Terahertz holography for imaging amplitude and phase objects

: A non-monochromatic THz Quantum Cascade Laser and an uncooled micro-bolometer array detector with VGA resolution are used in a beam-splitter free holographic set-up to measure amplitude and phase objects in transmission. Phase maps of the diffraction pattern are retrieved using the Fourier transform carrier fringe method; while a Fresnel-Kirchhoff back propagation algorithm is used to reconstruct the complex object image. A lateral resolution of 280 µm and a relative phase sensitivity of about 0.5 rad are estimated from reconstructed images of a metallic Siemens star and a polypropylene test structure, respectively. Simulations corroborate the experimental results.


Introduction
Quantum Cascade Lasers (QCL) in the THz regime provide sources of reasonable power and coherence on a small footprint to be used in interferometric experiments [1][2][3]. Imaging technologies based on THz coherent radiation are becoming a powerful tool for material science and mechanical engineering [4]. Scanning the object through a focused THz beam is still a popular way for obtaining images [5][6][7], especially for sources based on optical antennae. Alternatively, the focused beam may be scanned across the object, or a point detector is scanned across an object beam [8]. Experiments are often designed in analogy to optical techniques, such as imaging in transmission and reflection, interferometry in Mach-Zehnder [9] arrangements for digital holography or optical tomography [10].
Digital holography is of special interest if resolution has to be pushed below the diffraction limit for small objects. A precursor to digital holography with direct THz imaging was realized by Federici et al. [11] a decade ago by using a number of single detector elements from which the original field distribution from the target was reconstructed. Recently, Wang has reported a digital Terahertz holographic system [12] based on transforming the pulsed THz signal by optical rectification in a birefringent ZnTe crystal. Interferometric images were obtained when passing the THz wave through the non-linear crystal and probing the induced birefringence with an optical beam which then is used in a classical interferometer. The authors report reconstructed intensity images but no phase-maps of the object.
Different area detectors are available for direct imaging of THz radiation. Microbolometer arrays of 160 × 120 to 384 × 288 pixels have been shown to work with THz radiation [13][14][15]. A Pyrocam array detector with 124x124 pixels on a 100 µm pitch was reported to perform in-line holographic measurements at 2.52 THz without optical rectification [16]. The set-up was based on a collimated incoming beam, and image reconstruction was made under the assumption that the amplitude of the diffracted beam was much smaller than that of the reference beam. The reconstructed images suffer from edge diffraction artefacts. Although iterative algorithms can be used to obtain a phase distribution from intensity distributions in an in-line holographic set-up [17], a direct measurement of the phase distribution holds great promise to increase the resolution and accuracy of the holographic reconstruction. Phase shifting THz interferometry with a CW source was reported at wavelengths below 0.7 THz so far [18,19], and that work was based on a single point detection set-up, while imaging was achieved by sample rastering. Off-axis holography using a beam splitter, a Pyrocam array detector of 124x124 pixels and Fourier retrieval of the phase distribution was reported by Ding et al. [20] for metallic objects; the lateral resolution was estimated to be better than 0.4 mm for an object size of 8 mm. Li et al. [21] assessed the resolution by improved zero-order diffraction suppression and higher numerical aperture (NA), viz. smaller recording distance. However, the pixel size of 100 µm limited the useful range of off-axis angle, and Laplacian differential reconstruction was needed to separate the overlapping zero and first-order peaks in Fourier domain. A minimum resolution of 0.17 mm was estimated from the cut-off frequency, while experimental tests based on a 10 mm diameter Siemens star yielded a resolution of 0.28 mm.
In contrast to those papers, we present a digital THz holography set-up that (i) uses a QCL at 3 THz; (ii) needs a single mirror, i.e. is free of a beam-splitter element; (iii) uses a highresolution 480x640 pixel, uncooled micro-bolometer array with 25 µm pitch for recording the holograms; and is (iv) capable of measuring both the amplitude of a diffracted object wave as well as the phase through the Fourier method introduced in holography by Takeda et al. [22]. With our equipment, off-axis configurations can be implemented at larger angles, as the THz wavelength of around 100 µm is four times the pixel size of the detector. The proposed setup aims at applying THz holography with relatively low cost and easy-to-use equipment for the feature analysis in objects in the millimeter and centimeter range. Potential applications range from non-destructive testing of plastics, composites, electronic devices to the investigation organic material.

