The van Cittert–Zernike theorem states that the Fourier transform of the intensity distribution function of a distant, incoherent source is equal to the complex degree of coherence [1–3]. Methods for measuring the degree of coherence can be broadly classified into two types. The first type is amplitude interferometry, which is based on the measurement of interference patterns produced by light coming from different regions illuminated by the source. The Michelson stellar interferometer belongs to this type and records interference patterns in the far field obtained by a two-pinhole Young interference configuration [4]. The second type is intensity interferometry, which does not measure light amplitudes but intensities [5]. The advantage of the intensity interferometer is that it is less sensitive to phase changes of the electromagnetic wave, allowing measurements in changing environmental conditions (e.g., atmospheric turbulences). Intensity interferometers can only obtain the modulus of the complex degree of coherence that is not enough for imaging; however, methods have been developed to obtain the phase and enable imaging with intensity interferometry by adding a reference [6] or using ptychographic detection [7]. We focus on amplitude interferometry methods because they allow a direct measurement of the complex degree of coherence and therefore are well suited for imaging. Computer simulations reported in [8] show that three-dimensional imaging can be implemented by the Michelson stellar interferometer. The complex degree of coherence is determined by the fringe visibility and the phase of the interference pattern versus the separation between the pinholes. Another method using sub-apertures to realize optical incoherent imaging is reported in [9]. Two small sub-apertures are sequentially open over the full aperture, and they transfer light from the observed object to the image sensor. The image of the object is obtained as a superposition of the recorded patterns. Imaging of incoherent or partially coherent light sources can also be carried out by holographic methods. These full-field methods [10–15] are based on the interference between two sheared (usually radial shearing) or rotated copies of the wave field arising from the object. Diffractive optical elements (DOEs) or lens/beam splitter systems are used for producing and superimposing the sheared copies.

In this Letter, we propose a lensless method for measuring the complex degree of coherence in one shot and from it calculating the intensity distribution of the light source. The method is based on the interference produced by multiple slit pairs. For explaining the method, we need to introduce some theoretical concept. We use the notation reported in Refs. [1,2]. $V({{P_1},{t_1}} )$ and $V({{P_2},{t_2}} )\; $ are the fluctuations produced by an extended incoherent source $\sigma $ at points ${P_1}$ and ${P_2}$ at times $t,\; \; t + \tau \; $. As shown in Fig. 1(a), the source $\sigma $ is located in the plane $({\xi ,\; \eta \; } )$ and the points ${P_1}$ and ${P_2}$ are located in the plane (X,Y); the distance between these planes is R. The complex degree of coherence is defined as

 

Fig. 1. (a) Notation for explaining the complex degree of coherence $j({{P_1},\; {P_2}\; } )$. (b) Setup for measuring $j({{P_1},\; {P_2}\; } )$. (c) Calculation of the phase and the contrast of $j({{P_1},\; {P_2}\; } )$ from the interference pattern (Fig. 1(a) is adapted from Refs. [1,2]).

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In Eq. (7), $j({{P_1},{P_2}} )$ is expressed as a Fourier transform of the source intensity (van Cittert–Zernike theorem). When ${P_1}$ and ${P_2}$ satisfy the relation $({X_1^2 + Y_1^2} )- ({X_2^2 + Y_2^2} )= 0$, $({\psi = 0} )$, then $j({{P_1},\; {P_2}\; } )= j({{X_1} - {X_2},\; {Y_1} - {Y_2}\; } )= j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$ depends on the pinhole separation only. Figure 2 shows cases where this relation is satisfied. In Fig. 2(a), ${P_1},\; {P_2}$ are symmetrically located around the origin O’; more generally the relation is satisfied for points located at the same distance from O’ (see Fig. 2(b)). $j({{P_1},\,{P_2}\,} )= j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$ can be determined experimentally by using the two-pinhole Young experiment (see Fig. 1(b)), where the phase and amplitude of $j({{P_1},\; {P_2}\; } )$ are calculated by processing the pattern obtained by the interference between the light transmitted through the pinholes located at ${P_1}$ and $ {P_2}$. Figure 1(c) shows how the phase ϕ is obtained from a line profile across the interference pattern by measuring the shift of the interference with respect to a reference point ${P_R}$ located at equal distance from the pinholes $ ({r_1} = {r_2}$). The amplitude $|{j({{P_1},\; {P_2}} )} |$ is given by the contrast of the interference (I_max − I_min)/(I_max+ I_min); see, e.g., Ref. [8]. $j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$ could be determined by processing interference patterns sequentially recorded with pinhole pairs located at different positions. In practice a spatial light modulator could be applied for this purpose but the method would be time-consuming.

