Terahertz referenceless wavefront sensing by 1 means of computational shear-interferometry

: In this contribution, we demonstrate the ﬁrst referenceless measurement of a THz 13 wavefront by means of shear-interferometry. The technique makes use of a transmissive Ronchi 14 phase grating to generate the shear. We fabricated the grating by mechanical machining of 15 high-density polyethylene. At the camera plane, the ‚ 1 and (cid:0) 1 diﬀraction orders are coherently 16 superimposed, generating an interferogram. We can adjust the shear by selecting the period of 17 the grating and the focal length of the imaging system. We can also alter the direction of the 18 shear by rotating the grating. A gradient-based iterative algorithm is used to reconstruct the 19 wavefront from a set of shear interferograms. The results presented in this study demonstrate 20 the ﬁrst step towards waveﬁeld sensing in the terahertz band without using a reference wave. 21 the test wavefront was reconstructed with a root mean square wavefront error of 𝜆 (cid:157) 12 . Our 320 results are the ﬁrst demonstration of full complex waveﬁeld measurements in the THz regime 321 without using a reference wave. 322


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Over the last four decades terahertz (THz) radiation has found a variety of applications [1] 24 owing to the combination of two important characteristics. Firstly, it is capable to penetrate 25 many materials, allowing non-destructive testing in many contexts. Secondly, unlike X-rays it is 26 non-ionizing and, therefore, safe for humans [2]. These advantages have fueled the development 27 of THz-devices of many types, including sources and detectors [3], quasi-optical components 28 such as lenses [4] and wave-plates [5]. Accordingly, THz-radiation has found a wide range of 29 applications including biomedical imaging [6], quality control of food and agricultural products 30 [7] and other forms of non-destructive testing [8]. Many of these applications benefit from the 31 capacity of sensing the terahertz electromagnetic wave provided by time-domain spectroscopy 32 (TDS), a technique introduced in the 1980s, which involves the use of ultrafast lasers. The 33 detection of the electromagnetic wave is acquired pixel by pixel and is performed by the cross-34 correlation between a relatively short (∼ 1 ps) broad-band terahertz electromagnetic transient 35 and a femtosecond optical pulse that either gates a photoconductive detector or probes the 36 birefringence induced in an electrooptic crystal by the THz electric field among other methods [9]. 37 The pixel by pixel acquisition is time consuming and causes a significant limitation, some 38 schemes have been introduced in order to do simultaneous multi-pixel electro-optical sampling 39 which helps to overcome the acquisition time problem, however, they require the use of free-40 space ultrafast pulses, which limits the applicability of the technique to laboratories with high 41 demands on the required operating temperature stability and laser safety [10,11]. 42 Since full-field coherent imaging requires less optical components and is fast compared to 43 single point scanning, full-field methods such as digital holography (DH) have the potential to 44 broaden the applicability of THz in many areas, such as biology [12]. DH includes two main holographic configurations, i.e. in-line as well as off-axis THz digital holography (THz-DH), 46 which are capable to record the amplitude and phase of monochromatic and spatially coherent 47 THz-radiation in a single interference pattern. Both distributions, amplitude and phase, can then 48 be recovered from the interference pattern using spatial [13]  plane of a 4 -imaging system. This concept has been used in the visible spectral range [33].

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Yet, to the best of our knowledge, this is the first realization of such a system for the THz-regime.

