1. Introduction

Imaging of the constructs of the samples is of fundamental importance for a wide range of investigations in material, biological and optical sciences. In recent years, lensless imaging techniques have been developed and attracted a growing interest, partially because the potential to overcome the resolution limits introduced by the optical system and partially because of the scarcity of high-efficient lenses and mirrors in the short wavelength x-ray region. Coherent diffractive imaging (CDI) [1] is one recently developed lensless imaging technique. In this scheme, the diffraction pattern is oversampled and then converted to the object by using an iterative phase-retrieval algorithm [2,3]. This approach has been successfully utilized to reconstruct the biological and noncrystalline specimens (see the review [4] and references therein). The great advantage of CDI is that the resolution is, in principle, only limited by the illuminating wavelength. Therefore, very high resolution is possibly realized using the x-ray sources. Nevertheless, we have to face the difficulty of the convergence of phase-retrieval algorithms [5], which start from a random trial and iterate between real and reciprocal spaces. To converge to a unique and satisfactory image, typically, several thousand iterations are required [3], which is very time consuming. On the other hand, a priori knowledge of support constrain, which is the boundary of the sample, in real space is required. Even though the shrinkwrap algorithm [6] can speed up the convergence, a tight support is hardly known and then a large number of simulations with different initial trials are still required to confirm the convergence of the reconstruction.

Another lensless imaging scheme is Fourier transform holography, in which the interference of the object and reference lights is recorded and the object can be directly retrieved by an inverse Fourier transform. Conventionally, the reference beam is created using a pinhole. The optical system of this approach is quite simple and has been successfully utilized in many applications [7]. The drawback lies in the low visibility of the interference fringe due to the weak reference wave passing through the pinhole. To alleviate this problem, multiple references [8, 9] and specially-designed references [10] have been used. A recently developed technique, called holography with extended reference by autocorrelation linear differential operation (HERALDO) [11, 12], has significantly alleviated this problem by taking an extended slit or polygonal reference instead of the pinhole. The effectiveness of this approach has been demonstrated both in the visible [13] and x-ray regions [14–16]. In HERALDO, because the reference emerges from boundary waves produced by sharp corners of the polygon (or the edge of the slit), the photon flux can become comparable with that of the object wave, therefore high contrast interference can be observed. Moreover, image can be directly retrieved by applying a noniterative differential or integral operation and inverse Fourier transform. Nevertheless, compared with CDI, the resolution of HERALDO is limited by the sharpness of the edge and thickness of the reference. On the other hand, HERALDO depends very sensitively on the straightness of the edge [17], which therefore requires subtle fabrication of the reference. Moreover, HERALDO requires a priori knowledge of the orientation angle of the corner or slit reference. This information can be estimated from the streaks in the diffraction pattern, unfortunately the accuracy is dependent on the experiment.

In this work, we demonstrate a rotation-free approach of holography by using an extended arc reference, which is called ARC-HERALDO. We show that the object can be easily retrieved with a two-step algorithm without a prior knowledge of the information of the sample and reference. Moreover, this scheme in principle enables us to overcome the resolution limits introduced by the reference and optical system and therefore promises to achieve the diffraction-limited resolution. Also high contrast interference fringe can be recorded.

