We present an imaging technique in which the broadband frequency information of terahertz (THz) pulses is transformed into spatial resolution. Efficient blazed diffractive gratings spread the individual frequency components over a wide and defined spatial range and f-theta optics are employed to focus the individual components onto a one-dimensional image-line. Measuring the time domain waveform of the THz waves allows therefore for a direct reconstruction of spatial sample characteristics as the spatial domain information is encoded in the terahertz spectrum. We will demonstrate terahertz imaging on selected samples with an improvement in acquisition speed up to two orders of magnitude.
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The development of the terahertz (THz) technology has progressed steadily over the last decades and yielded a variety of different optoelectronic THz system architectures. Particularly, the THz time domain spectroscopy (TDS) is of high interest for various applications [1, 2]. Here, femtosecond lasers are used to drive photoconductive antennas, typically made of low-temperature grown gallium arsenide . Within the antennas, the fs laser pulses excite free carries which are accelerated by a bias field and thus, short current transients result. These transient currents are the source for ps-long THz pulses which consist of an almost continuous spectrum that spans from a few tens of GHz to several THz . Therefore, the information delivered from a single measurement allows a broadband characterization of the complex dielectric properties and the thicknesses of samples at the same time . Hence, the TDS technique is ideal for applications where broad spectral information is required.
However, while TDS systems are being steadily improved and a number of commercial systems is already available [6, 7], fast THz imaging is still a challenge. Common TDS systems exhibit only one single spatial channel. Hence, to spatially analyze the sample, a time consuming pixel-by-pixel scan has to be performed. An emerging promising alternative to pixel-by-pixel scans is the use of compressed sensing methods  where spatial light modulators [9, 10] are employed to encode information on the THz wave that can later be used to reconstruct the sample geometry from a series of THz time-domain waveforms. It has been shown that due to the compressing, it is possible to reduce the number of measurements by up to a factor of 100 while still being able to reconstruct the elementary shape of the sample . Yet, for an almost complete reconstruction, the number of required scans is still high and in the same order of magnitude as the number of pixels itself. Other approaches for fast THz imaging contain often a broad THz illumination of the sample and detection is performed with electro-optical sampling or by the use of a pyroelectric detector array [11, 12]. Disadvantage of these methods is the need of sources with high THz output power, typically driven by femtosecond regenerative amplifiers.
In this paper, we present an alternative approach of THz imaging which is based on transforming the broadband frequency domain information of the THz signal into spatial information on a sample. We make use of efficient THz blazed diffractive gratings which spread the individual frequency components of the terahertz pulse over a defined spatial range. Therefore, they encode the spatial information into the frequency domain. In combination with an f-theta-lens system , the measurement of a single time domain waveform allows a direct reconstruction of the one-dimensional spatial transmission pattern. As every waveform-scan delivers the information of an entire line, the speed of THz imaging is drastically enhanced.
2. Experimental setup
The measurements are performed using a fiber coupled THz TDS system based on an Er-doped fiber laser emitting 100 fs long pulses with a repetition rate of 100 MHz and a pulse energy of 3 nJ. Commercially available photoconductive antennas based on InGaAs are used to convert the laser light into THz waves as well as for the coherent detection. To sample the THz waveform, a fiber stretcher is employed. It provides a time delay window of 190 ps with a scan rate of 12 Hz. A DFB-laser based interferometer accounts for thermal or mechanical drifts and ensures an accurate time axis of the scan .
2.1 Diffraction gratings
As diffractive elements, we fabricated blazed gratings made of aluminum as illustrated in Fig. 1 . The line density and the blaze angle are chosen for an efficient diffraction of the spectral region between 300 GHz and 500 GHz where the dynamic range of the employed TDS system is the highest.
The measured angular dispersion of the used gratings is shown in Fig. 2 together with simulation results. The simulation was performed using conventional Fourier optics. For the measurement we mounted the transmitting and receiving antenna together with the grating on a goniometer and recorded a THz time domain waveform for each output angle. As can be seen in the figure, we achieve an angular dispersion of about 15° per 100 GHz. The first diffractive order covers the frequency interval between 300 GHz and 600 GHz, the second diffraction order the range between 700 GHz and 1100 GHz. Due to the blazed design, we achieve a high diffraction efficiency of about 85% for the first order.
2.2 Reference measurement
For the THz imaging setup in transmission geometry, which is illustrated in Fig. 3(a) , a pair of two identical blazed gratings is used. The first grating induces an angular dispersion and spreads the THz radiation over the aperture of the following THz lenses. The gratings are illuminated with a beam of 25 mm full width at half maximum diameter. The diffracted THz waves are imaged by an f-theta lens system such that each focal spot of the individual frequency components lies in the same line between the two focusing lenses. To recombine the frequency components, a second grating follows the f-theta-lens system. The lens system consists of two lenses made of high-density polyethylene. This material exhibits a refractive index of 1.54 and offers a low absorption at THz frequencies. The lenses have a diameter of 10.6 cm. The first lens has a biconvex surface and the second is a convex-concave formed lens. The system consisting of grating and the two lenses has a total length of 35 cm.
