1. Introduction

In recent years, it has been demonstrated that terahertz radiation offers a great potential for possible niche applications in various industrial sectors. However, while there is no doubt that terahertz radiation holds some aces in its hands, the technology has struggled in its development due to low data acquisition rates, low power and high cost of the measurement systems. In particular, the lack of a fast data acquisition concept has significantly impeded progress in the research of terahertz spectroscopes with imaging capabilities, which is especially unfortunate, since the greatest potential of terahertz radiation has been identified in the realms of imaging spectroscopy. The reason for the assessment lies in the specific property that many materials, especially dielectrics, are transparent for terahertz radiation or have a unique spectral fingerprint. For this reason, inherently non-ionizing terahertz radiation is ideally suited for imaging purposes which could be widely used for quality control, medical and security applications [1–5].

Meanwhile, several measures have been taken to increase the data acquisition rate in terahertz imaging processes. Besides traditional detector arrays or improved raster-scanning [6–8], a promising step towards increased data acquisition rates was the exploitation of coded aperture imaging techniques for the terahertz technology. In this context, either compressive or fully sampled imaging methods have been used [9–11]. In coded aperture imaging, the terahertz source does not illuminate the object directly, but is previously spatially modulated by a modulator and the scattered radiation from the object is focused onto a single-pixel detector. By changing the modulation pattern, different projections of the scene are successively recorded and the image can be retrieved by either direct matrix inversion or corresponding retrieval algorithms. While coded aperture imaging relies on a well-known and optimized detection technology, i.e. terahertz single-pixel detectors, the highest hurdle in the implementation of such systems is the technological realization of spatial terahertz wave modulators.

In this respect, different approaches to modulate terahertz radiation have been investigated, as e.g. the use of metal masks [11,12], liquid crystals [13], optically pumped semiconductors and graphene [14, 15], or switchable metamaterials [16–21]. A promising step towards the reduction of data acquisition time was the introduction of frequency-division multiplexed imaging methods by Padilla et al. [22]. With respect to the modulation of terahertz waves, each of the aforementioned approaches displays advantages and disadvantages. While metal masks enable spectrally broadband terahertz wave modulation, they are usually not electronically reconfigurable and need mechanical realignment to generate different modulation patterns. In contrast, metamaterial-based modulators can usually be electronically controlled, yet lack of spectral bandwidth, which usually ranges in the order of tens to hundreds of GHz at a center wavelength of a few THz. In comparison, optically pumped semiconductors or graphene allow spectrally broadband modulation, but the modulation depth is limited when used as a spatial terahertz wave modulator, because of the spatial cross-talking between the pixels.

In our approach to spatial terahertz wave modulation, we consider the use of micromirror arrays (MMAs) for dynamic deflection of terahertz waves, similar to MMA spatial light modulators in the visible regime. While for visible light the micromirror size is of the order of 10 µm to avoid significant diffraction from the mirror, a terahertz wave modulator would require mirrors of the size of about 3 mm. However, micromirrors that large cannot be fabricated with state-of-the art technology. To circumvent this technical issue, we pursued a modified approach, in which an individual pixel is not defined by a single large-area mirror, but an ensemble of small micromirrors instead. Although the modification is seemingly straightforward at first sight, one has to take into account that the pixel-defining micromirrors don’t form a flat surface when they are tilted, but constitute a grating instead.

Here, we report the design and a thorough numerical investigation of the electromagnetic behavior of a spatial terahertz wave modulator that is based on a reconfigurable grating composed of micromirrors. The individual pixels of the modulator are defined by an ensemble of micromirrors with a length of 266 µm. A mirror size of such dimensions is required in order to minimize diffraction from the individual mirrors. It has been previously shown, that the fabrication of MMAs of similar size is feasible [23]. We show that such an MMA based approach allows us to modulate terahertz waves over a frequency range from 1.7 THz to beyond 3 THz at a modulation depth of more than 0.6.