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In the following sections we will discuss the concept behind the interferometer including the  THz source Optional aperture Lens Lens Input plane Image plane is the distance between the THz-source and the input plane where an optional aperture is placed and is the focal length of the two identical lenses of the 4 arrangement. 2Δ and are the period and the groove depth of the grating. +1 and −1 represent the first orders of the grating which are separated by a shear . The vectors ì, ì and ì are 2D vectors across the input, Fourier and image planes of the setup, respectively. The scheme to the right illustrates the sheared two copies of the test wavefront generated at the image plane of the camera and shows the region of interest (ROI) where the two copies are overlapping.
Here, the design parameters of the RPG-based SI, such as the focal length of the lenses and  Now we will describe the design parameters of the RPG-based SI. Let us assume that the 118 complex amplitude of the wave field at the aperture plane is ( ì). Since the configuration 119 represents a linear shift-invariant system, the complex amplitude of the wavefield ( ì) 120 generated across the camera plane can be written as where, ì = −ì are two parallel space vectors at the input and the camera planes, respectively, 122 is the wavelength of the beam, F refers to the Fourier transformation, ì = ( , ) is a vector 123 at the RPG plane, and (ì) is the RPG transmittance. If we assume that the grating vector 124 P r e -p r i n t . M a n u s c r i p t a c c e p t e d a t O p t i c s E x p r e s s ( 2 0 2 2 ) ì = (2 /2Δ ) ì with the unit vector ì lying in the grating grid surface gives the direction of 125 its periodicity, (ì) can be written in the form Here is the phase introduced by the grooves of depth of the RPG. It can be estimated as where is the refractive index of the RPG material. rect(· · · ), (· · · ) and comb(· · · ) are the where is an integer corresponding to the index of the diffraction order and ì = ì/ is a vector Consequently, the RPG has to be designed in order to modulate the field across the common 139 Fourier plane such that two laterally shifted copies of the input field centered around the positions 140 of the +1 and −1 diffraction orders are generated. If these copies overlap, an interference pattern 141 depending on the input wavefield and the shear appears. According to Eq. (1), the intensity of 142 the interference pattern ( ì) will take the form where, 0 ( ì) is the background intensity, R{· · · } refers to as the real part of a complex number 144 and * refers to the complex conjugate. It is noted, that R{· · · } represents the interference terms 145 which contain the phase information. The main feature of SI is that the measurements represented Eq. (6) can be used to recover 148 finite differences Δ ( ì) of the test wavefront (ì) using spatial or temporal phase-shifting 149 techniques. However, these finite differences must be integrated to reconstruct the wavefront 150 under investigation. In the following section, we present the iterative gradient-based scheme to 151 reconstruct the test wavefront. Reconstruction of a wavefront from finite differences is an ill-posed inverse problem [37] 153 which is communally solved using optimization theory by minimizing the following objective to find the wavefront estimate˜so that˜= arg min { }. The factor ( ì) given by ( ì) = 156 ( ì − ì/2) ( ì + ì/2) is a weighting factor which is inherently obtained from the spatial phase 157 shifting approach. Accordingly, Eq. (7), the sum of the distance squared error (SDS-error) is 158 minimized which yields the least-squares solution to the problem. For this purpose, shear 159 measurements are required. In Eq. (7), · · · 2 = ì ∈ | · · · | 2 and refers to the the camera's  for all the diffraction orders, = 0, +/−1, · · · , can be calculated from Eq. (4) using Notice that this equation provides the diffraction efficiency for all the diffraction orders .

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Optimal grating grooves require a phase modulation of which leads to +1 = 40.5% and 0 = 0. we expect that this will not significantly affect the main signal of the proposed RPG-based SI.

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In addition, we want to point out that the circular shape of the focused +1 and −1 orders at the 194 camera plane indicates that there is minimal wavefront distortion caused by the imaging system 195 since the spots have a circular shape.

Experimental setup of the proposed RPG-based SI
197 Figure 3 shows the experimental setup based on the scheme shown in Fig. 1. We use an impact   208 We can determine the propagation of a spherical THz beam through the setup in order to 209 calculate the expected interference pattern generated at the camera plane. By using the Fresnel 210 approximation, the spherical wave at the input plane of the 4 -system can be written as

Shear interferograms of a spherical wave
here, ( ì) is the real amplitude of the spherical wave, = 2 / is the wave number and is 212 the distance between the source and the input plane of the 4 -system, i.e. radius of curvature.

