Intensity interferometry (II), the landmark of the second-order correlation, enables very long baseline observations at optical wavelengths, providing imaging with microarcsecond resolution. However, the unreliability of traditional phase retrieval algorithms required to reconstruct images in II has hindered its development. We here develop a method that circumvents this challenge, which enables II to reliably image complex shaped objects. Instead of measuring the whole object, we measure it part by part with a probe moving in a ptychographic way: adjacent parts overlap with each other. A relevant algorithm is developed to reliably and rapidly recover the object in a few iterations. Moreover, we propose an approach to remove the requirement for a precise knowledge of the probe, providing an error-tolerance of more than 50% for the location of the probe in our experiments. Furthermore, we extend II to short distance scenarios, providing a lensless imaging method with incoherent light and paving a way towards applications in X-ray imaging.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
In 1950s, Hanbury Brown and Twiss (HBT) invented II and discovered the second-order correlations of incoherent light, which is named HBT effect . This discovery overturned the traditional view on “incoherence” of thermal light, and immediately invoked a hot debate , which led to intensively and extensively understanding of the quantum nature of thermal light [3, 4]. HBT’s work had a profound impact on the development of quantum optics . Meanwhile, the second-order correlation has heavily influenced imaging, such as ghost imaging [6, 7]. Instead of measuring the phase relation of light fields, the second-order correlation measures the variation of light intensities and seeks for the correlation of the intensity fluctuation , which therefore brings us many advantages including high resolution, the use of incoherent sources and turbulence-insensitivity. Thus, the second-order correlation imaging with incoherent sources has attracted a lot of attentions [9–13].
II is a landmark of the second-order correlation system. It consists of a set of telescopes far from each other. Each telescope independently records the time-varying intensity of an object temporally and spatially, and electronically (rather than optically) conveys the intensity information to a center where the correlations of the intensities are computed. This feature dramatically simplifies the detection process and reduces the cost . Moreover, II is insensitive to either atmospheric turbulence or telescopic optical imperfection. Thus, these great advantages enable very long baselines (kilometers) observations in short optical wavelengths with microarcsecond resolutions [15–18]. During 1963 to 1974, II had been used to measure the angular diameters of 32 star, with a resolution as high as 0.4 milliarcsecond [1, 19, 20]. Recently, an international project called Cherenkov Telescope Array has been being attempting to exploit air Cherenkov telescopes to construct kilometers baselines of II, in order to acquire a resolution approaching 30 microarcsecond [21–23].
Since II can only achieve the spatial Fourier magnitude of an object’s intensity distribution, it requires phase retrieval algorithms to recover the missing Fourier phase . However, the currently employed algorithms such as Error Reduction (ER) and Hybrid Input-Output (HIO) are unreliable and time-consuming, preventing from recovering complex shaped objects [25–27]. This challenge has become a major obstacle to the development of intensity interferometry imaging. Inspired by ptychographical iterative engine (PIE) [28, 29], we here introduce a ptychography-type measurement into the second-order correlation and create a new type of imaging method with a novel algorithm, which is named as ptychographical intensity interferometry imaging (PIII). Instead of detecting a whole object, we measure the object part by part in a ptychographical way that adjacent parts highly overlap . Along with the proposed algorithm, we circumvent the challenge and enable II stably and rapidly imaging the fine structure of objects. Moreover, taking the advantage of the second-order correlation of incoherent light, PIII can remove the requirement for the precise knowledge of the illumination function, such as the illumination’s shape, position and amplitude distribution [31–34]. For instance, in PIE that usually works with coherent diffractive imaging (CDI), the probe function needs to be pre-determined experimentally or post-calculated with a proper algorithm such as extended PIE (ePIE) [32, 34]. In contrast, in PIII, the probe is merely an aperture function. We only need to specify a size that is roughly larger than its actual value. Then the algorithm recovers high-fidelity images even when the estimation of the illumination’s location has a 50% deviation. This feature makes the realization of this method practical and simple. Furthermore, based on this technique, we proposed a lensless imaging system with incoherent light in short distance, which can be applied to X-ray imaging for complex-shaped object with high resolution, extending the application scope of II.
2.1. Physical mechanism
Figure 1 depicts a scenario potentially applicable to PIII, which images a nebula with a telescope array. The telescopes simultaneously detect the same section of the nebula (the lowest yellow dashed circle). After measuring the second-order correlation for the current section, the telescopes then change their angular areasand detect the next section. Usually telescope can provide a small field of view about several arcseconds. Repeat this procedure until all the sections have been detected. Here, the adjacent sections overlap with each other, which is a ptychography-type measurement.