2. Simulation model

The terahertz wave modulator proposed in this paper is based on a micromirror array (MMA). To date, most commercially available MMAs have been designed to work in the optical or infrared frequency range. In order to obtain a high spatial resolution, the mirror area must be as small as possible, i.e. of the order of the wavelength of the incident radiation on one hand. On the other hand, increased diffraction can be observed when the mirror size becomes too small, which automatically lowers the achievable spatial resolution. For this reason, an optimal mirror size must exist that limits diffraction and increases spatial resolution at the same time. Typically, the length and width of a micromirror is of the order of tens of micrometers in the visible and infrared range. In comparison with the wavelength of terahertz radiation, which is e.g. 150 µm at 2 THz, the mirror size of commercially available MMAs is much too small. Yet, scaling the mirror size up is challenging, since the distance between the base electrode, that controls the mirror position via electrostatic forces, and the mirror itself increases with the length of the mirror arm for a given inclination angle. To circumvent this issue, we chose an alternative approach that relies on the use of an ensemble of small micromirrors that define one pixel (see Fig. 1(a)). As an additional advantage of this approach, the pixel size can be deliberately varied by changing the number of individual micromirrors in the ensemble, even after fabrication of the device. In direct consequence, it is possible to dynamically adopt the resolution of an image.

 

Fig. 1 Modulation principle. The modulator consists of micromirrors that can be switched between two states. A pixel is comprised of a defined number of mirrors. In this figure, a pixel consists of an ensemble of 3 mirrors. This number can be deliberately changed. In the on-state (left part of Fig. (a)), the mirrors of one pixel are all aligned in a flat surface and reflect the terahertz beam. In the off-state (right part of Fig. (a)), the mirrors of a pixel constitute a reflection grating in Littrow configuration (b), for which the terahertz beam is mainly diffracted to the 1^st order back into its origin.

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Although the basic idea is seemingly simple, it must be noted that the multi-mirror pixel has the form of a grating when the mirrors are tilted at the same angle, whereas the ensemble of mirrors of a pixel lies in a flat plane for the non-tilted case (Fig. 1(b)). In the remainder of the study, we refer to two different states of a (multi-mirror) pixel. In the on-state, all mirrors of a pixel are aligned in a flat plane. In this case, all incident radiation is reflected under the reflection law into the detector. In the off-state, all mirrors of a pixel are inclined at an angle of 20^◦ and form a diffraction grating in Littrow configuration. In this case, almost all radiation is diffracted back into its origin.

In order to study our multi-mirror pixel approach, we first numerically evaluated the electromagnetic performance of a large multi-mirror pixel consisting of 32 individual mirrors. For the calculations, we used CST Microwave Studio Suite 2016. A sketch of the applied simulation model is shown in Fig. 2. The terahertz radiation travels from a waveguide port to the multi-mirror pixel consisting of 32 individual mirrors with a size of d = 266 µm each. These mirrors are arranged in a single row. The boundary conditions of the model are open at the edges of the incident plane, in the third dimension perpendicular to the incident plane electric boundary conditions are used. Effectively, the simulation model can be considered a 2-D cross section of a modulator. The mirrors can be switched between the on-state, in which the mirror surface is flat, and the off-state, in which all mirrors are inclined by an angle of 20^◦. The reflected or diffracted radiation is captured by an array of 320 probes that record the spatial distribution of the reflected or diffracted terahertz signal along a line perpendicular to the propagation direction of the reflected or diffracted terahertz beam, shown as a green line in Fig. 2. The recorded probe signals were summed up in order to obtain a single value corresponding to a single-pixel measurement.

 

Fig. 2 Schematic of the simulation model. The terahertz radiation of the port in the upper right corner is directed towards the MMA that modulates the beam. The reflected radiation is detected by an array of probes, depicted in green in the upper left corner.

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In order to obtain a high modulation depth between on- and off-state at the location of the probes, the MMA is utilized as a blazed grating in Littrow configuration for the off-state. This means that for a given design frequency, the incident radiation from the waveguide port is efficiently diffracted into the first diffraction order of the grating and thus directly back in the direction of the waveguide port [24]. At the same time, almost no radiation is diffracted into the zeroth diffraction order, which means that almost no diffraction travels into the direction of the probes. For the on-state, all mirrors are aligned in a plane and the incident radiation is simply reflected into the detector under the reflection law. As a benefit of this configuration, the flat plane guarantees that the phase fronts of the reflected radiation are not distorted even for short terahertz pulses, which stands in contrast to a grating configuration.

