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Abstract
The terahertz structured beams played a significant role in imaging. We utilized the transmitter with 0.1 THz to generate the quasi-Pearcey beam. The beam is produced by combining the self-designed parabola-slit modulated plate and Fourier lens, showing stripe-shaped pattern and self-focusing property. Based on that, introducing it into the testing of ptychography, we discovered there are decent effects in field reconstruction of the probe and sample with this beam by comparisons both in the simulations and the experiments. The beam has good spatial coherence through the analysis of the spatial frequency spectrums. It suggests that the beam with such features can take advantage of rapid reconstruction in full-field imaging.
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1. Introduction
The modulation of structured beams has attracted a lot of research interest. Owing to the specific distributions of structured beams, they show potential application in many aspects [1–4], e.g., micro-particle manipulating, optical microscopy, imaging, communication, and so on. In 1987, Durnin first realized the Bessel beam [5], which belongs to the family of nondiffracting beams, whose intensity distributions almost hold invariant as the propagating distance increases. The finding of the Bessel beam paved the way for research on structured beams. Subsequently, Vortex, Airy, and Mathieu beams were produced successively [6–11]. These beams either hold the property of nondiffracting or bear unique structured distributions, which aroused much more detailed poring. Actually, the Airy beam originated from the solution of wave equation within paraxial approximation, which is a diffraction pattern of caustic in +1 dimension based on catastrophe optics [12]. In the same condition, another beam emerged, the Pearcey beam, corresponding to the diffraction distribution from caustic of 1 + 1 dimensions [13]. Like the Airy beam, the Pearcey beam includes self-healing property, and its transverse distribution is characteristic of symmetry on the vertical axis. The Pearcey beam presents energy convergence at a specific position. It bears a depth of field due to the self-focusing feature, which could provide worthwhile application in particle manipulation and optical imaging.
The structured beams are deeply investigated in the optical band. Terahertz structured beams sprang up gradually with the development of THz technology. Terahertz structured beams have excellent imaging and communication potential since they bear special field distributions and THz features. In the THz region, Bessel, Airy, and Mathieu beams are produced successively, and their imaging effect is studied [14–18]. The Pearcey beam has not been explored in the terahertz band; thus, it is worth probing into the property of the THz Pearcey beam.
The work on imaging of terahertz structured beams has been extended gradually in recent years due to the high penetration efficiency of THz waves. These include transmission intensity imaging and full-field imaging related to computed imaging [19–23]. Full-field imaging is a noteworthy imaging mode that can reconstruct the topography of an object with information on both amplitudes and phases. As a type of full-field imaging, Ptychography also belongs to coherent diffractive imaging (CDI) [24,25]. It collects a set of diffraction patterns generated from overlapped scanning of object at different positions. It combines with computational algorithms to reconstruct the complex amplitude both in object and probe. This technique has also been explored in the THz band because of the advantages of the system that works in reference-less and lens-less modes [26–28]. After that, Xiang et al. implemented the long-distance diffraction-free beam into the Ptychography testing and acquired decent results [29].
In this paper, the terahertz Pearcey beam was generated at 0.1 THz by self-designing diffractive elements, amplitude-modulated plate, and Fourier lens, respectively. Combining simulations and experiments, we studied the intensity distributions of the generated beams. It is found that the Pearcey beam presents an apparent in-line striped structure at the self-focusing position, maintaining a certain depth of field to this position before and after. Based on that, the reconstructing effect of ptychography with the Pearcey beam has been investigated, and decent results have been obtained. Compared with the ptychography of the THz dot-spot beam, it appears in a good and fast reconstructing mode in that of the Pearcey beam.
2. Theory and design
The transverse field distribution of an ideal Pearcey beam follows the form of the Pearcey function, which can be written as follows [13]:
where the X and Y transverse dimensionless variables to the propagation direction z, and s denotes the variable in complex space. In free space, the Pearcey function can also be expressed as Pe (x/x0, y/y0), where x0 and y0 represent scale factors along x and y, respectively. The spectrum of the Pearcey beam can be obtained as Eq. (2) through acting on Pe(x/x0, y/y0) by Fourier transform [30].Here, kx and ky correspond to the spectrums of the x and y directions, and δ denotes Dirac function. The frequency spectrum of the Pearcey function is a trajectory along the parabola $k_x^2x_0^2 = {k_y}{y_0}$, which is modulated by the factor $\textrm{exp} ({ik_x^4x_0^4} )$. The spectrum according with parabola was produced through a spatial light modulator (SLM) by Ring, which was transformed via a 4f system to generate the Pearcey beam [13].