 

Fig. 2. Aperture distributions satisfying the relation $\psi = 0$. (a) ${P_1},\; {P_2}$ are symmetrically located around the origin O’. (b) ${P_1},\; {P_2}$ are located at the same distance from O’. (c) Multiaperture pairs.

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We will now describe the method that we propose for measuring $j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$ in one shot. Consider a mask with many apertures (see Fig. 2(c)), arranged symmetrically around the center O’. The main idea is to place such mask at a certain distance from the light source and record with a pixelated detector (CCD or CMOS) behind the mask the interference produced by the light arising from the apertures. Figure 3(a) shows the experimental setup used for the verification of the method. The sample used is the U character cut out from black paper with a laser cutter. Its size is approximately 0.7 mm × 1 mm. The sample was illuminated with light arising from a LED emitting at a central wavelength of $\bar{\lambda }\,$ = 625 nm and a bandwidth of $\Delta \lambda $ = 25 nm. The LED was very close to the sample (5 mm); furthermore, a diffuser (diffusing tape) was inserted between the LED and the sample for increasing the aperture angle of the illumination. The distance between the sample and the mask containing the slits was 650 mm. We designed a mask (see Fig. 3(c)) with apertures symmetrically arranged around the center. This mask was manufactured by Beijing Zhongke Shengze Technology Co., Ltd. For allowing higher light collection, we did not use round apertures but pairs of slits. Each single slit has a size of 100 µm × 20 µm. The distance between the slit pairs increases radially by 50 µm from one slit pair to the next one. Close to the center (first slit pair), the distance is $50\,\mathrm{\mu}\textrm{m}$, for the second pair is $100\,\mathrm{\mu}\textrm{m}$, and so on until the last pair which is ${\Delta _{\textrm{max}}} = $40 × 50 µm = 2000 µm (40 slit pairs arranged at different distances from the center). There are 12 radial pattern sequences azimuthally separated by 15°; thus, the total number of slit pairs in the mask is 40 × 12 = 480. Since $j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )= {j^\ast }({ - \mathrm{\Delta }X, - \mathrm{\Delta }Y} )$ (where * denotes a complex conjugation), it is sufficient to measure j in a half of the plane. For this reason, there are no apertures in the left part of the mask (see Fig. 3(c)). Figure 3(d) shows one part of the interference pattern recorded with a CMOS camera (QHY 163 M, QHYCCD Co., Ltd., 4656 × 3522 pixels, pixel size 3.8 × 3.8 µm²) located 30 mm behind the mask. The size of the camera sensor is only 17 × 13 mm² but the part of the mask containing the pinholes is larger (30.31 × 15.15 mm²); for this reason, the camera was translated six times in order to record all the interference patterns. There are larger sensors allowing the recording in one shot (e.g., 36 × 24 mm²), but they were not available for our experiment. The sampling theorem [16] requires that there must be at least two pixels per fringe for allowing the retrieval of the amplitude and the phase of a signal. The largest slit separation inside the mask was ${\Delta _{\textrm{max}}} = $2 mm and we have chosen the distance between the mask and the detector to be 30 mm in order to satisfy the sampling requirement. The shorter interference spacing produced by the double slits was 30 $\bar{\lambda }$ / ${\Delta _{\textrm{max}}}$ = 9.3 µm > 2 × 3.8 µm (2× pixel pitch). We see that the interference patterns produced by the different slit pairs are well separated; this allows to calculate the amplitude (from the contrast of the patterns) and the phase of the degree of coherence for different slit pair separations and orientations. Line profiles across each interference patterns (480 in our case) were chosen and their phases were determined by Fourier transforming these signals, filtering in the Fourier plane by keeping the first order followed by inverse Fourier transforming. These phase signals are then evaluated at the central points (reference points; see Fig. 1(c)) that can be identified by processing the pattern recorded by the camera (centers of the signal envelopes measured across the patterns). The contrast of the interference pattern that determines $|{j({\mathrm{\Delta }X,\mathrm{\Delta }Y\; } )} |$ is calculated from the maxima and minima of the line profiles. Figure 3(e) shows the position of some measured $j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$ points (central points corresponding to lower $({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$). Our mask had only 480 double slits; thus, only 480 values of $j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$ were measured (the values for the second half plane are obtained by complex conjugation), and the points used for the reconstruction are 2 × 480 = 960. The 480 measured $j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$ values (together with their complex conjugates) are arranged on circles around the center. We assigned the value 0 to all the other points. A Fourier transform applied to the sparse $j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$ allows to retrieve the image of the object (see Fig. 3(f)); no regularization or phase compensation of the measured $j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$ is required. The reconstruction in Fig. 3(f) has small size but by zooming in, we see the reconstructed U character (Fig. 3(g)). The reconstruction is done only on few pixels (15 × 20); by zero padding $j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$ before Fourier transforming, we reconstructed the sample on more pixels (see Fig. 3(h)). In this case, the image is smoother. The spatial resolution of this imaging method is inversely proportional to the maximum distance between the slits inside the mask [8]. Thus, the spatial lateral resolution with a maximum slit pair separation ${\mathrm{\Delta }_{\textrm{max}}}$ is approximately equal to the resolution $d = R\bar{\lambda }/\,{\Delta _{\textrm{max}}}$ of a conventional lens imaging system having such aperture. In our experiment, we had R = 650 mm, $\bar{\lambda } = 625\; \textrm{nm}\; $, and ${\Delta _{\textrm{max}}} = $ 2000 µm; this gives a resolution d = 203 µm. The linear dimension of the symbol U was approximately 1 mm and the width used for cutting it out from the black paper was approximately 0.2 mm (see Fig. 3(b)). The system was able to resolve the symbol U as shown in Figs. 3(g) and 3(h). Increasing the distance R between the source and the plate (the object and the plate remain the same) will decrease the resolution of the system. Decreasing R will increase the angular aperture and in principle improve the resolution, but will also decrease the contrast of the interference pattern (in particular, the pattern produced by the light arising from the slits with large separation). This could lead to incertitude in the contrast and phase measurement reducing the quality of the reconstructed image.

