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Abstract
Bessel terahertz (THz) imaging employing a pair of thin silicon multi-phase diffractive optical elements is demonstrated in continuous wave mode at 0.6 THz. A proposed Bessel zone plate (BZP) design – discrete axicon containing 4 phase quantization levels – based on high-resistivity silicon and produced by laser ablation technology allowed to extend the focal depth up to 20 mm with minimal optical losses and refuse employment of bulky parabolic mirrors in the imaging setup. Compact THz imaging system in transmission geometry reveals a possibility to inspect objects of more than 10 mm thickness with enhanced contrast and increased resolution up to 0.6 of the wavelength by applying deconvolution algorithms.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Employment of terahertz (THz) radiation to inspect thick objects is one the most considerable difficulties in THz imaging technology. As a rule, THz beam propagation in bulky media suffers phase perturbation and troublesome attenuation inducing thus deterioration in the image contrast. This outcome is a strong obstacle in implementation of versatile hand-held unattended package inspection systems [1], development of single pixel imaging methods [2] or spectroscopic imaging systems [3,4]. To circumvent the issue one needs to apply special optical means aiming to organize a self-reconstructing beam profile. Moreover, these advantages should be built-in in the setup preserving simultaneously features of convenient usage of terahertz imaging systems: they should be compact, preferably, free of optical alignment and cumbersome optical components, reliable, and providing an ability of relatively rapid scans. The first hurdle can be well-evaded reshaping conventional Gaussian beam profile into the Bessel beam [5,6], while reduced size of the THz imaging system can be kept using compact THz emission sources, e. g. quantum cascade lasers [7–9], or electronic emitters [10], compact sensitive detectors such as nanometric field effect transistors [11–13], microbolometers [14–16], diode structures-based sensors [17–20], but, in particular, via replacing bulky optical passive elements, for instance, parabolic mirrors by relevant diffractive optical components [21,22]. As it was shown recently, laser ablation technology was found to be powerful tool to fabricate effective Fresnel zone plates of different diameters [22,23], to produce anti-reflective and phase shifting structures [24] and apply them successfully for Fibonacci (bifocal) THz imaging [25]. The latter approach with two planes simultaneous image recording not only enables an increase in registration speed – imaging systems containing such kind of zone plates are capable to reach wavelength limited resolution even in case when the focal length is less than a few cm. However, it needs to be underlined that the image quality is strikingly sensitive of the sample position in respect to diffractive optical element. This circumstance, especially, in exploring thick objects in real environment, turns into an ultimate challenge as image becomes blurred resulting in complication in resolving and identification of the recorded data.
In this work, we introduce a novel route to overcome aforementioned issues – thin and compact Bessel zone plates (BZP) based on silicon multi-phase diffractive optics – are designed, produced and demonstrated in the THz transmission imaging system. In contrast to already known approaches in fluorescence [26] and light-sheet microscopy [27], where a single diffractive element is employed in combination with conventional lenses, we propose to use in the experimental setup a second BZP for enhanced imaging purposes in terms of increased compactness. The discrete axicons containing 4 phase quantization levels were fabricated from a high-resistivity silicon by employing laser ablation technology allowed to extend the focal depth up to 20 mm with minimal optical losses and refuse employment of bulky parabolic mirrors in the imaging set up. Compact THz imaging system reveals possibility to inspect objects of more than 10 mm thickness with enhanced contrast, weak dependence on the sample thickness and position due to long focal depth as well as increase its resolution up to high level by applying deconvolution algorithms [28].
2. Design and fabrication
The choice of discrete axicons relies on their advantages being more compact and exhibiting much less optical losses in the material if compared with classical cone axicons [29]. Moreover, their manufacturing is much easier because the accuracy of the cone requires special attention, especially, in the central zone with the tip of the cone. As an important detail in the design served our previous finding that at least 4 phase quantization levels are required for rational trade-off between the effective focusing performance and the fabrication complexity [22] . Silicon discrete BZP design was simulated using three-dimensional finite-difference time-domain (3D FDTD) method.