However, there is no adaptable SLM for generating the Pearcey beam in the terahertz band. Inspired by the previous works, we combined the self-designed static parabola-slit modulation plate and the Fourier lens to create the terahertz Pearcey beam. The related schematic diagram is shown in Fig. 1.
The plane wave beam modulated by the parabola-slit plate can be approximated as a parabola wave source, which may form the Pearcey beam through a Fourier lens. Assume that p is the parameter of the parabola. The parabola wave source can be expressed in the form of Eq. (3). After the parabola beam passes through the lens, with focal length f, its field distribution at a specific position can be described by Collins matrix [31] (seen in Eq. (4)), whose expression can be simply written as Eq. (5) (the detailed derivation is listed in Supplement 1).
In Eq. (5), ξ and η represent the coordinate position of the source; x and y are the positions of the receiving plane; z denotes the distance (it starts from the parabola-slit plate) that is defined as z = L + f (L is the distance from the lens); λ is the wavelength, and k = 2π/λ is wave-vector. It is manifest that Eq. (5) presents the form of the Pearcey function. In particular, when L = f (z = 2f), it can be seen that the field distribution of the Pearcey beam evolute into the form of Eq. (6). Simultaneously, the energy of the beam increases abruptly, and the position (z = 2f) is the self-focusing point of the beam.
However, it is impossible to realize an infinite narrowed parabola wave source. Here, the diffractive plate we used is an amplitude-modulated element notched with a finite width parabola-slit. It is assumed that the parameter of a parabola is p, and the center slit width is d. The parabola-slit can be described via two parabolic curves, both y1 and y2, as expressed in Eq. (7). Hence, the transmittance function of the plate is T(x,y), shown in Eq. (8). The material of the parabola-slit plate is aluminum alloy; the Fourier lens is fabricated by 3D-printed technology. The material used is LY1001 ABS (acrylonitrile butadiene styrene), a photosensitive resin, and the lens refractive index and absorption coefficient are 1.655 and 0.49 cm-1, respectively. Such a diffractive lens is designed by the principle of phase folding, which can reduce the absorption of beam energy and is equivalent to the corresponding refractive element in terms of the optical path. The beam produced from the system, including the two elements mentioned above, can be defined as a quasi-Pearcey beam.
3. Generation and property of quasi-Pearcey beam
The setup for generating the THz quasi-Pearcey beam is shown in Fig. 2. The transmitter is comprised of InP Gunn diode (GKa-100, SPACEK LABS) and horn antenna, delivering a 0.1 THz continuous wave with an output power of 25 mW. The original beam through a planoconvex lens, with a focal length of f = 100 mm, is shaped into a collimated beam, whose waist radius is ω0 = 30 mm. The collimated beam sequentially passes through the parabola-slit plate and Fourier lens to be converted into the quasi-Pearcey beam. The Fourier lens is shaped into a phase diffractive element with a focal length of f = 100 mm, and the lens is placed behind the plate of 100 mm. The receiver, a Schottky diode joint with a horn antenna, detects the generated Pearcey beam. The receiver is mounted on a three-dimensional motorized translation stage, whose scanning scope is 100 mm × 100 mm in the transverse plane (xy plane) and 100 mm × 200 mm in the xz and yz planes, respectively.
Here, this terahertz system optical power output efficiency is approximately 13.7% to 25.57%, primarily determined by the parameter p of the parabola-slit. However, compared to systems generating optical Pearcey beams, it exhibits a noticeable improvement in optical efficiency. This is due to avoiding the significant absorption and losses of optical power caused by SLM and the lengthy 4f mode present in optical light system.
Fig. 2. Schematic of the experimental setup for the generation of the quasi-Pearcy beam. The insert: the normalized intensity distribution of the collimated beam.