Experimental
We use a THz Quantum Cascade Laser (QCL) emitting around 3 THz (Alpes Lasers, Neuchâtel, Switzerland) with 1.2 mW peak power and 5% duty cycle cooled with liquid nitrogen. The laser emits on a few narrow wavelengths in the range from 95 to 108 µm with dominant peaks at 99.0 µm, 100.0 µm and 102.5 µm. The detector is an uncooled Vanadium Oxide micro-bolometer array (DelMar Photonics, San Diego, USA) with 640x480 pixels on a pitch of 25 µm. Images are recorded through a CameraLink interface. To eliminate the thermal background, frames with laser off are subtracted from frames with laser on using a LabView program. For increasing the signal-to-noise ratio, some 1000 frames were averaged leading to a fringe modulation of up to 7 bit.
The QCL emits a cone beam with an approximate 60° full angle which allows us to use an off-axis digital holography set-up without beam-splitter, see Fig. 1, like a Lloyd's mirror interferometer which is used in optics [23,24] or x-ray lithography, but -to our knowledgehas not been reported for Terahertz holography so far. Part of the beam is reflected from a metallic mirror and serves as reference beam, while the direct cone beam illuminates the object, and the diffracted beam represents the object beam. This off-axis holographic set-up generates a fringe spacing of around 6 pixels (Fig. 2), i.e. 7 line pairs per mm (LP/mm), which corresponds to an interference angle of around 40°. The visibility of the fringes in this image vanishes for regions where the contributions of the different wavelengths cancel out.
where subscripts ref and obj refer to the reference beam and the object beam, respectively. φ is the respective phase of the waves, while δφ denotes a phase step controlled by mirror translation. In order to reconstruct the object, knowledge of the amplitude and the phase of the object wave is a prerequisite. While its amplitude is readily available from the diffraction pattern (taking the square root of the intensity of the object wave measured without reference beam), the phase of the object wave is retrieved applying the Fourier transform method to the carrier frequency frames. In order to reduce systematic errors we use differences of the phases derived from measurements with and without object. Furthermore, to cope with beat frequencies from the non-monochromatic source, the quality of the phase map was improved by tiling areas with the most reliable signals from phase maps taken at different mirror positions.

Forward and backward propagation
The diffraction pattern of the object at the detector plane is simulated by propagating the field distribution at the object plane (numerical mask) to the detector (image plane) using the convolution method based on the Fresnel-Kirchhoff integral [25]. We use a spherical Gaussian beam and superpose diffraction patterns from the three main emission wavelengths at 99.0, 100.0 and 102.5 µm with respective weights of 0.25, 0.25 and 0.5 corresponding to their relative strengths taken from a measured emission spectrum of the QCL. Then, the field at the image plane is sampled with the pixel resolution and size of the detector. The reconstructed complex field at the object plane is obtained using the inverse Fresnel-Kirchhoff diffraction formula. For a metallic object, the reconstructed amplitude yields the object contour with random phase at opaque areas, while for a phase object the phase distribution of the original object is obtained at the appropriate reconstruction distance.

Estimation of resolution
The reconstructed field A R can be written as the convolution of the original object wave A with the pixel function Π p and intrinsic resolution function W [26].
where is a complex constant. For an N × M pixel array, the functions are where g is the reconstruction distance, the wavelength, and p the pitch of the square pixels. For large N, the intrinsic resolution function W is a sinc-function with a resolution d W in Xdirection estimated by the first zero occurring at Nξ = π, which is and correspondingly for Y. Loosely speaking, the intrinsic spatial resolution in the reconstructed field is limited by the numerical aperture of the system and the wavelength. It dominates the resolution implied by the pixel function for distances larger than 2 p gN λ ≥ In our experiment the bolometer array has p = 25 µm, N = 480, while = 100 µm. Thus, for object distances g≥3 mm the intrinsic resolution function dominates. The spectral spread s of the THz source also limits the obtainable resolution. An upper limit to its contribution is estimated assuming a source with two equally strong emission lines at 0 (1 ± s). Back propagation to a distance g using the average wavelength 0 will correspond to reconstruction distances which are off focus by ± sg. The spectral resolution limit d s is estimated from the intensity distribution of a spherical wave on a plane at a distance of z = sg from a point source, Eq.
As a measure for resolution we take the full width at half maximum of the intensity, which yields d s = 1.5 sg.