Silicon wafer of 0.52 mm thickness with specific resistance 10–1000 k$\Omega \cdot$ cm and refractive index of 3.44 served as the core material for the manufacturing of the BZP. Their design with phase quantization level of $P=4$ were processed employing laser direct ablation technology. The set up was composed of Pharos-SP laser (Light Conversion Ltd, wavelength 1028 nm, pulse duration = 350 fs, energy = 50 µJ, repetition rate = 100 kHz), IntelliSCAN-14 scanner by $ScanLab$ and F-Theta f100 lens by $Sill Optics$ with scanning speed = 1000 mm/s.
Design and fabricated BZP of 20 mm diameter are depicted in Fig. 1. Panel Fig. 1(a) displays the BZP peculiarities of four phase quantization design and its cross-section of the central part with indicated dimensions, while panel Fig. 1(b) shows photo of the fabricated discrete axicon was obtained by $Hirox$ KH-7700 microscope. Enlarged subzones of the ablated rings with detailed geometrical parameters are displayed in Fig. 1(c). Measured subzones depths of produced structures are $h_1=47$ µm, $h_2=106$ µm, $h_3=154$ µm, and these results are in a good agreement with calculated ones. The center hole in the BZP was used for the optical alignment.
Fig. 1. (a) Bessel zone plate design and its cross-section of the central part with marked dimensions in microns of four phase quantization levels. (b) The photo of thin silicon-based Bessel diffractive element for the 0.6 THz. (c) 3D reconstruction of the zoomed area in the center part displays ablated and polished silicon surface, h indicates the places where groove depth was measured using $Hirox$ digital microscope.
Point of departure for the beam profile modelling was the proposition that a unit amplitude plane illuminates a thin circular hologram with complex transmission function
where $k_\rho =k\sin \theta$, $k$ is the wave number, $\theta$ is the halfcone angle of the Bessel beam and $\rho$ is the radial coordinate in the $(x,y)$ plane. A practical realization of the smooth phase function described by the Eq. (1) is rather complicated. For the sake of the simplicity we implement a quantified phase coded hologram with a complex step-like profile, so the complex transmission function N levels can be written as3. Results and discussion
Silicon discrete BZP focusing properties were also modelled using 3D FDTD method. The simulation was performed with spatial resolution from 10 $\mu$m up to 100 $\mu$m. To simplify calculations, symmetry conditions of the structure were used, and the absorbing boundary conditions were set in all directions. Plane wave of 0.6 THz frequency served as an excitation source. The electric field was recorded in the whole simulation volume.
The simulated Bessel beam 2D cross-section at the center of BZP via normalized electric field reconstruction at 0.6 THz frequency is shown in Fig. 2(a). As it is seen, one can expect transformation of the plane electric field profile into a needle-like shape along the beam propagation.
Fig. 2. (a) Normalized THz radiation power distribution simulation using 3D FDTD method in $xz$ plane. (b) and (c) – Measured distribution of the THz radiation in $xy$ plane and in $xz$ plane, respectively, at 0.6 THz frequency focused by the Bessel zone plate. Insets in (b) and (c) show performances of the multi-phase Fresnel zone plates with 4 phase quantization levels (MPFL) for comparison in the same scale. (d) and (e) – Experimentally evaluated distribution (black line) of THz signal amplitude along $z$ axis and $x$ axis, respectively, compared with the results of the simulations using 3D FDTD method (orange lines) and MPFL focusing (green lines). Collimated radiation profile at 0.6 THz is also depicted for comparison.
To confirm the prediction, performance of the BZP was investigated by measuring Gaussian beam intensity distribution in the focal plane and along the optical axis at 0.6 THz frequency using modified experimental setup [22]. The THz radiation was registered by resonant THz antenna-coupled microbolometer detectors [14]. Results of the focusing performance are presented in panels Figs. 2(b) and 2(c). One can see clearly resolved focal spot of a needle shape starting from $z = 4$ mm and ending at $z = 21$ mm away from the BZP sample as expected. For comparison, insets in Figs. 2(b) and 2(c) denote focusing performance of multilevel phase Fresnel zone plates lens (MPFL) with 4 phase quantization levels and 1 cm focal length [22] in the same scale.