The parabola-slit plates, including three different p parameters, produce the quasi-Pearcey beams for comparison. The three plates are shown in Fig. 3, where we set the slit width of the center in a plate is 24 mm and the plate thickness is 2 mm. The origin of coordinates (0,0,0) is set in the center of the slit in the plate; the starting point of detection is set in z = 30 mm because the actual photosensitive plane is placed in the vertex of the horn, which is about 30 mm from the mouth of the horn. The xz plane normalized intensity distributions of the quasi-Pearcey beam behind the Fourier lens are demonstrated in Fig. 4, including the results of experiments and simulations. From the simulated results displayed in Fig. 4(a), it can be seen that the intensity distributions of the beams demonstrate profiles similar to parallel beams. Observing the process of beam propagation from the original position (z = 30 mm), the profiles of the beams present a trend of convergency, showing the concentration of the beam energy. When the beams access a particular position, it is the limitation of the convergency. This position is the self-focusing site, almost located in z = 95 mm, approximately to the back focus of the Fourier lens. Moreover, as the p value decreases, the beam shape gradually extends along the x-direction. Because as the value of p decreases, the gap of the parabola-slit becomes smaller, and the opening of the parabola narrows (analyzed in analogy to double-slit diffraction). These changes lead to an enhancement of diffraction effects, resulting in an expansion of the spatial distribution of the beam along the x-direction. From the experimental results in Fig. 4(b), it can be observed that the intensity distributions show some quantitative discrepancies compared to the simulation results, also with the self-focusing position nearly at z = 100 mm. However, the evolution patterns roughly match with the simulations.
Fig. 4. The normalized intensity distributions of the generated quasi-Pearcey beam in xz plane. (a) Simulated results; (b) Experimental results.
Figure 5 shows the intensity distributions of the beams in the yz plane. The beam propagation directions differ from previous cases; instead of being parallel to the z-axis, they tend to propagate diagonally. It indicates that the energy distributions of the beams shift along the y-direction with the propagation along the z-axis. This can be partly interpreted by Eq. (5). Similarly, the experiments align with the simulations in terms of the general evolution patterns.
Fig. 5. The normalized intensity distributions of the generated quasi-Pearcey beam in yz plane. (a) Simulated results; (b) Experimental results.
Figure 6 illustrates the transverse intensity distributions of the beams. From the simulation, at the initial stage of beam propagation, it exhibits a shell-shaped pattern where the vault faces upwards. As the propagation distance increases, the sector angle of the shell-shaped continuously flares, and the beam shape gradually forms a horizontal stripe. Meanwhile, the central energy of the beam spot increases nonlinearly, and the site of the peak is almost the self-focusing position of the beam. After passing it, the angle increases, equivalent to the decrease reversely. The spot demonstrates a shell-shaped downward, revealing the evolution of the spot reversing before and after the self-focusing position. It can also be represented in Eq. (5) and Eq. (6), and the transverse field evolutions correspond to both xz and yz plane propagation patterns. However, there are evident differences between the profiles in the experiment and simulation, which are affected by certain factors. Nevertheless, the evolution pattern of the beam remains similar, mainly between the two. In addition, the Gaussian profile modulates the generated beam; even at the self-focusing position, the intensity distribution is not an infinitely extended bright line, but it exhibits a finite-length stripe-shaped spot.
Fig. 6. The normalized intensity distributions of the generated quasi-Pearcey beams in the transverse plane. (a) Simulated results; (b) Experimental results.
The following is an analysis of the causes of the difference between the profiles of the Pearcey beam generated in the experiment and that in the simulation. The intensity distributions of the selected positions are employed to illustrate the discrepancies. Figures 7(a)–7(c) show the distributions of Pearcey beams generated by simulated Gaussian beam, experimental Gaussian beam, and simulated Gaussian beam after interfered specifically, respectively. The phases of the Gaussian beams in the first two sets are assumed to be zero, whereas the simulated Gaussian beam in the third set is assigned an arbitrary phase distribution. It can be observed that the profiles of the beams in the first two sets closely match during the propagation process, though there is a bit of difference between them. However, in Fig. 7(c), the profile of the Pearcey beam generated by the Gaussian beam interfered with the phase modulation, which is quite different from the previous two sets. According to the comparison, it can be shown that the phase distribution of the original beam has a significant influence on the evolution behavior of the generated beam. Based on this, it can be suggested that the considerable discrepancies between the experiment and the simulation are primarily attributed to the fact that the Gaussian beam in the experiment does not conform to an ideal Gaussian beam distribution at the waist, especially concerning the phase distribution, which is not a perfect plane wave distribution. Hence, some positions in the wavefront may be subjected to additional phase disturbances. Furthermore, the original beam may also produce some phase distortion after being interfered by the elements and the optical system. These factors are all reasons for the significant profile discrepancies observed in the Pearcey beams between the experiment and simulation.
Fig. 7. Comparison of the three intensity distributions originated from different Gaussian beams. (a) Ideal Gaussian beam from simulation; (b) Gaussian beam from experiment; (c) Gaussian beam of intervention with phase.
