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In single-pixel coded aperture terahertz-imaging, the individual pixel size in the spatial terahertz modulator is usually comparable to the terahertz wavelength in order to obtain a sufficient spatial image resolution. Therefore, diffraction plays an important role in the imaging process and must be accurately taken into account when the image is computationally retrieved. For this reason, we analyzed the impact of diffraction from the spatial terahertz modulator on the quality of the reconstructed image in single-pixel coded aperture imaging. We observed that the most important fraction of the image information is already contained in the zero order diffracted radiation. Higher diffraction orders do not contain enough information to retrieve the image from them solely, yet can contribute to an improved image quality when added to the zero order information.
© 2016 Optical Society of America
1. Introduction
In the recent decade, terahertz technology has raised increasing scientific and industrial interest due to its ability to easily penetrate almost any dielectric. For this reason, materials as e.g. textile, cardboard, paper, wood etc. are highly transparent to terahertz waves which implies that hidden objects behind such materials can be detected by means of terahertz radiation. Even more, many hazardous substances such as explosives, illegal drugs or poisonous material display characteristic spectral fingerprints in the typical terahertz frequency range between 0.1 THz and 10 THz. The sensitivity to spectral features makes the terahertz technology especially useful for quality control, medical purposes or security applications [1–5], since it is not only possible to detect e.g. hazardous substances without removing the covering material, but also to identify the substance as soon as spectroscopic terahertz data is collected by the measurement system. Yet, because the exact position of the hazardous substance cannot be known in advance, it is always necessary to spatially scan suspicious objects in order to spot and localize the position of the threatening substance. In other words, it is a requirement in such a scenario to spatially image a significant portion of the complete scene over a larger area and, if applicable, to record spectral information for any imaged pixel.
One of the major technological limitations of current state-of-the-art terahertz measurement systems lies in the low achievable data acquisition rate which especially carries weight in terahertz imaging systems. At this point, it is significant that the terahertz frequency range lacks of multi-pixel detectors, a technology that is readily available in other regions of the electromagnetic spectrum. Although detector arrays have been demonstrated in the terahertz range [6, 7], these detector arrays are usually only sensitive to a narrow frequency band and, thus, do not allow frequency-resolved detection. As a consequence, for frequency-resolved terahertz imaging, single-pixel detection is often applied in raster scan methods or multiplexed computational imaging techniques instead of detector arrays [8, 9]. The latter ones require a specific type of adaptive coded apertures that create multiple projections of the scene. Those projections are then imaged onto a single-pixel detector and the scene is reconstructed by computational algorithms. Coded aperture imaging can be combined with compressive imaging techniques in order to further reduce the acquisition time [10].
In coded aperture imaging, the limit of the spatial image resolution is set by two conditions. First, as in any other far-field imaging method, spatial details that are smaller than the wavelength of the used probe radiation cannot be physically resolved. In fact, this natural limit dictates the boundary for the highest spatial resolution in the far field. Second, the spatial resolution of the image is determined by the pixel size of the spatial terahertz modulator that creates the projections of the scene. At this point, it must be noted that the pixel size is comparable to wavelength of the used probe radiation to maintain a high spatial image resolution. This diffraction limit is not much of an issue for visible or infrared light coded aperture imaging systems, where the wavelength of the probe radiation is in the range from several hundred nanometers to a few microns and thus high spatial resolution can be even obtained with pixel sizes that are twice or three times the wavelength of the probe radiation. Yet, it is obvious that pixel diffraction must be taken into account for terahertz imaging systems, where the far-field resolution is already low due to a radiation wavelength of the order of several hundred microns. To avoid a further reduction of the spatial image resolution, it becomes necessary to optimize the pixel size of the spatial terahertz modulator and to establish a trade-off between pixel size and diffraction from the individual pixels. Furthermore, pixel diffraction does not only limit the spatial image resolution, but also significantly limits the quality of the image reconstruction in coded aperture imaging.
In this paper, we numerically study the impact of pixel diffraction in spatial terahertz modulators on both the quality of the image reconstruction and the obtainable spatial image resolution in coded aperture terahertz imaging. For this purpose, we consider a real-world terahertz coded aperture imaging system using off-axis parabolic mirrors to collimate and focus the terahertz beams, a spatial terahertz modulator that generates binary masks and a single-pixel detector with finite aperture. We also investigate the dependence of the image reconstruction quality on the detector aperture and detector position and show that the most significant portion of the image information is contained in the lowest order diffracted radiation from the spatial terahertz modulator, while higher order diffracted radiation only decently increases the quality of the reconstructed image.
