Coherent reconstruction of a textile and a hidden object with terahertz radiation: supplementary material

phase initial max acceptance the 1 . The diﬀraction patterns were acquired at equidistant planes separated by ∆ z from each other. Results are shown for two values of ∆ z , which have been normalized to a reference depth of focus While in the main manuscript the steps of the Single-Beam Multiple-Intensity Reconstruction (SBMIR) algorithm have been outlined, here we detail its optimum acquisition parameters and resolution limits, focusing on real and simulated reconstructions performed at terahertz wavelengths. Next, we define the stitching operator used in the main manuscript for the separation of the two objects. Finally, complementary reconstructions of a textile hidden behind an object, as opposed to the standard object-behind-textile reconstructions reported in the main manuscript, are presented.


Optimum acquisition parameters and lateral resolution of the SBMIR technique at THz wavelengths
The minimum recording distance of a SBMIR phase retrieval experiment can be obtained using the Nyquist criterion. The fringe pattern with the smallest period δ produced by the wavefield scattered by the unknown object must be reliably sampled by the detector pixel pitch p. For the case of an object of size S at a distance v from the detector, one obtains δ = λ/[2 sin(α/2)], where λ is the wavelength and α ≡ 2 arctan[S/(2v)] is the angle under which an on-axis point on the detector sees the object. Then the recording distance must be large enough to satisfy 2p < δ, or sin(α/2) < λ/(4p). In the visible range [1] this limits α in the range of a few mrad, as λ optical /(4p) ∼ 10 −2 for standard CCD cameras. However, in a typical experiment at THz wavelengths one takes advantage of the dramatic increase in the wavelength by almost three orders of magnitude, leading to λ THz /(4p) > 1, i.e. the object can be placed as close to the detector as experimental constraints allow, thereby increasing the numerical aperture of the imaging system. By analogy with ptychography, the SBMIR algorithm requires that each diffraction pattern provides both new and redundant information upon translation diversity. The propagation distance in the z-direction after which the diffraction pattern of the object has significantly changed from the pattern at the distance v is comparable with the depth of focus z of the imaging system, defined as z ≡ λ(v/w) 2 , where w is the beam size. Therefore a reasonable choice of the longitudinal shifts ∆z is on the order of z .
We investigated the lateral resolution ρ of the SBMIR reconstructions with simulations, and the results are plotted in Fig. S1. A binary amplitude object in the shape of a nine-spoked Siemens Star with an inner diameter of 4 mm was reconstructed from its diffraction patterns taken at N z = 5 planes perpendicular to the z-axis with 200 iterations, started with flat amplitude and phase distributions as the initial solution. The minimum recording distance v 1 spanned the range 1.6 -12.6 mm, resulting in a maximum numerical aperture N A max ≡ sin θ max between 0.35 and 0.95, where θ max is half the detector acceptance angle seen from an on-axis point at the distance v 1 . The diffraction patterns were acquired at equidistant planes separated by ∆z from each other. Results are shown for two values of ∆z, which have been normalized to a reference depth of focus

Coherent reconstruction of a textile and a hidden object with terahertz radiation: supplementary material
This document provides supplementary information to "Coherent reconstruction of a textile and a hidden object with terahertz radiation," https://doi.org/10.1364/optica.6.000518. While in the main manuscript the steps of the Single-Beam Multiple-Intensity Reconstruction (SBMIR) algorithm have been outlined, here we detail its optimum acquisition parameters and resolution limits, focusing on real and simulated reconstructions performed at terahertz wavelengths. Next, we define the stitching operator used in the main manuscript for the separation of the two objects. Finally, complementary reconstructions of a textile hidden behind an object, as opposed to the standard object-behind-textile reconstructions reported in the main manuscript, are presented. 0 = 0.87 mm calculated at λ = 96.5 µm, w = 4 mm and v = 12 mm. For each reconstruction, the lateral resolution was estimated with the cutoff frequency of the modulation transfer function, obtained by averaging the reconstructed intensity along concentric rings with decreasing radii and centered with the Siemens Star, as detailed in [2]. Each simulation point plotted in Fig.  S1 is the average of ρ/λ, performed over 8 wavelengths in the range 96.5 -400 µm, while their standard deviation gives the total length of each error bar. For a fixed longitudinally scanned length, set by ∆z and N z , the lateral resolution is shown to be bound by the resolution limits ρ/λ = (2N A max ) −1 (plotted with a black line) and ρ/λ = (2N A min ) −1 (plotted with the color of the corresponding ∆z) from below and above, respectively, where the minimum numerical aperture N A min is calculated at the farthest recording plane at the distance v 5 from the object. The light-blue square indicates the lateral resolution of the experimental reconstruction of Fig.  2(b1) in the main manuscript, obtained with a real 4-mm Siemens Star hidden behind a glass fabric sample and retrieved with the procedure described in the main manuscript. Its value is compatible with the SBMIR reconstruction of the simulated Siemens Star, where no cover object was present.

2(b1)]
] Simulations max Fig. S1: Lateral resolution ρ normalized to the wavelength λ of the SBMIR reconstructions simulated with N z = 5 diffraction patterns at relative distances ∆z, used in 200 iterations. Two values of ∆z, plotted with different colors, were chosen and expressed in units of a reference depth of focus 0 . The estimated normalized resolutions are found to be bounded by (2N A max ) −1 (in black) and (2N A min ) −1 (in the color of the corresponding ∆z) from below and above, respectively. The minimum and maximum numerical apertures N A min and N A max are calculated at the farthest and closest recording plane, respectively. Each simulation data point is the average of the resolution obtained at 8 wavelengths, and the total length of each error bar is one standard deviation. The light-blue square shows the resolution of the experimental reconstruction in Fig.  2(b1) of the main manuscript.

Definition of the stitching operation for the separation of the transmission functions of the objects
Let us consider the diffraction pattern of an object moving with respect to the camera. In a reference frame fixed with the object, we denote with ∆x the displacement of the camera and with f and g the detector images taken before and after translating the camera, covering the detection areas F and G, respectively. We define the sum operator acting on f and g, S( f , g), as follows: where \ and ∩ indicate set difference and intersection, respectively, and g is derived from g in a way that ensures that f and g have the same mean value in their overlap region, i.e. g (x) ≡ α g(x) + β. The gain α and the offset β are used to compensate for exposure differences during the acquisition of the images and are obtained through a pixel-wise linear regression in the overlap region F ∩ G.
If the two images f and g contain circular data, as for instance phase values bound in the range [-π, π), care must be taken when performing the sum in the overlap region. In order to avoid 2π-jumps, the stitching operation between f and g, S( f , g), must be calculated according to S( f , g) = arctan2[S(sin f , sin g), S(cos f , cos g)], where α = 1 and β compensating for a phase offset between f and g bearing no physical relevance. In all the other cases, the sum operator applied to the two images directly provides the stitched image, that is, S( f , g) = S( f , g).
Once the operator S, acting on two images only, has been defined, we can extend the stitching operation to more than two images by defining the new, extended stitching operator S , acting on the set of images f m . Denoting with 1 m the indicator function of each f m , which equals 1 in the corresponding detector area and 0 elsewhere in the total stitched image, we define: that is, the stitching is performed pair-wise between the new image and the already stitched image. Figure S2 shows the reconstruction of the amplitude and phase object hiding the glass fabric sample. The samples, the acquisition and the reconstruction parameters are the same as those used for the results shown in Fig. 2