, such as positive absorption, can be combined with sparsity constraint to further improve reconstruction accuracy and convergence speed.
This paper is organized as follows. Section 2 presents the experimental setup of the THz on-chip in-line holography. In section 3, an iterative optimization reconstruction framework is established and a sparsity-based multiple constraints algorithm is proposed to suppress the twin-image artifacts. Section 4 experimentally demonstrates the success of the sparsity-based THz sub-wavelength on-chip holography and section 5 makes a conclusion.
2. Experimental setup
A schematic diagram and a real picture of the THz lens-free on-chip holography setup are depicted in Fig. 1. The source is an OPTL developed by ourselves. CH3OD is chosen and pumped by a CW 9R(8) CO2 laser to generate a 5.24 THz CW beam corresponding to a wavelength of 57.25 μm. The output power is about ~40 mW with a beam diameter of ~10 mm. The detector is an uncooled microbolometer array (IRay Technology Co., Ltd., China) optimized for 0.7–5 THz with 640 × 512 pixels on a pitch of 17 μm; thus, the total size of the screen is 10.88 mm × 8.704 mm. The THz beam is expanded and collimated via two gold-coated confocal off-axis parabolic mirrors (PM1 and PM2) with the focal length of 50.8 mm and 101.6 mm, respectively. Next, the beam direction is changed by a reflecting mirror (M) to vertically align the array detector which is placed flat on an optical table. The sample is placed directly on the detector window so that the recording distance is as close as ~3 mm corresponding to a maximum detection NA of ~0.87. This is typical in-line holography where the reference wave and the object wave are not separate beams but a unique one. The wave scattered by the sample forms the object wave which interferes with the unscattered part of the illumination wave. By recording the interference pattern with a THz detector array, we obtain the hologram. Note that the Nyquist sampling theorem can be fulfilled when recording holograms with a wavelength of 57.25 μm and a 17 μm detector pitch, so the problem that the imaging resolution is limited by pixel pitch in the visible range does not exist in our setup, and the effective NA is the same as the detection NA. According to Rayleigh’s criterion (0.61λ/NA), the diffraction limit is ~40 μm. During the experiment, a hologram and an illumination background image are recorded with and without sample respectively, and then a normalized hologram can be obtained to suppress the non-uniform illumination background by dividing the hologram image by the background image. Note that this step is essential unless the intensity of illumination is uniform. All the reconstruction routines in this study are based on the normalized holograms.
Fig. 1 Schematic layout (a) and real picture (b) of the experiment setup. PM1 and PM2 are gold-coated confocal off-axis parabolic mirrors with the focal length of 50.8 mm and 101.6 mm, respectively. An output THz laser beam of ~10 mm in diameter is expanded and collimated to ~20 mm so that the detector array can be completely covered by THz wave. The wave scattered by the sample forms the object wave while the unscattered part of the illumination forms the reference wave. The resulting interference pattern is called an in-line hologram.
3. Sparsity-based reconstruction algorithm
Since the phase information is lost during hologram recording, the holographic reconstruction can be regarded as a phase retrieval problem. Here, an optimization framework, as well as a sparsity-based multiple constraints algorithm, are presented to solve this inverse problem. The estimation of the hyperparameter is also given.
3.1 Classical model
The conventional phase retrieval is an iterative procedure of field propagation back and forth between recording and object plane, where the magnitude of the normalized hologram is used as the constraint in the detector plane while object constraint, such as a finite-support, is used as the constraint in the object plane. This classical phase retrieval can also be formulated as a constrained minimization problem [27]:
where H is the magnitude of the normalized hologram, T(z) is the point spread function of diffraction propagation over a recording distance z, X ∈ Cm × n is the wavefront at the object plane, * denotes a 2D convolution and Ps denotes the projection operator for object constraint. In this work, the angular spectrum (AS) method is adopted to calculate diffraction propagation for its validity at a short distance. Inspired by the method in [24], we express the problem in Eq. (1) equivalently aswhere ʘ is the element-wise product of the matrix entries and W ∈ Cm × n corresponds to the missing phase information. Alternating minimization can be used to solve Eq. (2) iteratively by minimizing the following two subproblems sequentially in each iteration: The closed-form solutions for Eqs. (3) and (4) can be expressed as This is exactly the classical ER algorithm by Gerchberg and Saxton [26].3.2 Sparsity-based basic (SBB) model
According to Gabor, not more than about 1% of the illuminated field should be covered by the sample [28]. When performing an iterative reconstruction, this ratio can be relaxed to 25% [29], as required in coherent diffractive imaging. Therefore, reliable reconstruction for Gabor in-line holography requires wavefront at the object plane to be sparse, which can be used as an object constraint in reconstruction. Here we follow the sparse model proposed by Haeffele [24]. The sparsity of the signal is measured by L1 norm, and an additional term μ ∈ C corresponding to the uniform background is split from the wavefront at the object plane so that the value of the pixel that doesn’t contain an object is 0. This results in the reconstruction model expressed as