More detailed focusing features can be extracted analyzing THz signal amplitude distributions along $z$ axis and $x$ axis. Experimental results compared with the relevant simulation data are presented in Fig. 2(d). It is seen that the Bessel peak amplitude value is located at $z = 9$ mm with the 20 mm depth of focus evaluated at $1/e^2$ of the maximal amplitude value. For comparison, the MPFL with the same focal distance at 10 mm, reveals at least 6 times shorter focal depth reaching only 3 mm. It is worth noting that the simulated and the measured focused beam profile along the optical path are in a good agreement and displays only negligible deviation between each other as it can be observed in the peak around $z = 15$ mm. The focused beam profiles with BZP and MPFL as well as collimated beam profile cross-section in $x$ direction are given in Fig. 2(e). Our experimental realization of the Bessel beam is inline with standard finite energy approximation – Bessel-Gaussian beam. In this model the envelope has a Gaussian profile, therefore it is helpful to use semi-logarithmic scale to estimate the width of Gaussian envelope. And indeed, we see, that the width of a Gaussian envelope is approximately 3 mm. The estimated full width half maximum (FWHM) for both diffractive elements are $\sim$ 0.43 mm. One can note that the unfocused incident Gaussian beam before the Bessel zone plate exhibits FWHM = 7.2 mm, while the signal amplitude is nearly 15 dB lower.
While planning our experiment, we have carefully chosen parameters so, that the Gaussian envelope of Bessel-like beam had sufficient width to enable diffractionless behavior of the beam [31]. We found that our estimations are also in line with recent results [32].
Experimental results on the Bessel focusing performance are presented in Fig. 3. Panel (a) presents imaging resolution and its dependence on the target position relevant to the BZP lens from $z$ = −6 mm to $z$ = 6 mm by 1 mm steps. It was estimated using specially constructed resolution target consisting of periodic metallic stripes with different distances in between as shown in Fig. 3(b). It is worth underlying that the Gaussian beam was focused on the sample and transmitted radiation was collected in a novel way – without parabolic mirrors employing two BZP lenses instead, – and it is the main difference from the set up described elsewhere [22]. As one can see in Fig. 3(c), apertures of 2.5 mm to 0.6 mm periods in all images can be clearly resolved. Moreover magnified image of 0.6 mm period aperture indicates that spatial resolution of 0.3 mm can be reached applying deconvolution procedures. One can note that it amounts to $0.6~\lambda$.
Fig. 3. (a) Experimental demonstration of the Bessel focusing performance using resolution target imaging at 0.6 THz radiation along the beam propagation path. The cross sections of resolution target position along optical path represent the resolution alternation in the focal plane direction. The distance between cross sections was set to 1 mm. 2D images consist of 390 $\times$ 116 pixels. Pixel size: 150 µm $\times$ 300 µm. (b) Photo of the resolution target with indications of the apertures period in mm scale. (c) Enhanced contrast image of resolution target at 0.6 THz after the deconvolution procedure, where blue line represents the cross-section of each period of stripes (data lines shifted down for convenience of illustration). (d) Dependence of signal to noise ratios on z coordinate evaluated for apertures of 1 mm, 2 mm and 2.5 mm period. (e) Dependence of imaging resolution on aperture period, where the target position in $z$ direction is fixed at $z = -4$ mm. Enhanced image resolution after deconvolution procedure is shown additionally for comparison.