On each section, the second-order correlation is calculated from the product of two intensity fluctuations detected at two telescopes:35, 36],
Unfortunately, even for Γ(ρ), the Fourier phase is lost during the measurements, and the image can not be directly recovered by inverse Fourier transform. Phase retrieval algorithms are required to reconstruct the missing phase. However, the noise and imprecise measurements will corrupt the Fourier magnitude and make the phase retrieval algorithm struggle to converge to the correct result [37, 38]. We therefore introduce the ptychography-type measurement and achieve a set of Fourier magnitudes from different sections, with j = 1…Np (Np is the total number of sections). The j-th one is
In PIII, Pj(ρ) is merely an aperture function that specifies the location and area of the j-th section (see discussion for details). The adjacent sections must overlap, which is the key to connect different constraints together, providing redundant information. The more constraints for the same object, the more robust is the convergency of the phase retrieval. Because redundant constraints can effectively solve the problems of multiple solutions and converging to local minimums. Based on the above theory, we redesign an image reconstruction algorithm. The details are shown in the next subsection.
2.2. Image reconstruction algorithm
Pj(ρ) is actually an aperture function, which is equivalent to divides a small sub-object, i.e., Γ(ρ)·Pj(ρ). Therefore, in the algorithm, Pj(ρ) acts as a support similar to that in ER algorithm. The realness and non-negativity constrains are still applied, since PIII only work for amplitude objects. In the following, we can see that the PIII algorithm uses a similar way as that in ER to update the sub-objects in turns in each iteration. Figure 2 is the flowchart.
- seed the algorithm with an initial guess for the whole image, Γ(ρ). Then start the following iterative calculation from the first big iteration (k = 1) and the first section of the object (j = 1).
- calculate the Fourier transform of the j-th section: .
- compute the inverse Fourier transform to obtain the image of j-th section: .
- if j is the last section, goto step 8; otherwise goto step 2 and compute the next section, i.e., Θk,j+1(ρ). Here, we use rebounding scanning: in the iteration of odd k, the scanning is from 1 to np; in the next iteration (even k), it rebounds from np back to 1.
- after all the sections has been gone through, start the next big iteration (k → k + 1) and goto step 2.
- if a satisfactory image is obtained or maximum iteration number is reached, exit the calculation.
3. Simulation and experiment
In 1957, HBT revealed that the second-order correlation results from the intensity fluctuations of thermal light . The intensity of thermal light always fluctuates temporally and spatially. Temporally, its values at a location randomly vary from one moment to another moment, where ‘a moment’ is a time period much shorter than the coherence time of the light. Spatially, if we capture the intensity distribution within ‘a moment’, we can observe a stationary speckle-like pattern (the intensities sharply change from one location to another). Roughly, we may say that “incoherence” consists of a set of temporally and spatially random speckle patterns, but the correlation of the speckles yields the HBT effect. However, the coherence time of thermal light is on the order of femtosecond, and it is hard for a detector to follow such a rapidly varying intensity. In 1964, it is found that a laser beam scattered by a moving ground glass can perfectly mimic the intensity fluctuation of thermal light , but its coherence time can be artificially slowed down to be milliseconds or even slower, allowing slow response detectors (such as CCD or CMOS) to sense its intensity fluctuation. Thus, with the pseudo-thermal light, we can perform a close perfect II experiment . Since 1964, the pseudo-thermal light has become a standard and widely used source in quantum optics and imaging [41, 42].
This article aims to demonstrate the principle and the preliminary experiments of PIII. We therefore employ a pseudo-thermal source and a CMOS camera for the experiments. As shown in Fig. 3, the experimental setup mimics the scenario described in Fig. 1. The pseudo-thermal illuminating a hollow object simulates a light emitting object. The camera simulates the telescope array: each pixel mimics a telescope. Since the pixels of the camera does not have ability of adjusting the field of view like telescopes, we use a light cone to let all the pixels simultaneously detect a small section of the object. The cone is made of a thick black paper. Its big tail lies on a fixed rod, and is loosely tied to the rod with a rubber band. Its small head is attached to a pinhole mounted to a motorized stage. Since the cone’s body can deform a little bit and the rubber band can provide some extent movement, the small head can scan horizontally or vertically in a small distance (but big enough to cover the whole object in our experiments). Actually, the function of the cone is similar to using an aperture to scan the object. Meanwhile, it can prevent most of environment light entering the camera.
