1 Introduction

This paper aims to investigate unusual single-scan, two-dimensional space (2D) imaging capabilities in the bandwidth of terahertz (THz) frequencies [1], simply using a very basic setup of interest in the field of time-domain spectroscopy (TDS). Applications involved include, more especially, the characterization of small size, plus or minus complex assemblies of mixed compositions, and the detection or localization of defects in inhomogeneous pieces of materials. Here we do not just consider pulse shaping [2] and its related applications, but rather focus on the search of spatial resolution in view of a fast and robust acquisition process. This actually takes place among the current topics of major interest in the field.

Most current TDS setups in industrial applications nowadays are made of a single-cell source, to be coupled with a single-cell detector, and of a motorized delay line for step-by-step scanning. The search of 2D spatial resolution under these conditions implies sequential processes, very often, via time-consuming scan sequences. A number of works have been published recently, where the authors try to reduce or overcome this kind of issue. A first option [3] involves single line sourcing, which concerns with 1D imaging using a single-cell 1D detector. This concept enables rapid monitoring with moving objects. It has been proved to be very efficient, but at the expense of added complexity in terms of optical architecture. A second option involves fully multichannel detection, either using electro-optical sampling [4] or matrix antennas [5]. In the latter work, the authors experience a linear bar of photoconductive, closely coupled detectors. They evidence impressive performance and imaging speed in the 1D space. But this also implies a complex and expensive architecture, with significant inherent issues regarding calibration and alignment. Still consistent with more standard technological designs using single-cell active components, other innovative schemes have also been proposed on the basis of the coupling with dual-echelon optics [6]. Thanks to using 2D electro-optic conversion and 2D detection this efficient principle has been validated within the frame of single-scan triggering, referring to the so-called single-shot mode of operation. This also refers to the technique of space-to-time encoding (STE), still requiring 2D detection. In addition, it is worth to remind complementary routes by means of pure numerical processing. Starting again from 2D matrix detection, for example, various images from the same scene may be combined accordingly to the technique of aperture synthesis [7]. This has been validated in a THz configuration comprising multiple mirrors, thus enabling the optimization of spatially resolved data by means of an appropriate use of the intensity–phase relationships.

In order to go on with the more standard and robust TDS technology using single-cell active components, we propose hereby an original, fairly simple, robust and inexpensive, but still fully 2D configuration. Our measurement process enables the acquisition of multispectral images, in the normal plane to the propagation axis of the THz beam, by coupling appropriate silicon phase mask in the setup and taking advantage of a single scan. Phase masks have already been experienced in other works in the field, but, to our best knowledge, for pulse shaping [8] and not yet in 2D imaging configurations based on STE with pulse replication. We attempt to verify that an appropriate mask may help to generate equally spaced, coherent replicas from the seed pulse. Each of these replicas can be connected to a given location in the mask thanks to STE. The unique 2D correspondence between the specified locations and temporal windows gives access to efficient discrimination capabilities in the determination of local values of the complex index, simultaneously, during a unique scan.

This paper focuses on a benchmark analysis for the sake of a generic validation of the concept, including the explanation of basic limitations. Stacking small size bars of high-resistivity silicon, we make use of two kinds of masks with square patterns, either comprising four or nine pixels (2 × 2 and 3 × 3 phase masks). Our study is organized as follows: First, the experimental TDS setup and operating conditions are specified. Then we implement a number of preliminary tests for consistent benchmark to evidence STE and introduce actual limitations in the discrimination capabilities between adjacent pixels. This involves phase shifting and transmission issues pixel per pixel. Second, a variety of specific devices under test (DUTs) with different material combinations are coupled to a 2 × 2 phase mask, aiming to validate global performance for multispectral analysis by means of fast Fourier transform (FFT) and dedicated numerical computations. In a third step, thanks to 2D finite elements modeling, we discuss generic performance limitations due to diffraction and propagation in the presence of thick and sharp apertures. This will help to suggest possible upgrades of further concern in the optimization of the spatial resolution.