2. Numerical model for coded aperture imaging
Figure 1 shows the schematic of the investigated coded aperture terahertz imaging setup. The generated terahertz radiation from the emitter is collimated by a 90° off-axis parabolic mirror (Oapm) and modulated by a spatial terahertz modulator (Stm) before illuminating the imaging object. Experimentally, such an Stm can be implemented for example by use of reconfigurable metamaterials or photo doped semiconductors [11–17]. The scattered radiation from the object is then focused by a second Oapm onto a single-pixel detector with finite detector width. In our study, we considered terahertz radiation at a frequency of 1 THz, which corresponds to a wavelength of 300 μm. The imaged object in this study consists of a four level grayscale transmission profile as shown in Fig. 2. The object size is 4 cm. We chose the width of the individual plateaus to be significantly larger than the wavelength of the terahertz radiation in order to minimize diffraction at the individual boundaries between the 4 plateaus. As such, a terahertz image recorded by a camera with infinite spatial resolution would display the original gray scale pattern and diffraction from the individual boundaries would be negligible in good approximation. This is important, since we only intend to study the impact of diffraction from the spatial terahertz modulator on the image resolution and image reconstruction quality, which forces us to assure that the impact of modulator diffraction is not overcast by diffraction from the object.
Fig. 1 Schematic setup for single-pixel terahertz imaging. The radiation from the terahertz emitter is collimated by an off-axis parabolic mirror (Oapm) and illuminates the object. In order to detect and reconstruct the image, the terahertz beam is spatially modulated by a spatial terahertz modulator (Stm) and focused with a second Oapm onto the detector. The region which is numerically investigated in this study is represented by the grey region.
Fig. 2 Amplitude transmission curve of the object to be imaged. For the simulation, an object with a spatially varying amplitude transmission was used. The spatial dependence of the transmission can be described by 4 different transmission plateaus of different length and transmission amplitude.
In the numerical studies, we restricted the calculation domain to the grey shaded region in Fig. 1, since it defines the area where all diffraction processes and the imaging onto the single-pixel detector occur. In order to limit the simulation time to a reasonable value, we used a 2-D model. The additional diffraction that occurs in a third dimension could be treated similarly to the diffraction we described in our 2-D model, and thus, the results could be easily transferred to a real-world application in 3-D. We used COMSOL Multiphysics ® with the RF Module for the 2-D full wave simulations. Figure 3 shows a screenshot of the numerical model. After passing the Stm, the terahertz wave scatters from the object and is focused by an Oapm with a focal length of 10 cm onto a detector with finite aperture. We placed the detector exactly in the focal plane of the Oapm. For parametric studies, the model allowed us to change both, the aperture width and the lateral position of the detector normal to the propagation direction of the terahertz beam. We calculated the energy of the detected terahertz wave by integration of the electromagnetic field energy density over the width of the detector aperture. The aperture width is indicated by the white line in Fig. 3.
Fig. 3 Simulation model. Electric field of the terahertz waves after scattering from the object/Stm. The simulation was performed with COMSOL Multiphysics ®. The scattering from the object and the Stm is simulated by means of scattering boundary conditions. The scattered radiation is focused by an Oapm onto the detector (D). The detector is placed in the focal plane. The field energy on the detector is calculated by integrating the field energy density over the detector width.
In order to simulate the scattering from the object and the spatial radiation filtering by the Stm, we analytically calculated the spatial amplitude transmission function as a product of the greyscale level transmission function of the object and the transmission function of the Stm. Based on the product function, we defined appropriate scattering boundary conditions for the calculation domain.
We derived the masks used in this study from Hadamard matrices and random matrices, which are two commonly used matrix mask types in coded aperture imaging [10, 15]. To achieve the +1 and the −1 values of the mask, we ran two simulations for each mask in which the mask of the second simulation was inverted. By calculating the difference between the received field energy on the detector for the +1 and the −1 value of the mask, we determined the measured field energy as it was received by use of a ±1 binary mask. As in the usual Hadamard matrix approach for coded aperture imaging, we reconstructed the image by matrix inversion.
3. Results and discussion
3.1. Impact of the detector width on the image reconstruction quality
In a first step, we investigated the influence of the detector width on the image quality of the reconstructed image. For encoding and reconstructing the images, we used the Hadamard matrix approach. To investigate the influence of the detector width on the reconstruction quality, we performed numerical calculations for Hadamard masks with 32 pixels. With respect to a mask size of 4 cm, this corresponds to a pixel size of 1.25 mm, which is about four times the wavelength of the terahertz probe radiation.