The modulator is designed to have its maximum modulation depth at a center frequency of about 2.2 THz. Note that the frequency for maximum first order backscattering and the frequency of minimal zeroth order diffraction in the direction of the detector are not identical. As a result, the maximum modulation depth is achieved at a higher frequency than the frequency for which the Littrow condition is fulfilled.

The dependence of the intensity distribution of the diffracted radiation from a blazed grating on the frequency f can be found in [24] as

To evaluate the spectral behavior of the grating, we derived the following expression from Eq. 15 of [24] by taking the finite width L of the grating into account

In the following, this general intensity distribution I_L (f) will be analyzed for two different orders of interest, the 0^th order which is backreflection to the emitter and the 1^st order where the detector is placed.

From Eq. (3), we calculated the spectral intensity distribution of the 0^th order diffracted radiation for an incident angle α = θ_B under the constraint n = 0 inserted into Eq. (2), which corresponds to a diffraction angle of β = −θ_B. Thus, we obtained

For the calculation of the spectral intensity distribution of the 1^st order diffraction (n = 1) the diffraction angle yields β = θ_B and α = θ_B due to the Littrow configuration

With the two equations (4) and (5) an analytical description of the modulator is possible.

3. Results and discussion

In a first step, we numerically calculated the intensities of the diffracted and the reflected radiation from the MMA for the off- and on-state (see Fig. 3). In this initial study, the whole micromirror array, consisting of 32 micromirrors, served as a single-pixel. As a reminder, the MMA is in the off-state when the diffraction into the 0^th order, i.e. into the detector, reaches a minimum. This case is depicted as the lower curve ( ) in Fig. 3. Here, the 0^th order diffraction minimum is located at a frequency 2.2 THz. For the on-state, all 32 micromirrors of the single-pixel are aligned in a flat plane and the intensity of the detected beam is almost independent of the frequency, as is expected from the reflection law (upper curve, in Fig. 3).

 

Fig. 3 Intensity modulation. Intensity of the detected beam vs. frequency for all mirrors in the on-state (upper curve, ) and all mirrors in the off-state (lower curve, ). The data originates from the numerical simulations.

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In order to analyze and physically understand the diffraction behavior of the modulator in the off-state, we calculated the spectral intensity distributions of the 0^th order and 1^st order diffraction by means of Eqs. (4) and (5), which is shown in Fig. 4(a) as the lower curve ( ), and the upper curve ( ), respectively. As a result, the frequency of minimal 0^th order diffraction into the detector at 1.9 THz is higher than the corresponding frequency of maximal 1^st order diffraction into the source at 1.66 THz.

 

Fig. 4 Intensity distribution. (a) Analytic calculation (dashed) of the intensity distribution of the 0^th and 1^st order diffracted beams, the vertical lines mark the maximum and the minimum of the 1^st and 0^th order, respectively, (b) and the corresponding numerical simulation (solid). Both evidence that the frequency of the maximal 1^st order diffraction intensity is lower than the frequency of minimal 0^th order diffraction intensity.

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To further substantiate our analytic findings, we numerically calculated the diffraction off the grating in the off-state by means of 3-D full-wave simulations. In the numerical simulation, the amplitude of the 1^st order diffracted radiation into the source is measured by the S₁₁-parameter (shown as in Fig. 4(b)). The maximum of the S₁₁-parameter is at 1.66 THz, which is in good agreement with the analytic result. Yet, the minimum intensity of the 0^th diffracted beam into the detector is observed at a frequency of 2.2 THz ( in Fig. 4(b)). In comparison with the analytic calculation, the frequency of minimum diffraction into the detector is shifted by 0.3 THz towards larger frequencies.