2. Theoretical model

Figure 1(a) illustrates the configuration of the object and reference in our ARC-HERALDO scheme. Number 3 denotes the object under study. Different from previous holography, an extended arc is utilized as the reference. The arc is partial of a ring centered at the sample. Note that the ring is not necessarily centered accurately at the sample, our two-step algorithm can retrieve the true image as discussed below. However the distance between the sample and reference (edge to edge) should be larger than twice of the size of the sample and the length of the arc is larger than the size of the sample. These requirements are the same separation conditions as for the conventional holography and HERALDO [12]. For a “benchmark”, we simulated the intensity of the diffractive pattern, which is shown in Fig. 1(b). According to the Fraunhofer diffraction formula, the complex amplitude of the diffraction U_f in the far field is the Fourier transform of the wave in the object plane, which can be expressed as U_f = 𝒡(o + r), where o and r denote the object and reference waves, respectively. The inverse Fourier transform of the diffraction intensity is given by 𝒡⁻¹(|U_f|²) = o ⊗ o + r ⊗ r + r ⊗ o + o ⊗ r, where the cross correlation o ⊗ r is defined by ∫ o(x′)r^*(x′ − x)dx′. To reconstruct the object, a proper operation L can be applied to the extended reference such that L(r) = δ. For the arc reference shown in Fig. 1(a), we introduce an operator of the derivative with respect to the polar angle, $L=\frac{\partial }{\partial \varphi }$, which is equal to a hybrid of polynomial and differential operators in the Cartesian coordinates $x\frac{\partial }{\partial y}-y\frac{\partial }{\partial x}$. The object is reconstructed by two steps. In the first step, we apply the polar derivative L to the diffraction pattern. The retrieved images are shown in Fig. 1(c). We can know that L(r) = δ(x − x₀, y − y₀) + δ(x − x₁, y − y₁), where (x₀, y₀) and (x₁, y₁) denote the coordinates of two ends of the arc reference. Moreover, according to the definition of cross correlation and Fourier transform theory, we have x(r ⊗ o) = x ∫ r(x′)o^*(x′ − x)dx′ = ∫ x′r(x′)o^*(x′ − x)dx′ − ∫ r(x′)(x′ − x)o^*(x′ − x)dx′ = (xr) ⊗ o − r ⊗ (xo) and $\frac{\partial }{\partial y}\left(r\otimes o\right)=\frac{\partial r}{\partial y}\otimes o=-r\otimes \frac{\partial o}{\partial y}$. Then we can show that $x\frac{\partial }{\partial y}\left(r\otimes o\right)=\left(x\frac{\partial r}{\partial y}\right)\otimes o-\frac{\partial r}{\partial y}\otimes \left(xo\right)$, $x\frac{\partial }{\partial y}\left(o\otimes r\right)=-\left[\left(ox\right)\otimes \frac{\partial r}{\partial y}-o\otimes \left(x\frac{\partial r}{\partial y}\right)\right]$. From these relations, we get

 

Fig. 1 (a) Sample under study. The colorbar indicates the intensity in linear scale. (b) Simulated diffraction pattern of the sample. The colorbar indicates the intensity in logarithmic scale. (c) Image reconstructed by the polar derivative operator in step 1. The colorbar is same to (a).

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To remove the system noise, in the second step, the weak-polluted image obtained in step 1 [marked by the ellipse in Fig. 1(c)] is input to the phase-retrieval algorithms. In this work, we adopted the hybrid-input-output (HIO) method [3]. The iteration starts from the initial guess, here the weak-polluted image. A modulus and support constraints are applied at each step of the iteration in the Fourier and real spaces, respectively. In the Fourier space, we replace the modulus of the retrieved object with the true modulus, i.e., the square root of the diffraction intensity shown in Fig. 1(b). In the real space, we force the object outside the support close to zero. The image obtained in step 1 provides us a very good estimation of the tight support. In our treatment, we first convolve the image obtained in step 1 with a Gaussian function. Then the support is obtained by using a contour of the convolution at the 5% intensity level. Figure 2(a) shows the image retrieved after 55 iterative steps. In comparison with Fig. 1(a), we can see that both the object and reference agree quite well with the true image. To monitor the convergence process, we define the error E_r = ∑_∉S |ρ_n|²/∑_S |ρ_n|², where S denotes the real-space support and ρ_n denotes the image retrieved in n-th iteration. The red solid line in Fig. 2(b) shows the error as a function of iteration step of our two-step algorithms. We can see that the result rapidly converges and the error decreases to less than 10⁻³ in 50 steps. Note that in CDI scheme, a large static support is usually adopted. For comparison, we have performed the simulation by adopting a static support of the autocorrelation calculated by the inverse Fourier transform of the diffraction intensity. However, it requires approximately 1300 iteration steps to converge to the similar result [see the green dot-dashed line in Fig. 2(b)]. This simulation can be speeded up if updating the support with the shrinkwrap algorithm [6], nevertheless more than 200 iterations still are required [see the blue dashed line in Fig. 2(b)]. Note that it is more time consuming to treat the experimental data because of the noise in experiments. Previous identification has shown that a tight support can dramatically speed up the convergence and also play a crucial role to retrieve a unique image [18]. In our algorithm, the image obtained in step 1 and the deduced support are already very close to the true object. The convergence of the iteration becomes faster and stagnation and uniqueness problems are mitigated. Moreover, this algorithm does not require a prior knowledge of the orientation of the sample and reference. Due to the symmetry of arc, our ARC-HERALDO scheme is actually rotation-free. It allows positioning arbitrarily the sample in the illuminating beam as long as the holographic separation conditions are satisfied. This makes it much easier to align the optical system, especially for the microsamples. More importantly, we should emphasize that the ultimate resolution of the conventional holography and HERALDO is limited either by the size of the pinhole or the sharpness and thickness of the reference. On the contrary, in our scheme, the sharpness and thickness of the arc reference only influence the image obtained in step 1, however this influence is overcome by the iterative algorithm in the second step. Therefore, the two-step algorithm in principle promises us to achieve the diffraction-limited resolution. Of course, the resolution in practical experiments depends on many factors, such as the highest recorded spatial frequency, signal-to-noise ratio and measurement artifacts, etc.