Figure 4a shows the retrieved images for different detector widths in the range from 0.5 mm to 25 mm. In more detail, Fig. 4b depicts the dependence of the reconstructed image on the detector width as a color scaled plot. The ordinate displays a continuous range of detector widths between 25 mm and 0.5 mm, while the colors represent the spatial field energy distribution of the image. As can be seen, we could retrieve the image with high quality when we used a large detector with an aperture width of 25 mm. Not surprisingly, the image quality decreases with decreasing detector width, although it is notable that a high image reconstruction quality could be maintained up to a detector width of 5 mm. For detector widths below 5 mm, the retrieved images cease to display sharp steps at the boundaries between two grey scale levels, but rather show gradual transition regions with rising and falling slopes. This behavior is especially prominent at the boundary between the grey scale levels at the position of 3 cm, where the transition region rapidly covers half of the width of the plateau. Below a detector width of 2 mm, also the transition regions between the other boundaries of the grey scale pattern increase and the retrieval becomes more inaccurate, although the coarse position of the boundaries can still be identified. At the same time, the contrast between the grey scale levels significantly decreases.
Fig. 4 Reconstructed image for 32 pixel modulator. The image is reconstructed for a pixel size of 1.25 mm and different detector widths between 0.5 mm and 25 mm. In (b) the amplitude transmission is coded in the color of the plot.
In order to quantify the dependence of the image reconstruction quality on the detector width, we calculated the maximum absolute error between an assumed perfect image of the object and the reconstructed image. For this purpose, we plotted the maximum absolute error between the original and retrieved transmission profile of the object versus the detector width in Fig. 5. In the trivial case of a detector width of 0 mm no field energy is detected and the maximum absolute error is 1, which equals the maximum amplitude transmission in the transmission profile. The maximum absolute error rapidly decreases to about 20 % as soon as the detector width is as large as the focal spot size of about 1 mm after the Oapm. At this point it should be clearly noted that we defined the focal spot size as the half width at half maximum of the beam energy distribution of the zero order diffracted beam. When we further increase the detector width and thus include higher order diffracted modes of the focused beam, the maximum absolute error of the reconstructed image further decreases. We noticed that the maximum absolute error dropped below 10 % when we opened the detector aperture to a width of 24 mm which corresponds to the situation that the edges of the detector agree with the maxima of the second order diffracted beam. As an obvious result, most of the image information is contained within the zero order diffracted beam while diffracted beams of higher orders only slightly improve the image reconstruction quality.
Fig. 5 Image reconstruction error for 32 pixel modulator. The maximum absolute error between the reconstructed image in Fig. 4b and the predefined object in Fig. 2 is shown depending on the detector width.
3.2. Impact of the detector position on the image reconstruction quality
We further investigated the dependence of the image reconstruction quality on the lateral misalignment of the detector with respect to the propagation axis of the focused beam. While the lateral alignment was offset with respect to the optical axis, the detector was perfectly positioned in the focal plane with respect to the longitudinal distance from the Oapm. We determined the lateral misalignment perpendicular the propagation axis of the beam as the distance between the zero order intensity maximum of the focused terahertz beam and the center of the detector. Figure 6 shows the dependence of the maximum absolute error on the lateral detector offset for detectors with a width between 0.5 mm and 10 mm. As long as the detector aperture includes the maximum of the zero order diffracted beam, the maximum absolute error of the image reconstruction yields low values. Even for a small detector width of 1 mm, the maximum absolute error is below 20 %. Yet, as soon as the detector offset becomes larger than half the detector width the maximum absolute error of the image reconstruction surges up to a value of 1, which means the reconstruction completely fails. It is notable that the reconstruction error stays 1, even if the detector aperture collects radiation of the higher order diffracted beam. This further indicates that the significant portion of the image information, that is required for a satisfying image reconstruction, is contained in a small area about the zero order diffraction maximum. The higher order diffraction maxima contain some additional information for the reconstruction as shown before, but do not carry enough information to reconstruct the image without detection the zero order diffraction maximum.
Fig. 6 Image reconstruction error depending on the lateral detector position. The maximum absolute error between the reconstructed image and the object is shown depending on the lateral offset of the detector from the optical axis of beam propagation for different detector widths between 0.5 mm and 10 mm.