2 System design

2.1 Operating conditions

The optical architecture of any TDS setup is figured as shown in Fig. 1. A set of off-axis parabolas is used to ensure proper focusing, collimation and beam alignment between the single-cell emitter and detector. The motorized delay line is assumed to prevent any parasitic misalignment effect, but defining the suitable temporal excursion and scanning precision. Actual numbers of interest are about 80 ps and some thousand steps, respectively, which meet further needs in our study for small series of up to 10-ps long time steps. The diameter of the collimated THz beam is 10 mm FWHM. In order to implement the phase mask, we just provided free possible access, a few centimeters long in air, along the final section of the collimated beam path, facing the detector. This permits close coupling between the DUT and the phase mask, with easy installation and removal. Considering basically planar geometries, there is no critical requirement in terms of angular positioning. This is the reason why the only critical issue, in our situation of a rather small beam size, consists of the attention to be paid to careful centering of the mask aperture with respect to the beam axis. The precision needed is a couple of hundred µm.

Fig. 1
figure 1

Simplified, standard TDS setup with a 2 × 2 phase mask, as implemented into the collimated beam path

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The final adjustment of the image of the center of the mask into the detector antenna is realized via the rear parabolic mirror. This leads to minimal positioning uncertainties. Finally, in order to get free from too significant moisture effects and parasitic absorption, the whole beam path is closed and makes use of dry air when starting the system.

2.2 Definition of phase masks

We need a variety of phase masks with different geometries to be implemented by stacking silicon bars. Thanks to usual polishing and cleavage techniques, silicon wafers enable the realization of high-quality bars with a fine definition of geometrical tolerances. This makes the realization of step-like geometries very easy. Prior stacking, the geometrical uncertainties lie in the range of 10 µm and even less, given bar lengths from 10 to 20 mm. Furthermore, the physical properties of high-resistivity silicon enable a fairly low spectral dispersion and a low absorption throughout the whole THz bandwidth. This takes place among the most important criteria in our selection. The main penalty just arises from elevated Fresnel losses, to be afforded when using uncoated material. The real index of about 3.4, to be verified below, then leads to insertion losses in the range of 3 dB whatever the configuration of the mask. Anyway, due to the high dynamic range of usual TDS setups, typically in excess of 50 dB, this still remains acceptable. The main benefit of a high index consists in the definition of possibly large and finely controlled phase steps, using rather thin bars. Then we split an uncoated, optically polished wafer into a series of individual bars of which the length, the width and the thickness are 15, 1.5 and 0.5 mm, respectively. This matches the values we need below to adjust the pixel dimensions and the phase step in further masks, Δ = 1.5–4.5 mm and d = 0.5–1 mm, respectively. The required number of bars is stacked and glued together, accordingly to the suitable configuration of the mask (Fig. 2). Attention being paid to limit the thickness of glue below a few µm, typically, the stacking process does not implicate unaffordable penalties in terms of additional parasitic absorption, or even second-order diffraction effects. In comparison with bulky masks to be engraved when using industrial processes, the major drawback of this prototyping process consists in slightly larger geometrical uncertainties, about 50 µm throughout the whole aperture of interest.

Fig. 2
figure 2

Experimental conditions involved in the benchmark analysis for STE: a 2 × 2 phase mask using Δ = 4.5 mm and d = 1 mm with coupled silicon bars, b 3 × 3 phase mask using Δ = 3 mm and d = 500 µm. Darker gray colors in b figure bar stacks with more bars

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In order to ease benchmark and the implementation of masks, the pixel with the lowest order pixel just consists of a square air location (pixel #0, Fig. 2). The basic configuration of a 2 × 2 phase mask for benchmark comprises two perpendicular stacks of 1 × 3 and 2 × 3 bars, with Δ = 4.5 mm. This geometry takes advantage of the lower diffraction effects and lower geometrical uncertainties. The second configuration for a 3 × 3 phase mask with Δ = 3 mm involves a series of 1 × 2 up to 4 × 2 bars, simply varying the spatial overlap between bars in such a way that location numbering matches the actual number of stacked bars pixel per pixel. This is denoted by convention, in view of further use. In both cases, the overall aperture matches the diameter of the seed beam, about 10 mm FWHM.