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According to Eq. (6) the interference term, in this case, the third term, can be written as with ( ì) = ( ì − ì/2) ( ì + ì/2) / 2 . Accordingly, the generated interference pattern is  The two phase difference maps, obtained after applying the spatial phase shifting approach [44] 235 to the two interferograms, are shown in Fig. 5. For measurements, the corresponding phase 236 differences Δ resulting after applying the symmetric shears ì can be written as These phase difference maps are the input information required to recover the wavefront from the In order to experimentally verify this approach, we performed two sets of shear measurements.

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One for a test wavefront, which we obtained when all setup components were aligned to the 242 optical axis as shown in Fig. 1. For the second set, a wavefront tilt was added to the test 243 wavefront by shifting the THz source laterally from the optical axis of the setup. In each set, 244 seven measurements with shears of the same magnitude but different directions were performed.

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This was achieved by rotating the RPG at 15 • increments starting from 0 • and ending with 246 90 • . Accordingly, seven interference patterns including the two patterns shown in Fig. 4 were 247 recorded. Thus, the corresponding seven phase difference maps Δ were reconstructed based 248 on the spatial carrier frequency method as mentioned above. These phase maps were unwrapped 249 by using the PUMA phase unwrapping method [46]. The unwrapped phase maps, obtained for 250 the phase difference maps presented in Fig. 5, are shown in Fig. 6(a,b).

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In order to reconstruct the test wavefront from the unwrapped phase maps, we implemented 252 the iterative scheme presented in Eq. (8). As an initial guess the Kronecker delta was used.

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The iterative process was stopped when the change between two successive iteration fell below 254 certain threshold . In order to better understand the process, the convergence of the recovered Two examples of unwrapped phases calculated from the measurements presented in Fig. 5(a,b).
follow the direction of the gradient which minimizes the objective function . In Fig. 7c, the 259 SDS-error is plotted as a function of iterations is presented to monitor the convergence of the 260 iterative scheme.

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The solutions of the minimisation are shown in Fig. 8(a,b). For the two measured wavefronts,

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In order to quantitatively evaluate the reconstructed wavefronts shown in Fig. 8(c,d), a Zernike    for a test object showing large height or thickness variations, the contrast will be reduced if the 291 temporal coherence is less than the object's height or thickness variation. Accordingly, in SI 292 light diffracted from surface points separated by the shear is correlated if the temporal coherency 293 is larger than the surface height or thickness variation. Additionally, spatially separated points 294 are correlated if the shear is selected to be less than the spatial coherence as discussed in [49].

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This can be obviously controlled by adjusting the accordingly. However, the most critical 296 parameter which affects the measurement is the gratings grooves depth which should be optimally

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In conclusion, we presented a new THz common-path wavefield sensor based on a shear-304 interferometer. The setup consists of a phase Ronchi grating located across the common plane 305 of a 4 -imaging system. We fabricated the lenses and grating required for THz radiation at 306 a frequency of 280 GHz. Using confocal microscopy, we characterized the grating in order to 307 determine its groove depth and period. The Ronchi phase grating achieves a diffraction efficiency 308 of the ±1-orders of 39.5% compared with the 40.5% for the theoretically calculated ideal groove 309 dimensions. Since the wavefield across the input plane of the imaging system is convoluted 310 with the impulse response of the Ronchi phase grating, two laterally shifted copies of the test 311 wavefield are coherently superimposed, generating interference fringes across the overlap zone.

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The lateral shift, i.e. magnitude of the shear, depends on the focal length of the imaging system, 313 the wavelength of the THz illumination and the period of the grating. In addition, rotating 314 the Ronchi phase grating leads to a change of the shear direction. As a proof-of-concept we 315 have theoretically predicted and experimentally shown interferograms of a spherical wavefield.

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In addition, we used the spatial carrier frequency method in order to reconstruct the phase 317 differences from each single interferogram. In order to recover the test wavefronts, a set of seven