 

Fig. 2 (a) Image reconstructed with the two-step algorithms. The colorbar is same to Fig. 1(a). (b) The evolution of the error as a function of iteration step. Green dash-dotted line: HIO algorithms and a static support constrain are used. Blue dashed line: HIO and shrinkwrap algorithms are used. Red solid line: our two-step algorithm is used.

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3. Experimental results and discussions

To demonstrate our ARC-HERALDO scheme, we have performed an experiment using a linearly polarized 0.5-mW HeNe laser. The central wavelength is 543 nm. The beam size was amplified to 30 mm with a lens pair and an aperture was used to adjust the beam size. The sample was prepared by printing Fig. 1(a) on a transparent paper. The dark area is opaque and the white pattern is transparent. Of course, due to the quality of the printer, the dark area is not perfectly homogeneous, which may induce some experimental noises. To record the diffraction pattern in the far field, we used a lens with a 500 mm focal length. The sample was put at the front focal plane and the CCD camera (Hamamatsu C4742-95) was put at the back focal plane to detect the diffraction intensity. CCD has a dynamic range of 12 bits and an area of 1024 × 1280 pixels with each pixel of 6.7 × 6.7 μm². The exposure time was adjusted from 132 μs to 2 ms to find the high contrast diffraction pattern. However, all the figures shown in this work are single frame measurement by the CCD. Average with several different exposure times was not applied.

Figure 3(a) shows the diffraction of the sample number 3 detected by the CCD with an exposure time of 880 μs. To reconstruct the object, we applied the polar derivative operator in step 1. As shown in Fig. 3(b), 4 images are retrieved. These images are surrounded by some noises, which include both the system noises and experimental noises. The amplitude of the noise is about 20% of the image. Moreover, unlike the previous holography schemes [7–12], we should notice that the autocorrelation in the center of Fig. 3(b) is comparable with the images. The noises in the center of in Fig. 3(b) are very weak, nevertheless, the noises gradually increase as departing from the center and become stronger at the corners of Fig. 3(b). On the contrary, in previous holography schemes, the autocorrelation is much stronger than the images. This difference is because of the operator $L=x\frac{\partial }{\partial y}-y\frac{\partial }{\partial x}$ in our ARC-HERALDO scheme. The multiplication with the position x,y has significantly suppressed the autocorrelation and noise in the center part. To remove the system noise, in the second step, we started the phase-retrieval algorithm with the image obtained in step 1. As shown in Fig. 3(c), the object has been successfully reconstructed.

 

Fig. 3 (a) Experimental diffraction of the sample number 3 shown in Fig. 1(a). The colorbar in (a) shows the intensity in logarithmic scale. (b) and (c) are the images retrieved from the experimental diffraction in step 1 and 2, respectively. The colorbars are same to Fig. 1(a). (d), (e) and (f) are similar to (a)–(c), but for another sample of LAN.

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Figure 3(d) shows the diffraction intensity for another sample, called LAN, which was prepared with the same way. The sample is the pattern shown in Fig. 3(f). Figure 3(e) presents the 4 images obtained by using the polar derivative operator in step 1. Note that the noises in this result are very weak, less than 15% of the image. By starting from this result, the iterative algorithms quickly converge to a unique image shown in Fig. 3(f). We have performed the same experiment with lots of different samples, such as other numbers 1, 2, ... and alphabetic characters A, B, etc. Even though two-step simulations are required, the iterative simulation in the second step always can converge to a unique image in about 200 steps, which does not takes very long time (∼ 1 minute for Figs. 3(c) and 3(f) on a computer with a 3GHz processor). In comparison, several hundred and more than one thousand iteration steps are required to obtain the similar image with CDI method by adopting the shrinkwrap and static supports, respectively. Moreover, for most of the sample in our scheme, the images obtained in step 1 do not contain much stronger noises compared with the final image reconstructed in step 2. This indicates that the system error in step 1 is weaker than the experimental noise. Nevertheless, the quality of the image retrieved from step 2 is much better than that in step 1.