4. Ptychography of the quasi-Pearcey beam
Based on the analysis above, it can be observed that the intensity distributions of the quasi-Pearcey beams exhibit lengthened-striped spots at the self-focusing position. If introduced into the ptychography, the imaging speed would be significantly boosted compared to conventional point-array imaging. The system of ptychography with the quasi-Pearcey beam is shown in Fig. 8(a), and the tested sample placed at the self-focusing position of the beam is to be scanned, where the distance between the sample and the photosensitive plane is 50 mm. The test sample is made of 3D printing material (LY1101), whose refractive index is 1.655. The sample is engraved with three capital letters “THZ”, and the size of the area covering the letters is 45 mm × 100 mm. The notch depths of the letters “THZ” are 0.57 mm, 1.14 mm, and 2.29 mm, and according to Eq. (9), these correspond to the three phases π/4, π/2, π, respectively. Here, Δφ is the phase of lettering; n and n0 are dielectric and vacuum refractive index, respectively; h is the depth of the notch. The beam as a probe illuminates the text region block by block. The illuminating trace is demonstrated as the red dashed line in Fig. 8(b). The displacements between the beam and the sample are 10 mm and 5 mm along the u-axis and v-axis, respectively.
Fig. 8. Schematic of ptychography of the quasi-Pearcey beam. (a) Diagram of the experimental setup; (b) the ptychography sample with the illuminating trace.
This arrangement results in a sequence of 10 × 11 sequentially overlapping illumination regions. The detector scans each block with a range of 100 mm × 100 mm. The number of the diffraction patterns correspond to the illuminating blocks, and each diffraction pattern includes 100 × 100 pixels. Due to the stripe-like distribution of the beam, the number of illuminating blocks in the u-axis is reduced. This ensures sufficient coverage of the adjacent blocks and improves inspecting speed. In this experiment, we set the sample in the two-dimensional motorized translation stage to realize the displacement between the beam and the sample, whose moving steps accord with the aforementioned.
It is expected to make the quasi-Pearcey beam applied in imaging appear as more extended-striped. Based on the previous analysis, the tendency of the extending stripe is more apparent with the decreasing p. Hence, we employed the parabola-slit plate with parameter p = 0.5 to generate the beam worked in ptychography. The generated beam is used as a probe whose intensity profile corresponds to the distribution at the self-focusing position, as shown in Fig. 6. Here, we introduced the algorithm of an extended ptychographic iterative engine (ePIE) to reconstruct the complex amplitude of the probe and transmission function of the sample [32]. The recovered results of the object are displayed in the left part of Fig. 9, and as referred to the simulated amplitude information, the three letters “THZ” are manifested clearly; however, in the experimental amplitude information, the degrees of clarity from the “Z” to “T” shows a tendency of declining, and this is due to the successive reduction in the corresponding notch depths, leading to the relevant increases in thickness, which results in significant absorption of the beam energy. Regarding the reconstruction of phases in the object, the simulated results clearly show the three letters, whereas the letter “T” is a little vague; the simulated consequences generally agree well with the experiments. From the calculations, the average depths of the three letters “THZ” are 0.51 mm (0.49 mm), 1.09 mm (1.05 mm), and 2.16 mm (2.12 mm), respectively (the experimental datum listed in brackets). According to the Fourier Ring correlation (FRC) method [33,34], this imaging system estimates the reconstruction spatial resolution at about 3.76 mm. The retrieval results of the probe are shown in the right part of Fig. 9. As to the amplitude information, the results of the simulations and experiments match decently with the original parts, but in the consequences of the recovered phases, the simulated distributions are more approximate to the original results than the experiments. Nevertheless, the reconstructed phases information in the simulations and experiments follow the originals in the overall profiles while presenting the discrepancies in quantitative distributions.
Fig. 9. The complex amplitude information of the object and probe. (a) The amplitude profiles of the object; (b) the phase profiles of the object; (c) the amplitude profiles of the probe; (d) the phase profiles of the probe.
Here, we make a specific analysis of the differences between experiment and simulation in the reconstruction results. Assuming that the amplitude $\sqrt I $ in the experiment is preserved, its phase is set to a value of zero; then, the probe is a plane wave. The probe is introduced into the ePIE algorithm, and corresponding reconstruction results are obtained, as shown in Fig. 10. The object amplitude and phase information can be clearly exhibited. The probe information is also distinct and agrees with the original distribution. On the other hand, we use the original Pearcey beam from the simulation as the probe, but the corresponding intensities of the diffraction patterns through the object are interfered to some extent. The diffraction patterns disturbed are used to reconstruct, and the recovered results are also obtained, as shown in Fig. 10. However, it is found that the reconstructed information of the object appears relatively ambiguous; some distortion and interference accompany the corresponding probe information. The above indicates that the initial phase distribution of the Pearcey beam has little impact on the reconstruction results. However, when the probe continues to propagate after loading the object information, the accuracy of the diffraction pattern information will be affected if the carried information is interfered in this process. It has a certain impact on the final reconstructed information. In the experiment, after the probe interacted with the object, it may lead to scattering and be affected by imperfect coaxiality. These could disturb the information of the subsequent diffraction patterns to some extent. Therefore, the experiment information appears different from that reconstructed by the simulation.
