Rotation-free holographic imaging with extended arc reference

Imaging of the constructs of the samples is of fundamental importance for a wide range of investigations in material, biological and optical sciences. The recent development of coherent x-ray sources based on synchrotron, free electron laser and high harmonic generation provide us a powerful tool for high-resolution imaging. We have also developed the table-top high-power high harmonic x-ray sources in our laboratory. Nevertheless, compared with the visible and near-infrared lights, the high-quality optical lenses and mirrors are quite scarce in the x-ray region. This has stimulated an increasing interest to develop the lensless imaging technique. Fourier transform holography is one of the most simple and high-resolution lensless imaging technique. Conventionally, a pinhole is utilized as reference. The optical system of this approach is quite simple and has been successfully utilized in many applications. The drawback lies in the low visibility of the interference fringe due to the weak reference wave coming through the pinhole. A recently developed technique, called holography with extended reference by autocorrelation linear differential operation (HERALDO) [1], has significantly alleviated this problem by taking an extended slit or polygonal reference instead of the pinhole. Nevertheless, the resolution of HERALDO is limited by the sharpness of the edge of the reference. Moreover, HERALDO requires a priori knowledge of the orientation angle of the corner or slit reference. To overcome these problems, in our work, we demonstrate a rotation-free approach of holography by using an extended arc reference, which is called ARC-HERALDO[2]. 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 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.

Imaging of the constructs of the samples is of fundamental importance for a wide range of investigations in material, biological and optical sciences.The recent development of coherent x-ray sources based on synchrotron, free electron laser and high harmonic generation provide us a powerful tool for high-resolution imaging.We have also developed the table-top high-power high harmonic x-ray sources in our laboratory.Nevertheless, compared with the visible and near-infrared lights, the high-quality optical lenses and mirrors are quite scarce in the x-ray region.This has stimulated an increasing interest to develop the lensless imaging technique.
Fourier transform holography is one of the most simple and high-resolution lensless imaging technique.Conventionally, a pinhole is utilized as reference.The optical system of this approach is quite simple and has been successfully utilized in many applications.The drawback lies in the low visibility of the interference fringe due to the weak reference wave coming through the pinhole.A recently developed technique, called holography with extended reference by autocorrelation linear differential operation (HERALDO) [1], has significantly alleviated this problem by taking an extended slit or polygonal reference instead of the pinhole.Nevertheless, the resolution of HERALDO is limited by the sharpness of the edge of the reference.Moreover, HERALDO requires a priori knowledge of the orientation angle of the corner or slit reference.
To overcome these problems, in our work, we demonstrate a rotation-free approach of holography by using an extended arc reference, which is called ARC-HERALDO [2].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 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.
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 was put at the back focal plane to detect the diffraction intensity.Figure 1(b) shows the diffraction intensity of the sample number 3 detected by the CCD.To retrieve the object, we developed a two-step algorithm.In the first step, we introduce an operator, , which is a hybrid of polynomial and differential operators.We emphasize that this operator does not require a prior knowledge of the orientation angle of the reference and object.By applying this operation to the diffraction pattern, we can retrieve 4 images at each end of the arc reference as shown in Fig. 1(c).Note that to clear distinguish these 4 images, the length of the arc reference should be larger than twice of the size of the sample.The distance of between the sample and reference (edge to edge) is larger than twice of the size of the sample, which is the same separation condition as for the conventional holography.The image obtained in step 1 is very close to the true object.However, we can find some noises between the images, which include both the system noise and experimental noise.To remove the 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].In brief, 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 1(d) shows the final image retrieved after the phase-retrieval algorithm.In comparison with Fig. 1(a), we can see that both the object and reference agree quite well with the true image.We must emphasize that a tight support can dramatically speed up the convergence and also play a crucial role to retrieve a unique image in the phaseretrieve algorithm.In our algorithm, the image obtained in step 1 and the deduced support is already very close to the true object.Therefore, the iterative algorithms always can rapidly converge to a unique result.The problems of stagnation and uniqueness of the iterative algorithms are overcome.More importantly, the phase-retrieval algorithm, in principle, enables us to achieve the diffraction-limited resolution.
We have also compared our ARC-HERALDO scheme with the conventional holography and HERALDO by taking the arc instead of a pinhole and slit as the references, respectively.For the conventional holography and HERALDO schemes, even through the object and its autocorrelation can be reconstructed and separated in space with a noniterative algorithm, the object is too weak compared with its autocorrelation and is hardly seen in the linear scale.Moreover, the resolution is limited by the size of the pinhole or sharpness 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 may introduce some noises and degenerate the resolution of the retrieved images.On the contrary, ARC-HERALDO scheme enables us to overcome these disadvantages.The drawback is that the arc reference requires a very subtle fabrication.Such a feature is very difficult to manufacture, especially for the applications of micro-samples.Fortunately the two-step algorithm can relax the requirement of the subtle fabrication.It is because the misshaped reference remarkably influences the image obtained in step 1.But it plays a minor role in the final image retrieved by the iterative simulation in step 2.
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.Moreover, the ARC-HERALDO scheme promises to achieve the diffraction-limited resolution.This technique 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.

Fig. 1 (
Fig. 1 (a) The sample under investigation.(b) The detected diffraction pattern.(c) The images retrieved in step 1.(d) The final image retrieved in step 2.