To demonstrate the rotation-independence of our ARC-HERALDO scheme, we have changed the orientation of the reference. The sample is similar to Fig. 1(a), but the reference is rotated to 0 degree. Figure 4(a) shows the diffraction pattern detected by CCD. After the treatment with the polar derivative operator, as shown in Fig. 4(b), 4 images were retrieved with surrounding by some weak noises. However, these noises are less than 10% of the image. Therefore, as shown in Fig. 4(c), the noises are easily removed in the second step. Furthermore, the reference was rotated to 45 degree. Figure 4(d) shows the diffraction pattern. As shown in Fig. 4(e) and 4(f), the object has been successfully retrieved with the two-step algorithms. These results show that the ARC-HERALDO scheme does not depend on the rotation of the reference and sample. It is rotation free, agreeing with our theory. Moreover, we note that the polar derivative operator in step 1 does not require a prior knowledge of the orientation angle and the second step starts from the images retrieved in step 1. Therefore, our two-step algorithm retrieves the object without a prior knowledge of the information of the sample and reference.

 

Fig. 4 (a) Experimental diffraction of the sample number 3 with the reference rotated to 0 degree. The colorbar in (a) shows the intensity in logarithmic scale. (b) and (c) are the images retrieved from the experimental diffraction in step 1 and 2, respectively. The colorbars are same to Fig. 1(a), 1(d), 1(e), and 1(f) are similar to (a)–(c), but the reference is rotated to 45 degree.

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We have also compared our ARC-HERALDO scheme with the conventional holography and HERALDO by using a pinhole and slit instead of the arc reference, respectively. The experiments were performed with the same setup. For the conventional holography and HERALDO schemes, even though the object and its autocorrelation can be reconstructed and separated in space with a noniterative algorithm, the object is much weaker than its autocorrelation. Moreover, the resolution is limited by the size of the pinhole or sharpness and thickness of the slit. In HERALDO, the orientation of the slit reference has to be estimated from the streaks in the diffraction pattern. The accuracy is limited by the signal-to-noise ratio in experiments. A deviation from the true value can introduce some noises and degenerate the resolution of the retrieved images. On the contrary, the two-step algorithm used in ARC-HERALDO scheme enables us to alleviate these disadvantages. On the other hand, we have applied a similar two-step algorithm for the conventional holography and HERALDO schemes. The image was firstly obtained from the diffraction pattern with the inverse Fourier transform and linear differential operator and then was input to the phase-retrieval algorithms. We found that the image quality was improved by the two-step algorithm and the retrieved image becomes smoother. By adopting the two-step algorithm the resolution is similar to that of CDI, which depends on the wavelength of the illuminating light and the highest spatial frequency detected in experiments but does not depend on the reference. Nevertheless, we emphasize that the iteration in the second step can converge faster to a unique result compared with CDI.

Compared with the conventional holography, although high contrast diffraction pattern can be observed in ARC-HERALDO and also HERALDO schemes, one drawback is that the arc or slit reference requires a very subtle fabrication. This is because the ARC-HERALDO scheme depends on the roundness of the arc reference and HERALDO method depends very sensitively on the straightness of slit reference [17]. Such a feature is very difficult to manufacture, especially for the applications of microsamples. Fortunately the two-step algorithm can relax the requirement of the subtle fabrication. Our test simulation and experiment show that the misshaped reference remarkably influences the image obtained in step 1. Nevertheless, it plays a minor role in the final image retrieved by the iteration in step 2.

4. Conclusions

In summary, we demonstrated a rotation-free holography scheme, called ARC-HERALDO, by using an extended arc references. High contrast diffractive pattern were observed. From the diffraction pattern, the object can be reconstructed with a two-step algorithm without a prior knowledge of the information of the sample and reference. The algorithm also alleviates the convergence and stagnation problem of the iteration. Even though our proof-of-concept experiment is performed with a visible light, it can be straightforwardly extended to the short wavelength region with the synchrotron, free electron laser or high order harmonic x-ray sources. Because the x-ray light and microsample are invisible, our rotation-free scheme allows us to easily align the optical beam and sample. Therefore it is more attractive for the applications of imaging the microsample with x-ray lights.