where τ is a hyperparameter that weight the relative contribution of the L1 norm term.Based on alternating minimization, the closed-form updates for all the variables are as follows:
where denotes the inner product operator and SFTτ denotes the complex soft-thresholding operator [30], given byIf the phase of illumination at object plane is further assumed to be zero, Eq. (8) can be replaced byAnd from Eq. (11), one could see that, whether a pixel is a background is determined by the hyperparameter τ which is tuned by running multiple trials in reconstruction and choosing an optimal one. The baseline of τ, in our work, is estimated from the first iteration, given by3.3 Sparsity-based multiple constraints (SBMC) model
The models in Sec. 3.1 and Sec. 3.2 are established on the same framework, within which the sparsity constraint and the conventional object constraint can be combined, resulting in our final model:
The solutions for Eq. (14) is the same as the ones for Eq. (7) except that Eq. (10) is updated byIn our implementation, L1 norm and positive absorption constraint without finite-support are applied in reconstruction, which is performed by sequentially updating the variables described by Eq. (8), Eq. (9) and Eq. (15) from initializing with X = 0, μ = 0 and W = 1. The projection operator based on positive absorption follows the work by Latychevskaia [18] with the background of μ instead of 1 in the original work. The hyperparameter τ can be relaxed to values smaller than what is given by Eq. (13) in this multiple constraints model, making the parameter tuning more easily. Specifically, a factor of 0.5 is imposed on the parameter τ in our reconstruction. Typically, the reconstruction result will converge after 10-20 iterations. Figure 2 illustrates the complete process of the algorithm. It is worth mentioning that this algorithm is not limited by wavelength, so it can be also used in other wavebands, such as visible light, x-ray, etc.
Fig. 2 Flow chart of the SBMC algorithm. Projection operator Ps is referred to positive absorption constraint in our reconstruction.
3.4 Simulation
Numerical simulation was carried out to verify the feasibility of the SBMC algorithm. The pixel pitch and wavelength were set to 17 μm and 57.25 μm respectively, which were the same as those in experiments. Figures 3(a) and 3(b) shows a ‘THz’ pattern used as complex amplitude object in simulation, whose hologram was generated by AS method, as shown in Fig. 3(c). The illumination at object plane was assumed to be 1.
Figure 4 illustrates the results reconstructed by conventional AS backpropagation method, SBB method, and SBMC method respectively. It is obvious that the reconstruction by the conventional AS method suffers from the out of focus twin-image artifacts. After 20 iterations, twin-image artifacts still exist by the SBB method, while they have been completely removed by the SBMC method and the recovered distributions are very consistent with the original ones. After 50 iterations with the SBB method, the twin-image artifacts are also eliminated, but the reconstruction accuracy is not as good as the one with the SBMC method. The mismatch between the original complex amplitude distributions (t0 and φ0) and the recovered ones (t1 and φ1) are estimated by the errors [29]:
as shown in Fig. 5, which indicate that the SBMC method is superior to the SBB method in both reconstruction accuracy and convergence speed.Fig. 4 Reconstruction results. (a), (e) Distributions of amplitude and phase by conventional AS backpropagation method. The twin-image artifacts marked by the red arrow are very serious. (b), (f) Distributions of amplitude and phase by SBB method after 20 iterations. The twin-image artifacts still exist around the sample. (c), (g) Distributions of amplitude and phase by SBB method after 50 iterations. (d), (h) Distributions of amplitude and phase by SBMC method after 20 iterations. The recovered distributions are completely twin-image free and very consistent with the original ones.
Fig. 5 Amplitude error (a) and phase error (b) between the simulated object and the recovered one with SBB method and SBMC method respectively.
4. Results and discussion
To quantify the spatial resolution of this on-chip holography, we fabricated resolution targets of 50 μm and 40 μm consisting of three aluminum lines on a quartz slide, with the same width and separation. The digital optical microscopic images (10 × ) and their reconstructions by SBMC method after 20 iterations are given in Fig. 6. Three lines can be distinguished, meaning that the resolution of the setup can achieve a sub-wavelength scale of 40 μm, which is consistent with the theoretical value. Due to the instability of the THz source and the interference between the sample and the detector window, the background is not well normalized. This problem may be avoided by further optimizing the THz laser and sample placement.
Fig. 6 Optical microscopic images (10 × ) and corresponding reconstruction of resolution targets by SBMC method with 20 iterations. (a), (d) Optical microscopic images (10 × ) of 50 μm and 40 μm resolution targets. (b), (c) Reconstructed complex amplitude distributions of 50 μm resolution targets. (e), (f) Reconstructed complex amplitude distributions of 40 μm resolution targets. Three lines can be clearly distinguished.