By principle, the selected value for d must be large enough to get free from any hazard of temporal overlap between adjacent replicas, whatever the peak value of transverse gradients in the optical thickness across the DUT. If d = 500 µm, adjacent replicas are spaced by about 4.5 ps.

3 Experimental results

3.1 Benchmark measurements in the configuration of the 2 × 2 phase mask

Efficient benchmark to demonstrate STE essentially implies a fine control of discrimination effects, pixel per pixel across the whole aperture of the mask. This is made possible by displacing closely coupled, square pawns or rectangular bands, with similar characteristic dimensions as those of the pixels. Then, a set of pawns and bands have been shaped, either consisting of thin metallic plates or of 500-µm-thick silicon pieces. Latter pieces are made using the same silicon bars as those of the masks.

Fairly efficient discrimination capabilities are evidenced this way (Fig. 3). As expected, this can be verified as well in terms of local transmission in the case of metallic pieces and of local phase shift with silicon pieces. Our process then enables a very fine monitoring to identify the location and the nature of pieces involved, when closely coupled behind the 2 × 2 phase mask. The relevant four temporal windows are enlightened, darkened or temporally shifted as shown, a consistent way when displacing metallic and silicon pieces.

Fig. 3
figure 3

Discrimination of a metallic pawns and bands to evidence (a) intensity encoding, or of b silicon pawns and bands with d = 500 µm to validate phase encoding by means of the 2 × 2 phase mask. Depending on the selected positions and pieces, pulse replica can be transmitted, canceled or temporally shifted

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For illustration (red plot, Fig. 3a), a metallic band being positioned on pixels #3 and #4 and the only two replicas which remain lightened are those connected to pixels #0 and #1. A single metallic pawn being positioned on pixel #2 (black plot, Fig. 3a), and this also obviously determines the only replica which experiences complete darkening. Then replacing the metallic pieces with d-thick silicon pieces of the same sizes, we verify the consistency of the measured value of the local phase shift applied with the selected phase step in the same mask. This enables us to move or superimpose various replicas, or make them disappear (Fig. 3b). For illustration, coupling a silicon band onto pixels #1 and #3 (blue plot, Fig. 3b), a fifth replica clearly appears besides the fourth initial replica and complete darkening can be observed, apart from initial other replicas. Coupling a unique silicon pawn onto pixel #4 (black plot, Fig. 3b), the fourth initial replica is temporally shifted toward the expected time window, which matches quite well the expected delay of about 4.5 ps. Our 2 × 2 phase mask then enables easy play in the 2D space, as defined in the closed area besides the mask, normally to the propagation axis.

3.2 Upgrade of the spatial resolution in a 3 × 3 phase mask

Useful complementary results are obtained, in view of the analysis of STE limitations, when considering the 3 × 3 phase mask and decreasing Δ down to 3 mm (Fig. 4). We make use of the same phase step as above, d = 500 µm. Here again, true discrimination capabilities are evidenced the same way. The total number of replicas and spacing still remains consistent with the geometrical data. But discrimination capabilities start to decrease slightly, due to larger noise and distortions. Some spurious variations in the peak intensity and distortions appear between neighboring peaks.

Fig. 4
figure 4

Monitoring the displacement of metal pawns thanks to the 3 × 3 phase mask for d = 500 µm

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These distortions may be partly explained by possible deleterious effects of larger geometrical uncertainties, by comparison with the former 2 × 2 phase mask, arising from bar stacking. This is consistent with other results, which are not pictured here due to the lack of any significance for STE, as provided by unsuccessful tests attempted when reducing the size of pixels down to Δ = 1.5 mm. But they mainly arise from inherent limitations, which need to be clarified by means of a comprehensive theoretical analysis. Indeed, it will be shown that basic STE limitations are governed by coupling effects due to edge diffraction between adjacent pixels and to possibly deleterious spatial overlap phenomena between delayed parasitic waves. Prior going on with modeling, it is now time to verify the effectiveness of multispectral imaging when experiencing a number of dedicated DUTs.