Fig. 10. The complex amplitude information of the object and probe. (a) The amplitude profiles of the object; (b) the phase profiles of the object; (c) the amplitude profiles of the probe; (d) the phase profiles of the probe.
For the comparison, the results of Gaussian beam in ptychography are presented, as shown in Fig. 11. In the recovered results of the object, both the amplitude and phase information are adequately presented in simulation. Whereas in the experiment, these details are essentially vague and indistinguishable. In the retrieval results of the probe, for the amplitude and phase in the simulation, their basic morphology, which is close to the distribution of the original Gaussian beam, can be presented. However, the amplitude information in the experiment is severely distorted, and the phase information is chaotic, which can be hard to reproduce. Moreover, the Gauss beam had scanning steps of 5 mm along both the x and y directions during the imaging process. Compared with the results of the Pearcey beam, the overall reconstruction quality of the Gaussian beam is much lower than that of the Pearcey beam, and its scanning time is also twice that of the Pearcey beam. Therefore, the Pearcey beam can perform much better than the Gaussian beam in ptychography.
Fig. 11. The complex amplitude information of the object and probe. (a) The amplitude profiles of the object; (b) the phase profiles of the object; (c) the amplitude profiles of the probe; (d) the phase profiles of the probe.
Next, the properties in ptychography of the quasi-Pearcey beam are analyzed in terms of the spatial frequency spectrum. We displayed distributions of the three-dimensional spatial frequency spectrums of the quasi-Pearcey beam at the self-focusing position in Fig. 12, giving simulated spectrum, recovered simulated spectrum, and recovered experimental spectrum, respectively. The general structure of the three spectrums is similar to an asymmetric sail. The original simulated distribution is much closer to the recovered simulated, while the profile of the recovered experimental is close to the counterparts of the two formers except for the distributions quantitatively. The maximum intensity of the spectrum is set to be located at the origin (0,0,0), and the full width at half maxima (FWHM) of the three spectrums along the fx and fy directions are as shown in Table 1. In the reconstructed spectrum from the experiment, the central frequency overall width appears slightly narrower than the two formers. Because the beam in the experiment extends more in the spatial domain along the x and y directions than the first two, resulting in the narrowing of the corresponding spectrums. Overall, the energy of the spectrum is mainly concentrated at fx = 0 and fy = 0, ensuring a fundamental appearance of the sample morphology in imaging. The central spectrum shows a narrower width in the fx direction, resulting in a longer coherence length in the x direction in the spatial domain. This allows for larger steps along the x direction in actual imaging while ensuring high quality in coherent diffraction imaging.
Fig. 12. The three-dimensional graphs of distributions with the spectrum amplitude of the quasi-Pearcey beam from the three results.
Table 1. FWHM of the three types of spectrums in the quasi-Pearcey beam
5. Conclusion
In this work, we first generated the quasi-Pearcey beam in the terahertz region. The optical system includes two core elements, parabola-slit plate and Fourier lens, which are fabricated by laser processing technologies and 3D printing, respectively. The properties of the terahertz quasi-Pearcey beam are investigated by combining simulations and experiments. This beam is a self-focusing beam, and at the self-focusing point, the beam appears to be nearly extended stripe; the extent of the spot extension along the x direction is inversely proportional to the parameter p of the modulated plate. Based on the characteristic of stripe-shaped distribution, we tested the field information reconstruction with the quasi-Pearcey beam in ptychography. The reconstructed results showed a good accordance between the recovered simulations and experiments, and the analysis of the spectrums suggested that the Pearcey beam has well spatial coherence. Efforts can be made in beam shaping to generate high-quality Pearcey beams in the future, which will enhance the performance of Pearcey beams in rapid reconstruction of full-field imaging.
Funding
Fundamental Research Funds for the Central Universities (2017KFYXJJ029).
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.
Supplemental document
See Supplement 1 for supporting content.
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