Raw images recorded with BZPs displays blurred characteristic due to the conical structure of the Bessel beam and its side lobes, which is seen in Fig. 2(e). This distinctive feature of the circular transversal structure of the Bessel beam apparently distorts the recorded images. This can be understood taken into account the fact that the intensity in the first ring of the Bessel beam is equal to that of the central spike. Due to the propagation invariance of the Bessel beam, the shape of this blur is relatively independent of the position in the field-of-view. Nevertheless, the whole recorded stack of images can be approximated as a convolution of the object function with a spatial intensity distribution of the Bessel beam. This understanding enables to use standard deconvolution techniques allowing thus to recover the transversal resolution of the Bessel beam imaging. It is evidenced in Fig. 3(c), where the reconstructed image of all raw images in $z$ direction is depicted. The blue line represents cross sections of the target apertures. As it is seen, the smallest period of stripes, which was clearly resolved is 0.6 mm. As the wavelength at 0.6 THz is 0.5 mm, therefore, the wavelength limited spatial resolution and good quality can be expected in recording THz images independently of the sample distance from the BZP. Dependence of signal noise ratio (SNR) evolution on the target position along $z$ axis for target aperture periods of 1 mm, 1.6 mm and 2 mm are depicted in Fig. 3(d). It should be noted that SNR for 2 mm aperture exceeds 2 times and is weakly dependent of the target position in $z$ direction. The same trend is kept for other apertures, too, however, the SNR value is reduced and varies around 1.5. Furthermore, sample position does not provide significant effect for the different aperture periods. It illustrates the Bessel beam advantage to be insensitive for inaccurate sample positioning with respect to BZPs location. By applying deconvolution technique contrast enhancement up to 5 times is reached for aperture periods starting from the wavelength scale up to 1.25 mm. It is apparent in Fig. 3(e), where enhancing SNR is visible for larger features. Combination of deconvolution algorithms and the Bessel beam enables high resolution imaging, as it could be seen from Fig. 3(e), where 0.3 mm aperture is resolved with SNR=5 without the necessity of the accurate position of the object under test. We note that the structured illumination (Bessel and other non-diffracting beams) was successfully applied for imaging purposes in fluorescence [26] and light sheet microscopy [27], where resolution less than wavelength was achieved. Details on the deconvolution algorithms and filters are available in Supplementary material.
To illustrate the quality of the Bessel THz imaging, a specially designed stack of targets for thick media imaging measuring 30 mm$\times$60 mm$\times$3 mm was 3D printed with PLA filament using Ultimaker 2 printer. Figure 4(a) demonstrates principle of the imaging procedure, where the object, with thickness varied from 3 mm to 12 mm, was placed between two silicon BZP, i. e. no parabolic mirrors were employed in image recording. For comparison, the same imaging procedure was also performed using conventional MPFL with $f = 10$ mm and $P = 4$ characterized in Figs. 2(b) and 2(c). Imaging results of the different sample thicknesses using Bessel zone plate and MPFL are given in Figs. 4(b) and 4(c). Varying the content of the stack from 1 to 4 identical targets, the Bessel imaging advantage is revealed via obtained horizontal profiles up to stack thickness of 12 mm in Figs. 4(d) and 4(e). As we can see in Figs. 4(d) and 4(e), in the case of 3 mm aperture and thickness of 1 sample SNR = 8 is achieved with MPFL while SNR = 4 was obtained with BZP is worse due to self-healing effect [6] and much longer depth of field of Bessel beam. The advantage of BZP appears when the stack is composed of 2 or more samples: SNR achieved with BZP increases from 2 to 4 times better values than MPFL.
Fig. 4. (a) Part of Bessel imaging setup displaying object under test placed between two silicon BZP. (b) and (c) – Stacks of 1 to 4 identical targets were imaged using Bessel zone plates and conventional MPFL, respectively, aiming to evaluate THz imaging performance of thick objects. (d) Beam profiles at y=17 mm (position is marked as dashed black line in (b) and (c) of obtained images using BZL. (e) MPFL beam are shown for comparison.
4. Conclusions
To summarize, Bessel terahertz imaging of more than 10 mm thickness objects via silicon multi-phase diffractive optics is designed and demonstrated in continuous wave mode at 0.6 THz. Discrete axicon containing 4 phase quantization levels based on high-resistivity silicon allowed to extend the focal depth up to 20 mm with minimal optical losses and record images with to up 5-fold enhanced contrast and increased resolution up to $0.6~\lambda$ by applying deconvolution algorithms. Experimental data agree well with the modelled results obtained using three-dimensional finite-difference time-domain method.