The disagreement between the analytic and the numerical model is caused by the use of a finite source, finite grating and finite detector in the numerical model, whereas the analytic model presumes infinite plane waves, with one single angle of incidence and a detection which is only sensitive to one specific angle in the far field. Due to aperture effects and the implied divergence of the beams in the numerical calculations, shifts of the interference maxima and minima are expected and caused by multi-angle interference, which explains the slight disagreement between analysis and numerical simulation.

Figure 5 illustrates the result of a numerical calculation of the electric field of the incident and diffracted waves for the off-case at frequencies of 1.66 THz (Fig. 5(a)) and 2.2 THz (Fig. 5(b)). At a frequency of 1.66 THz (Fig. 5(a)), the major fraction of the radiation is diffracted back into the 1^st order and thus into the source, while a minor part is diffracted into the 0^th order towards the detector. As expected, residual diffraction into the detector evidences that the off-state is not optimized for a frequency of 1.66 THz. Moreover, the diffracted intensity into the 0^th order has its minimum at a frequency of 2.2 THz, as shown in Fig. 5(b). In this case, most of the energy of the incident beam is diffracted back into the 1^st order towards the source. In addition, the −1^st order diffracted wave also carries a minor portion of the diffracted energy. Yet, since the −1^st order diffraction is directed away from the detector, it does not impair the modulation capabilities of such a grating modulator. The inset in Fig. 5 shows the spatial intensity distribution, which is a flat beam profile. This allows homogeneous illumination and is important for imaging applications.

 

Fig. 5 Diffraction orders. The electric field distribution is shown for two different frequencies, (a) the Littrow frequency at 1.66 THz and (b) the frequency of the highest modulation at 2.2 THz. The directions of the diffraction orders are indicated by the white arrows. The inset shows the spatial intensity distribution, which shows a flat beam profile even for 1.66 THz

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In order to evaluate the suitability of our structure as a modulator for terahertz radiation, we quantified the Michelson contrast as a measure for the modulation depth. In this respect, the modulation depth is defined as

As before, we first calculated the modulation depth for a single-pixel modulator that is comprised of 32 micromirrors. As a reminder, all 32 micromirrors are inclined at a given angle in the off-state, whereas they lie flat in the on-state. In such a case, the MMA is expected to provide the maximally achievable modulation depth. As shown in Fig. 6, a maximal modulation depth of 0.97 can be observed at a frequency of 2.2 THz. For a wide frequency span from 1.7 THz to beyond 3 THz, which is the highest frequency simulated, the modulation depth is higher than 0.6, which is an acceptable value for many applications [10,11].

 

Fig. 6 Modulation depth. The upper curve ( ) shows the modulation depth of a single-pixel modulator, in which the pixel is composed of 32 micromirrors. The modulation depth is determined between the states, where all 32 micromirrors are in the on-state and all mirrors are in the off-state. The middle curve ( ) shows the modulation depth of modulator pixels consisting of 8 individual micromirrors. The lowest curve ( ) depicts the modulation depth between on- and off-pixels for a 4-pixel-modulator with an alternating pattern of on- and off-pixels, in which each pixel consists of 8 micromirrors.

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For a configuration of 2 pixels, one in on-state and the other in off-state, with each pixel consisting of 16 individual micromirrors, the bandwidth is only slightly reduced, whereas the maximum modulation depth decreases to 0.87 and is shifted towards higher frequencies around 2.3 THz. For a configuration of 4 pixels with alternating pattern between on- and off-state, in which each individual pixel consists of 8 mirrors in this case, we observed a similar qualitative frequency-dependent behavior of the modulation depth as in the 32 micromirrors per pixel case. As expected, the modulation depth between on- and off-pixels is decreased due to edge effects at the interface between two neighboring pixels. The maximum modulation depth of 0.8 is observed at a frequency of 2.4 THz. In a frequency range from 1.9 THz to beyond 3 THz, the modulation depth is higher than 0.6. For pixels consisting of 6 or less individual mirrors the effect of the grating disappears and no clear distinction between the pixels can be observed, which results in a contrast below 0.6 over the whole frequency range observed.