4 Analysis of multispectral imaging capabilities

4.1 Test of material assemblies with various compositions

In order to minimize the influence of geometrical uncertainties, we go on with the 2 × 2 phase mask, simply upgrading the phase step up to d = 1 mm. This ensures no spurious temporal overlap, given peak variations of the local optical thickness of our DUTs. A variety of inhomogeneous assemblies have been carefully packaged, by filling the four locations contained in a very thin plastic grid with properly cut material samples. Depending upon the material of interest, the sample thickness ranges from 0.6 to 1.5 mm.

The geometry of the grid being precisely matched with those of the pixels in the mask, we pay a particular attention to proper alignment between samples and facing pixels. Consistent benchmark measurements can be made this way, under properly managed conditions, by replacing the former metallic and silicon pieces with a DUT to be selected on demand. We aim to identify the nature of the four material inclusions involved in a given DUT, simultaneously, operating the setup in a single scan. Figure 5 presents a number of raw temporal traces measured behind the mask, including various combinations of leather, rubber, grease and air holes. This selection takes advantage of very different spectral characteristics in the bandwidth of THz frequencies, which eases comparison and validation. According to standard material properties, rubber is reputed to exhibit much more linear absorption than grease or leather. On another step, the real part of the refraction index in leather should exceed by a large amount its value in grease.

Fig. 5
figure 5

Temporal measurement results to validate multispectral imaging capabilities, using various combinations of material inclusions with a leather and rubber, b leather and grease, facing specified pixels in the 2 × 2 phase mask

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For illustration, it can be verified on Fig. 5a that the intensity of replicas connected to rubber inclusions is reduced by a significant amount below the corresponding values for grease and leather inclusions. Similarly, the time shift of replicas for grease, leather and rubber inclusions is higher than its value for air holes. But in any case, to make sure of no confusion, it remains lower than its value connected to former silicon pieces with the same thickness. Any one among the sets of transmitted replicas can be clearly distinguished from its neighbors. These plots contain all the experimental data required prior the extraction of relevant multispectral characteristics, pixel per pixel.

4.2 Validation of imaging capabilities thanks to numerical computations

At this step, we need to develop appropriate numerical calculations. These calculations must be operated within each of the four temporal windows defined hereby, in order to extract the pixellized value of the complex index n(ω) as a function of the angular frequency ω. The computational principle is based on standard FFT, aiming to determine the local expression of the spectral transfer function pixel per pixel in the range of 0.1 and 1.5 THz frequencies. The FFT being a linear operator, there is no problem to deal with variations in the temporal distortions from replica to replica, due to edge diffraction, as observed prior positioning the DUT. Calculations to be operated along a specified temporal window just need to refer to the actual input pulse shape contained, as connected to the replica of interest. It is worth to underline that the computational process also makes use of preliminary spectral compensation, to remove parasitic distortions due to residual humidity along the beam path from the local value of the raw transfer function. This provides an effective value of this transfer function, which gets free from unwanted absorption lines due to the ambient vapor. Compensation is operated thanks to the determination of the ratio of the total transmission to air transmission, prior positioning the DUT. In terms of calibration requirements, it is also worth to underline that the combination of air and of silicon pixels in the same mask needs to make use of a particular numerical process. This either involves one or two preliminary transfer functions, depending upon the air or silicon nature of the selected pixel. This way, a proper correction can be applied in view of true spatial homogeneity. In addition, we implement zero-padding prior operating local FFT. This helps us to make sure of preserving the spectral resolution required and to guaranty relevant variations for consistent outputs, despite the limited width of usable temporal windows between neighboring replicas. The technique provides satisfactory results in any case of no unwanted temporal overlap. Denoting T exp(ω) the effective value of the local transfer function, as determined from Fig. 5, several kinds of algorithms may be selected for the extraction of n(ω). In our situation, referring to [9] with notations c, d DUT and T(ω) to figure the speed of light in vacuum, the local thickness of material and the complex value of the theoretical transfer function related to local transmission, respectively, we consider the following expression:

$$T(\omega ) = \frac{4n}{{(n + 1)^{2} - (n - 1)^{2} \exp \left( { - 2jn\frac{{\omega d_{\text{DUT}} }}{c}} \right)}}\exp \left[ { - j(n - 1)\frac{{\omega d_{\text{DUT}} }}{c}} \right]$$
(1)