We are aware that the results in Fig. 3 show only the reconstruction of a very simple object and that the built system cannot be used for imaging complex samples. We already pointed out that there were only 480 slit pairs in the mask; this limits the spatial frequencies of the recording. In order to increase the performances of the system, it would be necessary to design and manufacture a mask with more slit pairs. By increasing the density of the slit pairs in the mask, we will have unwanted overlapping of light beams diffracted by different apertures; this means that the interference produced by a single pair will be disturbed by unwanted light arising from other slits. We are also aware that the presented system has a very low light collection and that only a very small amount of the light reaching the mask is transmitted through the apertures and detected by the sensor. There is also the problem that the light diffracted by each slit produces an almost cylindrical waves and that this wave needs to travel a certain distance before it overlaps with the cylindrical wave arising from the other slit; this means that the overlapping area is small (see Fig. 4(a)) and most of the light transmitted through the slits does not contribute to the interference. Figure 4(b) shows how these problems could be reduced. By using prisms or diffractive optical elements (DOEs) behind the apertures, it would be possible to increase the superposition of the beams. These elements could be beneficial for the light collection. They can be designed and manufactured in a way that the light transmitted by each slit pair is directed toward a limited area of the detector where there is no light arising from other slits. This could also increase the density of the aperture pairs and thus the recording of more spatial frequencies allowing a better imaging.

 

Fig. 3. Experimental verification of the proposed method. (a) Experimental setup. (b) sample, (c) mask with slits, (d) interference patterns recorded behind the mask, (e) some measured points of $j({\mathrm{\Delta }X,\mathrm{\Delta }Y} )$, (f) reconstruction of the intensity of the sample, (g) enlarged reconstruction, (h) reconstruction obtained by zero padding_.

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Fig. 4. (a) Superposition of light diffracted by slit pairs. (b) Increase of the superposition by using prisms or DOEs.

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The method described in this Letter can be used for imaging sparse self-emitting objects. Compared with a conventional imaging system, it has some disadvantages (e.g., low light collection). On the other side, the method has the advantage to be scalable; this means that the mask with slit pairs (prisms, or DOEs, could be place behind the slits for increasing superposition of the diffracted beams and directing the light on the detector) could have different sizes to meet user needs. For imaging distant objects (e.g., astronomical applications), a very large mask (e.g., 10 × 10 m²) with many apertures could be used. The apertures could be large (some centimeters or meters) for higher light collection and the slit pair separation could also be large (centimeters, meters) for increasing the resolution. Many detectors could be placed behind large masks for recording the interference produced by the light arising from different slit pairs. The proposed method is in practice equivalent to a system composed by many (1000, 10,000, or more) Michelson stellar interferometers [4] that measure the complex degree of coherence in one shot. Furthermore, the system can be easily calibrated by using a known intensity distribution. The equal time complex degree of coherence can be applied for imaging (Eq. (3)), when the path difference ${R_1} - {R_2}$ is small compared to the temporal coherence of the source. In our experiment, the temporal coherence of the used LED was =25 µm (${\mathrm{\lambda }^2}/\mathrm{\Delta \lambda }$ = 625² / 25) and this requirement was satisfied. When using light sources with low temporal coherence, a more complex theoretical treatment and measurements of $\mathrm{\gamma }({{\textrm{P}_1},\,{\textrm{P}_2},\mathrm{\tau } \ne 0} )\,$ are required for obtaining an image of the source.