For the reflection imaging geometry (c.f. Fig. 3(b)), the reflected signal from the sample is guided through the same f-theta lens and grating as the incident wave. To direct the reflected signal to the detector, an uncoated silicon beam splitter is used which reflects 50% of the THz wave.
A typical reference spectrum of the imaging system is shown in Fig. 4 . The signal energy is condensed into two spectral regions spanning from 300 GHz to 500 GHz and from 650 GHz to 900 GHz corresponding to the first and second order of diffraction, respectively. Due to the higher signal-to-noise ratio we used the first order of diffraction for obtaining the images.
2.3 Frequency space relation
To derive the frequency-to-space relation of the setup, we measure the frequency dependent focal position as shown in Fig. 5(a) . For this purpose we placed the detector at the focal line and used a linear stage to measure a full THz waveform with a step size of 1 mm. A third order polynomial is fitted to the data which is used to transform the frequency axis to the corresponding spatial position.
As next step, we place a 35 mm thick metal bar between the lenses and compare the resulting spectrum with the one of a reference scan through air. As each spectral component corresponds to a defined spatial position along the focal line, the insertion of the bar results in a pronounced stop band in the transfer function shown in Fig. 5(b). After the frequency-to-space-conversion, we can utilize the transfer function, given by the amplitude ratio of the sample spectrum and the reference spectrum, to map the position of the metal bar and its width as shown in Fig. 5(b). As can be seen in the figure, the stop band has not an ideal step function shape but a finite slope due to the sizes of the THz focal spots. This limits the spatial resolution of the present setup to about 2 mm.
To validate the practical applicability of the proposed technique, we recorded images of samples by measuring them vertically line after line. The spatial information of each horizontal line was extracted from the individually recorded waveforms. To increase the image quality, we applied a one-dimensional wavelet denoising to the recorded images . Figures 6(a) and 6(b) exemplary shows photographs of a metal and a dielectric target, respectively. The THz images of these samples are shown in Figs. 6(c) and 6(d). As first sample, we attached characters (“THz”) made of aluminum foil to a sheet of paper. As can be seen in the figure, the metallic letters can clearly be identified in the THz image as they are opaque for the THz waves. The second sample consists of a bar of polyethylene with different drilling holes and inclusions. Also in this case, the THz image reveals the main features of the sample. In the measurements, we noticed that the right part of the images exhibits a slightly higher resolution than the left part. We attribute this to the non-linear frequency-to-space relation as shown in Fig. 5(a). Based on this relation, the spatial scales have been converted to a linear scale.
As we encoded the spatial information into the frequency domain, we reduced the number of measurements significantly because the two-dimensional image could be recorded by a one-dimensional scan. The number of scans was reduced by the factor corresponding to the number of frequency data points used to reconstruct each line. This pixel number is given by the frequency resolution and thus, the time window of the measurements. In the presented case we used a fiberstretcher providing a time delay window of 190 ps. This results in a resolution of 36 pixel for each line, resulting from the available delay multiplied with the bandwidth used for the image of 190 GHz. Modern fiberstretchers are able to acquire a larger time window and as a consequence the resolution will improve. In preliminary investigations not shown in this work using a conventional mechanical delay line a factor of approximately 100 in spatial resolution was reached compared to the measurements presented here.
The employed THz system exhibits a scan rate of 12 Hz and therefore, the acquisition time for the images was 20 s for approximately 240 lines shown. The use of THz TDS systems with scan rates in the kHz range  in combination with Galvo-scanning elements  is part of ongoing work to realize real-time THz imaging with video-rate scan rate.
Furthermore, we utilized the imaging system in reflection geometry to record an image of the “THz” sample shown in the inset in Fig. 6(c). Also in this case, our method allows a fast spatial characterization of the sample. However, due to the use of a beam splitter in zero-degree reflection geometry, the signal energy is reduced by 75% which affects the signal-to-noise ratio. Still, the spatial characteristics of the sample can unambiguously be identified in the measurement.
In conclusion we have presented a THz imaging technique that makes use of transforming spectral information into the space domain. Using efficient blazed gratings to induce an angular dispersion for the THz waves allows encoding the spatial characteristics of a sample to the spectrum of a broadband THz signal. Therefore, a complete one-dimensional line can be reconstructed from a single THz waveform leading to an up to two orders of magnitude increase of the imaging speed. To validate the concept, we presented data of recorded THz images in transmission as well as in reflection geometry.
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