This model does not require multiple echoes, as sometimes needed in other computational processes [10] to account for the presence of Fabry–Pérot effects in the DUT, but applies to a unique seed pulse in the setup. In comparison with other sophisticated processes dedicated to ultimate optimization [11], usually at the expense of complex and time-consuming algorithms, it takes advantage of a true consistency in our study. The expression (1) provides relevant values of n(ω) in any case, whatever the values of the optical thickness and of the absorption ratio in the sample. There is no particular issue with possibly large gradients in the optical length or in the optical absorption between adjacent pixels. Due to the mathematical expression (1), in the form of the product of a periodic function with an exponential, the main assumption for possible application only consists in the availability of appropriate numerical tools. Despite the apparent simplicity of (1), the search of vector solutions n(ω) does not still remain as obvious in terms of numerical analysis. Writing the identity T exp(ω) = T(ω), we define a local equation of reference, pixel per pixel. The complete solution to this equation still remains much more complicate to determine than that with more usual approximations for T(ω), which often apply to the absorption, or to the sensitivity to Fabry–Pérot resonances in the situation of optically thin DUTs.

After developing a specific code based on a combination of MATLAB™ and MAPLE™ resources, we get the computational results presented in Fig. 6. First of all, it is worth to notice that the extraction process has been applied to the determination of the actual spectral properties of actual pieces of high-resistivity silicon used to build the masks. This was done, as well as for reference in terms of material data and for preliminary control of the effectiveness of computations. Here we just verify the nearly constant value of n(ω) in this material, throughout the whole bandwidth 0.2–1.5 THz. It can be verified that the absorption remains quite low, as expected. These results must be connected to the notations n(ω) = n o (ω) − jκ(ω), which applies to the real and the complex parts of the index in (1), and to \(\alpha (\omega ) = \frac{2\omega }{c}\kappa (\omega )\) which applies to the definition of the related coefficient of linear absorption. They are consistent with previously published data dedicated to other high-resistivity silicon materials [12], thus confirming a real index of about 3.5.

Fig. 6
figure 6

Extraction of local material properties from former temporal data, in terms of variations of a the real and b imaginary parts of the computed complex index, together with c the absorption coefficient

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Rubber reveals the highest average value for absorption, as expected, with a broad, smooth peak near 0.7 THz. Even though a straightforward comparison with published data from the literature is not as evident as for silicon, due to the strong influence of additives possibly included in this kind of material [13], we still verify that α(ω) lies in the relevant range of 30–100 cm−1. Relevant values can also be noticed, at least in terms of orders of magnitude, in the other two plots figuring calculated data for leather and grease. Considering leather and the applicability of the method for industrial needs, it is worth to remind the sensitivity to moisture effects [14]. Grease is characterized by a low value of the real index, which matches rather well those of published values for specific compositions [15], typically 1.5–4 from 0.2 to 1.8 THz. Here again, highly variable absorption coefficients may be obtained, depending upon the basic composition and the nature or density of inclusions. The location and width of broad peaks in rubber and leather, near 0.5–0.7 THz, could be connected to a detailed analysis of the chemical composition of our samples. But this is out of the prime topic of the paper. Finally, our computations involve a pre-determined value of the sample thickness (d DUT). But other numerical methods [16, 17] should exhibit a particular interest in the more generic case of impossible pre-determination, for the sake of a simultaneous determination of n(ω) and of the unknown value for d DUT.

Following this benchmark of the concept of phase masks for multispectral imaging and the validation of discrimination capabilities, we still need to get back to the discussion of ultimate limitations in the optical performance.