3.3. Impact of the pixel size in the spatial terahertz modulator on the image reconstruction quality
In a next step, we studied the dependence of the image reconstruction quality on the number of pixels in the Stm. Since we held the total aperture of the Stm fixed at 4 cm, an increase of the number of pixels directly translates into a decrease of the individual pixel size. In our study, we increased the number of pixels from 32 to 512, which is related to reduction of the pixel size to 78 μm. Since the pixel size corresponds to about one fourth of the wavelength, we expect the incident terahertz wave to be notably diffracted from the individual pixels of the Stm. In order to asses the impact of diffraction on the image reconstruction quality when using a 512 pixel Stm instead of a 32 pixel Stm, we performed similar investigations on the impact of the detector width on the image reconstruction as for the 32 pixel mask. In this respect, Fig. 7a shows the transmitted field amplitude of the reconstructed image for detectors with a width between 0.5 mm and 25 mm. Comparing the amplitude transmission obtained with the 512 pixel Stm in Fig. 7a to the corresponding transmission detected in a 32 pixel Stm scheme, as shown in Fig. 4a, reveals that the diffraction at individual pixels becomes more prominent when the pixel size is reduced. The amplitude transmission displays a strong interfering modulation and the boundaries between the greyscale levels of the imaged object become more washed out and more difficult to identify when measured using a 512 pixel Stm. The increasing role of diffraction from individual pixels becomes also visible in Fig. 7b, where the amplitude transmission is shown as a color plot for detectors widths from 0.5 mm to 25 mm, similar to the graph in Fig. 4b for the 32 pixel case. As a further indication for a negative influence of individual pixel diffraction on coded aperture imaging, we noticed that the maximum absolute error of the image reconstruction could not be driven below 20 % even for a detector width of 25 mm. This can be seen from Fig. 8, where the maximum absolute error of the image reconstruction is plotted in dependence on the detector width. In comparison, the maximum error could be reduced to less than 3 % for a detector width of 25 mm in the 32 pixel case, as can be seen from Fig. 5. This indicates that information is lost due to diffraction and that it is not possible to improve the quality of the image reconstruction by increasing the detector width within a reasonable range.
Fig. 7 Reconstructed image for 512 pixel modulator. The image is reconstructed for a pixel size of 78 μm and different detector widths between 0.5 mm and 25 mm. In (b) the amplitude transmission is coded in the color of the plot.
Fig. 8 Image reconstruction error for 512 pixel images. The maximum absolute error between the reconstructed image in Fig. 7b and the predefined object in Fig. 2 is shown depending on the detector width.
We finally investigated the dependence of the maximum absolute error of the image reconstruction on the number of pixels for different detector width. In this study, we varied the number of pixels from 16 to 512, i. e. pixel sizes between 2500 μm and 78 μm, and chose detector widths between 1 mm and 25 mm. As can be seen from Fig. 9, the maximum absolute error is low for a small number of pixels and thus pixels with a large size, while the error increases for an increasing number of pixels, i.e. pixels of small size. Figure 9 also suggests that the gain in reconstruction accuracy by increasing the detector width is not as high for small pixels as for larger pixels. One reasons for this lies in the influence on small-scale noise on the reconstruction quality of the image.
Fig. 9 Image reconstruction error depending on the pixel size. The image reconstruction error is analyzed depending on the number of pixels used to project the object of 4 cm size onto the detector. The maximum absolute error is calculated for detector widths between 1 mm and 25 mm. The abscissa is in a logarithmic scale. The terahertz wavelength is for the imaging is at 300 μm.
3.4. Impact of the mask type on the image reconstruction quality
In a further study, we evaluated if the influence of diffraction explicitly stemmed from the regular construction of the Hadamard mask patterns that result in regular diffraction gratings. For this purpose, we eliminated the regularity of the Hadamard approach and performed similar studies to the previously reported ones for the case of random binary masks as often used in compressed sensing. We observed that the resulting image reconstructions and their corresponding maximum absolute errors did not show significant differences compared to the results of the Hadamard masks. This suggests that the loss of information for small pixel sizes is due to the diffraction at the individual pixels and that there is no further loss caused by the superordinate arrangement of the pixels.
4. Conclusion
In conclusion, we studied the impact of diffraction at the individual pixels of a spatial terahertz modulator on the image reconstruction quality in coded aperture imaging. We investigated the dependence of the maximum absolute error of the image reconstruction on the detector width, the lateral detector position, the pixel size and the mask type of the spatial terahertz modulator. In order to reduce the reconstruction error, it is necessary that the detector width is large enough to at least collect the zero order diffracted radiation from the spatial terahertz modulator. This basic condition also dictates that the lateral position can only be shifted from the optical axis of the beam propagation until the zero order diffraction maximum is no longer covered by the detector aperture. The detection of higher order diffracted terahertz radiation only slightly improves the image reconstruction quality. The study also shows that it is necessary to reduce the impact of diffraction by the implementation of pixels in the spatial terahertz modulator whose width is sufficiently larger than the wavelength of the used terahertz radiation. In comparison, spatial terahertz modulators with pixels sizes of the order of the wavelength inhibit a high image reconstruction quality even if higher order diffracted terahertz radiation is collected.
Funding
The Air Force Office of Scientific Research (AFOSR) (FA9550-15-1-0488).
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