Appendix A: Description of the deconvolution algorithm
A standard minimum mean square error (Wiener) filter to process the 3D-stack of recorded data (more information can be found in [28]) was employed. Bearing in mind the fact that the convolution can be understood as a multiplication performed in Fourier (reciprocal) space, each spatial frequency of the object was multiplied independently with the point spread function of the imaging system $H (x, k_z )$. The deconvolution can be performed in the Fourier space by correcting the spatial components of the recorded image stack $I(x, y, k_z )$ by multiplying it with Wiener filter $H_W (x, k_z )$ – the filter, given by the equation [28]:
minimizes the expected mean square error for a signal-to-noise of the model, $SNR (k_z )$.The main drawback of such approach is the limited knowledge on the optical transfer function, which requires some additional experimental measurements in our setup. In order to overcome this issue we have compared results obtained using an experimentally measured intensity profile of the Bessel beam with results obtained using the equation, see Ref. [33]:
It enabled us to estimate the signal-to-noise function SNR. One can note that due to limited knowledge of the actual spatial distribution of the Bessel beam some inaccuracy and artifacts can be expected.
Implementation of the deconvolution algorithm mainly follows procedures described in [27,28] with implementation containing two parts. The first one is calculation of the deconvolution of the filter, see Eq. (4) and its application to the 3D stack of images, while the second part is dedicated to the deconvolution of the object image.
Appendix B: Calculation of the deconvolution filter
The point spread function (PSF) of the discrete axicons used in the setup was experimentally determined. The digitally scanned 3D intensity distribution of the Bessel beam $u(x,y,z)$ corresponds to the PSF function of the focusing system, see Fig. 5. Integration of the intensity along the $y$-axis allows to arrive at the function $F(x,z)= \int _{-\infty }^{\infty } u(x,y,z)dy$. From this function the optical transfer function $H (x, k_z )$ was calculated by performing one dimensional Fourier transform $H(x,k_z)=F_z\{u(x,y,z)\}$.
As a next step, a model for a signal-to-noise ratio was introduced, and a power law distribution $SNR(k_z)=ak_{cutoff}/k_z$, was used initially, where $a$ is some experimentally determined constant needed to adjust the signal-to-noise ration of the recorded 3D image stacks and $k_{cutoff}$ denotes maximum spatial frequency which can be optically transmitted through the focusing system. It can be readily approximated from the numerical aperture of the used axicons and from the wavelength of the Bessel beam.
Appendix C: Deconvolution of the object image
Deconvolution algorithm calculation itself takes a few minutes, however we additionally do estimate effects due to a) signal-to-noise-ratio, b) slight misplacements of the sample during the z-scan and c) imperfections in experimental determination of Bessel beam profile. Additionally, these estimates are performed iteratively and may take up to 10 minutes. Here, we would like to point out that those estimations are given without any optimizations to the code.
Possible deconvolution artefacts were minimized by implementing cubic interpolation of the images in order to compensate probable lateral shift due to the axial translation. This also enabled to increase the spatial resolution.
Numerical simulation were performed using $I(x,y,k_z)=F_z\{i(x,y,z\}$, where $i(x,y,z)$ is cubically interpolated 3D stack of images. Deconvoluted image $i_{dec}(x,y,z)$ of the object was calculated as inverse Fourier transform after applying the Wiener filter to $I(x,y,k_z)$. Examples of the deconvoluted images using different noise parameters of the object are presented in Fig. 6.
Funding
European Regional Development Fund (01.2.2-LMT-K-718-01-0047); Lietuvos Mokslo Taryba (DOTSUT-247).
Acknowledgments
The authors acknowledge Vladislovas Čižas for his precious assistance during the preparation of figures.
Disclosures
The authors declare no conflicts of interest.
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