The wavelength of the laser beam is 457 nm (DPSS 85-BLS-601). Its beam size is ~ 10mm. The pseudothermal light is collimated by a lens with a focal length of 300 mm. The distance between the ground glass and the collimated lens is 300 mm. The ground glass (220 grit) rotates at a speed of 1rpm(round per miniute), resulting in a coherence time of 100ms. The distance between the Lens and the object is 100 mm. The CMOS camera (Infinity 3-1M) with 1392*1040 pixels, is located 300 mm away from the object. It captures the speckle patterns at 20 ms exposure time. The size of a pixel is 6.5µm. The light cone, mounted on a motorized stage, is used to scan 16 sections of the object.
First of all, we carry out a numerical simulation to investigate how well the imaging mechanism and the proposed algorithm work. In the simulation, we simply model the light cone as an aperture function, P(ρ − Rj), which defines a circular aperture function centered at Rj with a diameter of a. Thus, with the aperture function, we can precisely select a section of object to be measured in each ptychographic run.
The pseudo-thermal light is modeled as electromagnetic fields with spatially random phases. For j-th sub-object, the E-field propagates onto the object plane and passes through the object and the j-th probe. It thus carries the information of the sub-object. On the detection plane, the intensity of the transmitted field forms a speckle pattern. We use 100 sets of random phases and generate 100 different speckle patterns. We calculate the autocorrelation of each speckle pattern, and then take their average. After subtracting the background from the average autocorrelation, we also apply a rectangular window to filter out the noise. We then obtain the second-order correlation of the sub-object (see Eq. (1) and Eq. (3)), which yields , the j-th Fourier magnitude. With different aperture functions, we obtain the whole set of with j = 1…Np. By applying into our algorithm, we recover the whole image, as shown in Fig. 4(a). Note that the initial guess of the object is an array of random numbers.
To investigate the performance of PIII, we compare it with the traditional imaging method with ER and HIO algorithms. The simulations are similar to PIII, but the speckle patterns are produced from the whole object rather than from sub-objects. 100 sets of random phases are used to generate 100 different speckle patterns. Without the ptychographic process, it is difficult for ER and HIO to reconstruct the images, as shown in Fig. 4(b) and Fig. 4(c). Both ER and HIO do not have any sign of converging to a correct image after 1000 iterations (see Fig. 4(e) and Fig. 4(f)). In contrast, PIII converges to the clear image within 6 big iterations (which is equivalent to 96 small iterations since 16 sub-objects are used in each big iteration), as we can see in Fig. 4(d).
We then perform the experiment with the setup depicted in Fig. 3. With the stage, the head is shifted to 16 locations in a step of 0.8 mm, dividing the object to 16 sub-objects. In each location, we capture 100 speckle patterns, one of which is shown in Fig. 5(a). The calculation process of the autocorrelation is the same as that in the simulation. The result is shown in Fig. 5(b). By using the proposed algorithm, we achieve the image in 9 iterations, as shown in Fig. 6. As shown Fig. 6(c), the covergence is stable without oscillation. Note that the initial guess of the object is an array of random numbers.
In realistic applications, especially in long distance scenarios, it is difficult to precisely determine the location of the probe. The inaccurate knowledge would result in blurred images or even unacceptable reconstruction. As shown in Fig. 7(a), assuming that the locations of probes are even and denoting the distance of two adjacent probe as ΔR = Rj − Rj−1, the location of j-th probe is Rj = R1 + (j − 1) ∗ ΔR. If our estimation is ΔR′ with an absolute deviation h = ΔR′ − ΔR, the probe will be inaccurate located to (the dashed circles in Fig. 7(b)). Let us define a shift deviation in percentage as sd = h/ΔR∗100%. In the iterative calculation, if we intentionally add a shift deviation of 25%, the algorithm easily fails to recover a clear image (Fig. 8(a)).