Next, we experimented on a dragonfly wing, whose optical microscopic image, normalized hologram, and reconstructions at z = 2.41 mm are shown in Fig. 7. The width of the costa, radius vein, median vein, cross veins and secondary veins measured from the microscopic image are approximate 60-200 μm, 50-70 μm, 40-50 μm, 35-40 μm, and 25-30 μm, respectively. The focus region is marked in Fig. 8(a). Compared with the conventional AS backpropagation method, reconstruction quality is improved by SBB method, but there are still some bright twin-image artifacts around the object. Benefiting from the positive absorption constraint, the parts with the amplitude greater than background are eliminated, so that the reconstruction quality, as well as the convergence speed, are further improved by SBMC method. Since the sample is not completely flat, the best reconstruction distances for different parts are slightly different, as shown in Fig. 8. When reconstruction distance z = 2.41 mm, the costa, radius vein, median vein, and cross veins are very clear (Fig. 8(a)), while pterostigma is in focus at z = 2.53 mm (Fig. 8(b)). In addition, the network structures of the secondary veins whose width are about 30 μm, are observed in Figs. 8(b) and 8(c), although the contrast is relatively low.
Fig. 7 A dragonfly wing and its constructions at z = 2.41 mm. (a), (e) Optical microscopic image (5 × ) and the normalized hologram. (b), (f) Reconstructed complex amplitude distributions by conventional AS backpropagation method. The obvious out of focus twin-image artifacts are marked by red arrows. (c), (g) Reconstructed complex amplitude distributions by SBB method after 50 iterations. The reconstruction quality is enhanced, but some bright artifacts around the object still exist. (d), (h) Reconstructed complex amplitude distributions by SBMC method after 20 iterations. Benefiting from the positive absorption constraint, the reconstruction quality, as well as the convergence speed, are further improved.
Fig. 8 Reconstructed phase distributions at z = 2.41 mm (a), z = 2.53 mm (b) and z = 2.60 mm (c), respectively. The regions marked by the red dotted lines are in focus.
Finally, a bauhinia leaf was imaged. Figure 9 shows the leaf’s optical microscopic image (Fig. 9(a)), normalized hologram (Fig. 9(e)) and the complex amplitude distributions constructed by conventional backpropagation AS method (Figs. 9(b) and 9(f)), SBB method (Figs. 9(c) and 9(g)), and SBMC method (Figs. 9(d) and 9(h)) respectively. Obviously, the proposed SBMC method performs better than the other two methods. As shown in Figs. 9(d) and 9(h), the twin-image artifacts are well suppressed, the contour of the leaf is clearer, and the leaf veins are easier to identify. But from the hologram and the reconstructions, we could see that the transmission of the leaf is not high at 5.24 THz. Although the high-frequency THz can bring the resolution improvement, its penetration also decreases. Therefore, in practical THz applications, comprehensive trade-offs should be made in terms of resolution, penetrability, and thickness selection during sample preparation, etc.
Fig. 9 A bauhinia leaf and its constructions at z = 2.2 mm. (a), (e) Optical microscopic image (5 × ) and the normalized hologram. (b), (f) Reconstructed complex amplitude distributions by conventional AS backpropagation method. The reconstruction is disturbed by the unwanted twin-image artifacts. (c), (g) Reconstructed complex amplitude distributions by SBB method after 50 iterations. There are still some bright artifacts. (d), (h) Reconstructed complex amplitude distributions by SBMC method after 20 iterations. The twin-image artifacts are well suppressed, and the contour of the leaf is very clear.
5. Conclusion
In conclusion, we demonstrated that based on a self-developed high-power CW THz laser at 5.24 THz (λ = 57.25 μm) and a high-resolution bolometer detector array with a pitch of 17 μm, a THz on-chip holography setup was implemented and 40 μm sub-wavelength resolution corresponding to ~0.7λ was achieved for the first time. This is the best resolution of THz digital holography to date. In addition, to suppress the twin-image artifacts, a constrained optimization framework was established and alternating minimization method was employed to iteratively reconstruct the complex amplitude distribution of the object wavefront. The classical ER algorithm was well explained within this framework. Based on it, we proposed a sparsity-based multiple constraints algorithm by combining conventional object constraint and L1 norm, and the estimation of the hyperparameter was also given. The feasibility of the algorithm is verified by simulation. The sparsity constraint requires that the value of the pixel that doesn’t contain an object is a uniform background. However, the illumination in the experiment is usually not uniform, and in this case, the hologram normalization before applying the proposed algorithm is necessary. At the same time, as required in coherent diffractive imaging, the ratio of the area occupied by the object in the image should not exceed 0.25. Note that the proposed framework and the SBMC algorithm can also be used for holography in visible light, or x-ray, etc. Finally, we demonstrated the success of this sparsity-based on-chip holography by imaging biological samples of a dragonfly wing and a bauhinia leaf, and the structures of the ~30 μm wide secondary veins were observed. Overall this sub-wavelength on-chip holography, as well as the sparsity-based phase retrieval algorithm may open new opportunities towards THz imaging applications in biomedical, material characterization, and others.
Funding
National Natural Science Foundation of China (NSFC) (61575031, 61605184).
Acknowledgments
Authors acknowledge IRay Technology Co., Ltd. for providing a high-resolution THz detector array. The authors also acknowledge Xuemin Wang and Zhiqiang Zhan for their kind assistance with the resolution samples.
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