Amplitude object
To determine the spatial resolution of the THz holography set-up the diffraction pattern of a Siemens star with an inner diameter of 4 mm is considered. In Fig. 3 the corresponding numerical mask and the numerically reconstructed intensity and phase maps are presented.
The experimental results are given in Fig. 4. The Siemens star has been fabricated from a metal sheet of 0.1 mm thickness using laser ablation, Fig. 4 (left), together with the reconstructed intensity (center) and phase map (right). The lateral resolution after object reconstruction is estimated from the Modulation Transfer Function (MTF) around concentric circles with decreasing diameter intersecting the spokes of the Siemens star [27]. The circumferential modulation of the reconstructed intensity of the object, averaged over the nine spokes, is shown in Fig. 5 (left) for several circle diameters from 3.0 to 0.7 mm around the center of the Siemens star. The 10% modulation level can be defined as the resolution limit and is found at 3.5 LP/mm, corresponding to a resolution of 280 µm. For simulations using a polychromatic and a monochromatic source the 10% modulation level is at around 5.8 LP/mm and 6.4 LP/mm, respectively, see Fig. 5 (right) corresponding to lateral resolutions of 170 µm and 155 µm.   For an object distance of 18 mm the theoretical resolution due to the NA alone, Eq. (4), is d W = 150 µm. The resolution limitation expected from the spectral separation of two emission lines of our QCL yields d s = 340 µm for s = 0.0125 and g = 18 mm, Eq. (6). This rather conservative estimate is a factor of two larger than the simulated values for the polychromatic source. This can be explained by the limited NA of the reconstruction.

Phase object
To determine the phase resolution of the THz holography set-up the diffraction pattern of a structured polypropylene plate with rings of different depths is considered. In Fig. 6 the numerically reconstructed intensity and phase maps are presented, while Fig. 7 shows the object image and experimental results.  The phase difference values are estimated from averaging the phases in the center of the rings, in the crossings of the rings and on a background line. Table 1 compares these values with the predicted phase difference values obtained from optical depth measurements and the refractive index of polypropylene n (3 THz) = 1.51. The estimated depth of the ring crossings has a higher uncertainty because the lateral resolution of them is rather low. From the standard deviation of the THz measurements we can estimate a phase difference resolution of 0.5 rad, corresponding to a 16 µm thickness variation in polypropylene. Note that for the mapping of phase differences to depth prior knowledge of the approximate object profile and index of refraction are needed to resolve phase ambiguity. Measurements using dual wavelengths [28] could resolve phase ambiguity.
An estimation of the achievable phase resolution for our measurement geometry was again made with simulations for a polychromatic and a monochromatic source, yielding resolutions of 0.20 rad and 0.12 rad, corresponding to a thickness of 6 µm and 4 µm in PP, respectively. These resolution values compare favorably to pulsed THz systems, where thickness is measured by optical delay lines and typically achieve a resolution of 10 µm [29,30] while high-resolution time-of-flight methods achieve micron resolution [4], both at the expense of scanning.

Conclusions
In THz holography, lateral resolution is limited by three factors: numerical aperture, sensor pitch, and spectral spread of the source. Simulations of polychromatic sources show that resolution scales with distance g and the spread of the wavelength distribution. For our configuration the resolution loss due to polychromatism is of the same order as the intrinsic resolution from the restricted aperture. Both could be improved using shorter distances to the detector. However the off-axis geometry limits the distance, since the reference beam must not interfere with the sample. This is alleviated in our set-up because the detector pixel size is only one fourth of the wavelength, which allows much larger angles for the off-axis reference beam to be used.
Phase information is essential for a faithful reconstruction of the object from its diffraction pattern using THz holography. For a phase object the reconstructed intensity image is of minor importance, as intensity contrast is low. But it can provide additional evidence for the correct reconstruction distance, as the barely visible structure gets sharp contours only at that distance. The determination of a phase map with an area detector obviates the need for scanning.