5 Performance limitations and further upgrades

5.1 Theoretical design of propagation effects throughout arbitrary phase masks

A consistent analysis of the theoretical limitations of STE must be based on the comprehensive description of coupled effects due to diffraction and to propagated reflections along the beam path, versus the geometry of the phase mask involved. We aim to discuss the consistency of former experimental results, prior defining the optimization routes and possible upgrades. A number of authors already designed the spatiotemporal transformations experienced by THz pulses in the presence of phase changes and spatial filtering effects during propagation [18]. Others works in the field have also been dedicated to the understanding of diffraction effects across thick apertures [19], thus providing data of interest for orders of magnitude. But a comprehensive understanding of scaling phenomena, in our particular situation of silicon phase masks with square patterns and sharp silicon edges between neighboring pixels, will help to specify more relevant numbers and look at the generic trends. This will also help to determine further needs for complementary pre-calibration, pixel per pixel across the mask.

We develop a complete numerical model based on finite elements computations, in the 2D space–time domain. We are looking for a unique tool to analyze the gradual formation and the propagation of small size wavelets with distorted wave fronts, versus the characteristics of the seed pulse. This model leads to consistent estimates for coupling effects between adjacent pixels in the presence of delayed trails issued from neighboring pixels, accounting for the effect of pulse distortion. It makes use of an input plane wave, in the form of a short Gaussian pulse E(t, x, y), which seeds a combination of silicon pixels with adjustable size and phase step. The seed pulse comes from the left boundary of a rectangular computational domain, comprising air and silicon areas. This pulse propagates accordingly to a standard, time-dependent equation in the form of \(\nabla X(\nabla XE) + \sigma \frac{\partial E}{\partial t} + \frac{\partial }{\partial t}\left( {\varepsilon_{o} \varepsilon_{r} \frac{\partial E}{\partial t}} \right) = 0\). An estimate is specified as a function of the position in the domain, finely meshing silicon and air areas in order to figure the local value of the material permittivity ε(x, y). We set up absorbing conditions on the other three boundaries of air to prevent spurious numerical reflections. Under these conditions, Fig. 7 shows the variations in the computed distribution of the electric field versus the size of pixels, after launching a 500-fs-long pulse from the left side. The time delay between adjacent replicas is about 11 ps, when considering d = 1 mm. An arrangement with three pixels has been selected as a simplified, but fully representative pattern. This geometry matches a self-consistent approximation to figure the four phase steps in 2 × 2 phase masks, namely air-1 × d, air-2 × d, 1 × d-3 × d and 2 × d-3 × d steps of Fig. 2a. Higher-order effects related to coupling phenomena in the 3D space domain are not included in this description. But our 2D space domain takes advantage of easier implementation and of reduced computational times, saving memory allocation to enable a fine resolution for meshing. This must be finely controlled for physical consistency, in order to remain consistent with the spectral content of a short seed pulses.

Fig. 7
figure 7

Spatial variations in the distribution of the electric field, gradually increasing the width of pixels (gray blocks) Δ from 1 to 3 mm, given a constant phase step d = 1 mm. Spatial distributions of the electric field in the air path ensure a time delay of 45 ps, after the launch of a 500-fs-long pulse from the left side

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Figure 7 then provides an example of our results to illustrate the variations in the effectiveness of STE, while gradually increasing the size of the pixels. As long as Δ does not exceed 1.5–2 mm (Fig. 7a), individual wavelets associated with the expected replicas do not separate properly. Some residual, but rapidly decreasing coupling effects remain visible toward Δ = 2 mm (Fig. 7b). Values of Δ in excess of 2.5 mm (Fig. 7c) enable fairly efficient discrimination between adjacent replicas. Individual wavelets clearly start to separate one from each other, enabling proper spatial discrimination between neighboring components. This can be verified despite the remaining trail waves which still go on propagating from place to place, arising from the lateral areas and from co-propagating Fresnel reflections. When integrating the total electric field along the vertical axis of the computational domain, time after time, we can figure the temporal variations of total intensity to be coupled into the aperture of the detector in the Fig. 1. Neglecting the contribution of its inherent transition time, this may be reputed to fit the expected response of the TDS. The relative total intensity of parasitic waves varies approximately as the reciprocal of Δ. This participates in the relevant criteria, in order to set up the acceptable size of smaller pixels in the range of 2.5–3 mm and verify a rather good agreement with former experimental results.