We found that, if we internationally use a larger number rather than their actual size for the supports in the calculation, i.e. a′ → a + ΔR∗rl (the red dotted circles in Fig. 7(b)), this difficulty can be overcome. We name this process “loose support”, where rl defines the “loose rate”. Fig. 8(b) shows that with a 25% loose rate, the algorithm can successfully retrieve the image under the 25% shift deviation. Even for a shift deviation of 50%, PIII can still retrieve a good result after introducing a 75% loose rate, although it takes relatively more iterations (see Fig. 9). since, the deviation of the j-th probe will accumulate, i.e., . The needed loose rate increases relatively more quickly than the shift deviation does. Note that, in the calculation, a is 280 pixels, and ΔR is only 20 pixels, meaning that the loose support is a small expansion to the size of the probes. This approach not only provides correct results, but also reduces the experimental requirement.
4. Applications in short distance
Besides the applications in long distance imaging, PIII also has promising applications in short distance scenarios. Similar to CDI, II is a natural lensless imaging system. II works with not only incoherent sources, but also most coherent sources, since in most situations it is easy to turn a coherent source to an incoherent source by scrambling the phase of light with a diffuser. In contrast, it is difficult to change a source from incoherent to coherent. From this sense, if we are concerned for amplitude objects (rather than complex-valued objects), II has a wider application scope than CDI. In the following, we will use the laser of 457 nm wavelength to demonstrate a PIII experiment for short distance applications such as X-ray imaging.
The schematic is described in Fig. 10. Different than Fig. 3, the probe is replaced with a circular aperture plate and placed right in front of the object, which makes it very easier to determine the size and location of the probe. The size of the aperture in the probe is 3 mm. We move the probe in a step of 0.32 mm. The lens is also removed. The distance from the ground glass to the probe is adjusted to 150 mm. The distance from the object to the camera is 200 mm. Thus, the imaging resolution is 0.457µm ∗ 200mm/3mm ≈ 30µm. Besides the object of “51201816”, we also tested a USAF 1951 resolution chart board. The recovered images are shown in Fig. 11(a) and 11(b), both of which are done in 9 iterations. In Fig. 11(b), element 6 of group 3 in the resolution chart can be resolved, corresponding to a resolution of 35.08 µm, consistent with the theoretical estimation.
Actually, PIII is different than PIE both in mechanism and in algorithm. “Ptychographical” in our method rather indicates the ptychography-type measurement. PIE is based on the first-order interference, where the probe information is critical to reconstruction. The amplitude and phase distribution has to be accurately determined, which may be pre-calibrated or post-calculated using algorithms such as ePIE [32, 43], since a small phase change of the probe may significantly change the diffraction pattern. In contrast, PIII is based on the second-order correlation. The phase and amplitude distribution of the transmitted field from the object varies temporally and spatially. Thus, the probe merely acts as a support that just specifies the area of a sub-object. In the algorithm, we do not need to precisely determine the position, size or shape of the probe, but just use a little large support to replace it, which is similar to HIO and ER. This is the mechanism of “loose support”.
To investigate the detail of the mutual coherence function, we rewrite it as
Therefore, the second-order correlation gives the same Fourier magnitude pattern both in near-field and far-field . Thus, we are able to move the camera closer to the object to increase the detection sensitivity. Above all, the second-order correlation provides a simple way to obtain a Fourier magnitude for an amplitude object. We would like mention that, our method can not recover the phase distribution of a complex-valued object, which limits its applications.
On the other hand, the pixels of the camera must be fine enough to clearly image the shape of the speckles on the detection plane. Meanwhile, the number of speckles must be big enough to reach a sufficient ensemble average, which can be realized by either increasing the samples of speckle patterns (in the cost of detection time), or using more pixels distributed in a larger area. The good thing is that, the pixels can be sparsely distributed on the detection plane . If the number of pixels is sufficiently large, a single shot is able to resolve the Fourier magnitude of the object’s intensity distribution, saving a lot of detection time.
By introducing the ptychography-type measurement and the corresponding algorithm, PIII gains the ability of reliably and efficiently recovering a complex shaped object which is inaccessible with the traditional intensity interferometry imaging. By utilizing the loose support approach, PIII removes the requirement of the precise knowledge of the probe’s location and size, by an error-tolerance more than 50% (defined in shift deviation). Recently, incoherent light has become an attracting source for imaging, such as ghost imaging with X-ray [9–11]. We can expect that, PIII would be another promising lensless imaging method.
National Natural Science Foundation of China (NSFC) (11503020), 111 Project of China (B14040) and National Basic Research Program of China (973 Program)(Grant No. 2015CB654602).
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