With respect to application in computed imaging, it is beneficial when the modulated radiation energy of the modulator scales linearly with the number ratio between on- and off-pixels. For example, if a modulator consists of 4 pixels and two pixels are switched on and two pixels are switched off, the detected energy must increase by a factor of two in comparison with a configuration, where only one pixel is switched on and three pixels are switched off. This behavior must be observed independently of the specific location of the two on-pixels. To investigate the energy modulation properties, we numerically calculated the diffracted energy from a 4-pixel grating modulator into the detector in dependence on the ratio between the number of on- and off-pixels. In this consideration, we also accounted for various possible on-off pixel patterns at any given ratio between the number of on- and off-pixels. In Fig. 7, the detected energy is shown in dependence on the on-off pixel ratio for different modulation patterns. Assuming that every pixel contributes one quarter of the energy to the total energy in the all-on-configuration, the slope of the increase of the detected energy with increasing on-off pixel ratio is expected to be 1, which is confirmed by the simulation results in Fig. 7. A maximum deviation of the energy curve from the expected energy value has been observed for the on-off-off-on-configuration, where the pixels at the edge of the MMA are switched on and the pixels in the center are switched off. In this case, the relative deviation of the detected energy from the theoretically expected value was about 7 %, which can be used as an estimate of the accuracy of the modulator.

 

Fig. 7 Detected Energy. The detected energy is shown for different modulation patterns of the MMA at a frequency of 2.2 THz. The patterns are all possible permutations of 4 pixels. The dashed line shows the hypothetical energy that is expected under the assumption that every pixel contributes one quarter of the total energy, when all 4 pixels are in the on-state. The energy is normalized to the maximum energy when all pixels are switched on.

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In a proof-of-principle study, we numerically studied the suitability of a grating-based modulator for application in coded aperture imaging. For this purpose, we calculated the image of an absorbing block that is illuminated by a source, as shown in Fig. 8. The width of the block corresponds to 1.5 pixels and obscures half of the area of the second pixel and all of the area of the third pixel of our 4-pixel grating modulator. We modulated the grating and retrieved the fully sampled image by matrix inversion.

 

Fig. 8 Schematic of the simulation model with object. The object is placed between the waveguide port and the MMA. It conceals half of the area of the second pixel and the total area of the third pixel.

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For this purpose, we performed simulations for four different modulation patterns (all-on, on-off-off-on, off-on-off-on and off-off-on-on), which permit a unique retrieval by matrix inversion of a four-pixel image. Figure 9 shows the retrieved image in terms of the retrieved intensity for each pixel. The red crosses ( ) depict calculated reference intensities for each pixel without object in the beam path. As can be seen, the reference intensity distribution is reasonably flat with a slight decrease of the intensity towards the edges. With the object in the beam path (shown as blue circle symbols in Fig. 9), the intensity of the second pixel drops to the half of the intensity of the reference calculation, while the intensity of the third pixel is almost zero. This was expected, since the object obscures half of the area of the second pixel and the total area of the third pixel. The fourth pixel agrees with the expected value of the unblocked beam, which is also expected due to the object. Only the first pixel shows an unexpected behavior, as its intensity is larger than in the unblocked case. A reason for that might be scattering at the edge of the investigated object. Yet, in principle the calculations evidence that a grating modulator can be potentially used for computed imaging.

 

Fig. 9 Retrieval of the object. The object is retrieved from 4 measurements with a 4-pixel modulator and a single-pixel detector. The image is retrieved by matrix inversion. The red crosses ( ) denote the reference intensity profile of the beam without object in the beam path, whereas the blue circles depict the intensity distribution of the image of the object ( ).

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4. Conclusion

We numerically investigated the electromagnetic performance of a spatial terahertz wave modulator based on a reconfigurable grating composed of micromirrors. We evidenced that the device is capable of modulating terahertz waves over a wide frequency range from 1.7 THz to 3 THz at a modulation depth larger than 0.6. Since individual pixels in the modulator are defined by an ensemble of micromirrors, the pixel sizes can be dynamically changed and need not be homogeneously distributed over the modulator area. We have shown for the example of an object with binary terahertz transmission that the image, i.e. the spatial distribution of the terahertz transmission through the object, can be successfully retrieved.