From the viewpoint of a more generic interpretation, we must also underline that former modeling results fully take into account the limitations of our assumption of a nearly constant beam size. This we made for simplification, when replacing its frequency-dependent value with a constant despite the impact of broadband spectral dispersion. But it can be shown that such a simplification does not affect the validation of the concept, at least in the first order. If w o and z figure the respective values of the reference beam waist, starting from the emitter of the TDS, and the position of the DUT in the beam path, the beam size is expected to vary with the selected angular frequency ω according to a relationship of the form of \(w(z) = w_{o} \sqrt {1 + \left( {\frac{2cz}{{w_{o}^{2} \omega }}} \right)^{2} }\). Higher frequencies actually involve a slightly smaller beam size. Considering the upper and lower boundaries of interest throughout the whole spectrum of interest, from 1.5 down to 0.1 THz, w(z) undergoes theoretical variations from 1.1 to 15 mm at the location of the mask. Its value of 8.5 mm at the central frequency involved is not too far from our specified value of a constant 10 mm FWHM, i.e., 17 mm at 1/e 2 near the image focus of the parabola. On the other step, the design of our setup, which combines metallic reflectors with integrated antennas, helps to take advantage of a fairly low chromatic dispersion in the normal plane to the beam axis.

5.2 Possible routes for optimization

First, we may replace our current parabolas with other ones of a larger optical aperture. An increase in the input beam size by a factor of 2–3 should provide a rather simple option to upgrade the optical architecture of the TDS setup, including the replacement of our computer-controlled delay line with a longer one in order to preserve consistent scanning conditions. This way the spatial resolution should be possibly extended by a factor of up to 10, or even more. The subsequent penalty in terms of dynamic range, 10 dB in this case, remains consistent with inherent numbers in the range of 50 dB. Highly resolved, relevant multispectral measurements are still permitted in an optimized setup.

Second, the replacement of our current prototypes with industrial phase masks of improved quality should help to reduce residual geometrical uncertainties and minimize diffraction effects. Indeed, bulky masks may be realized using standard industrial processes, with precise engraving and possible edge apodization. This should help to lower diffraction effects at the location of silicon edges, thus removing higher spatial frequencies due to the superimposition of trailing edge components on useful replicas. Furthermore, anti-reflective coatings should be added on both sides of these masks, to reduce the overall insertion loss and residual reflections.

Third, it should be preferable to extend the operating bandwidth as far as possible, over the limit of 1.5 THz which was available in our TDS setup at the time of this study. In addition to appropriate apodization as a function of the bandwidth involved, this will help to reduce the limitations due to pure diffraction. Bandwidths of at least 3–5 THz remain consistent with cost-effective designs for industrial needs.

Finally, further improvements in the direction of smaller values of Δ, i.e., 2 mm or even less will also benefit from the development of complementary tools for calibration and post-processing. This mainly applies to the analysis of possible trade-offs in the pre-compensation of spatial overlap effects between delayed trails and useful replicas from the main pulse, thus becoming acceptable under appropriate assumptions for a fine control. The main issue, to be designed by means of numerical methods for ill-posed problems, should imply additional limitations in the attainable dynamic range. But this was out of the scope of our study.

This concept may be coupled to more sophisticated options, as those of 1D designs cited in the references above.

6 Conclusions

The principle of space-to-time encoding with high-resistivity, silicon phase masks then consists of an efficient option to implement 2D imaging with spatially resolved, multispectral measurement capabilities in a standard TDS setup. The setup being operated in a single scan, the concept enables the development of fast, still inexpensive and fairly robust equipment. This enables the extraction of self-consistent values for the local value of the complex index, in order to characterize planar, inhomogeneous assemblies with various material compositions. Alternatively, our configuration also applies to the determination of spatial gradients in the thickness of homogeneous pieces of bulky material. The demonstration was made within the frame of a benchmark analysis, thanks to prototyping masks with 4 and 9 pixels. Accounting for basic theoretical limitations, as identified thanks to the numerical model provided, a number of routes still remain to be explored in view of further optimization. A realistic prospective indicates possible resolutions in the range of 9–16–36 pixels, of which the size may be reduced down to 2–2.5 mm typically. This should contribute to the development of improved designs with a higher spatial resolution and of interest for new generations of multispectral imaging systems.