Single-shot aperture-scanning Fourier ptychography

Xiaoliang He; Cheng Liu; Jianqiang Zhu

doi:10.1364/OE.26.028187

1. Introduction

Fourier ptychography (FP) [1–10] has been developed as an efficient phase retrieval approach capable of reconstructing the high-resolution (HR) complex distribution of a sample. A typical FP employs an angle-varied plane wave generated by a light-emitting diode (LED) matrix illuminating the sample and records the corresponding low-resolution (LR) image at each angle. The FP method originated from ptychography [11–18], a type of scanning lensless coherent diffraction imaging (CDI) [19–24] technique for high-resolution complex image recovery, which laterally scans a sample at several points with respect to a localized illumination beam. In FP, angle-varied illumination, which is equivalent to the scanning of the two-dimensional (2D) Fourier spectrum of the sample with respect to the coherent transform function (CTF) of the employed objective lens, is combined with a Fourier ptychographic phase retrieval algorithm [5–8] to recover the complex expanded Fourier spectrum, resulting in reconstructed images with a resolution higher than the resolution limit of the objective lens. In recent years, FP has rapidly developed for applications, such as quantitative phase imaging [9,10], aberration removal [25], high-resolution imaging [26], and spectral multiplexing imaging [27].

One of the major limitations of the implementation using angle-varying illumination in FP is that the sample being observed needs to be thin [1] to satisfy the requirement that the change in the illumination angle is equivalent to the shift in the 2D Fourier spectrum. If the sample is not sufficiently thin this requirement is not satisfied, the overlap constraint of the Fourier spectrum cannot be imposed and the phase retrieval algorithm is not able to recover the sample. Recently, inspired by three-dimensional (3D) ptychography [28,29], the multi-slice Fourier ptychography method for 3D intensity and phase imaging was introduced [30], in which the thick sample was modeled as a series of thin slices. Horstmeyer et al. [31] proposed a method known as Fourier ptychographic tomography (FPT) suitable for thick samples. However, the computational complexity is significantly increased for these two implementations. Dong et al. [32] demonstrated a detection path based imaging scheme known as aperture-scanning Fourier ptychography (ASFP) to eliminate the limitation on the sample thickness. In their reported scheme, an aperture was placed at the Fourier plane of the 4f imaging system while a thick sample was illuminated by a single plane wave, and multiple intensity images were recorded when the aperture was shifting in the Fourier plane. The recorded images were stuck together in the Fourier domain by a conventional FP algorithm to recover the complex wavefront exiting the thick sample. However, the method is based on the scanning of the aperture, which results in a long overall acquisition time. Moreover, the scanning scheme increases the stability requirement of the imaging system. A spatial light modulator (SLM) was introduced to replace the mechanically scanning aperture in ASFP by Horstmeyer et al. [33] and Ou et al. [34], which provided a faster and more accurate scanning. Although replacing the mechanical displacement with an SLM significantly increased the speed of data acquisition, owing to the response time of the SLM and the scheme of multiple measurements, this method still cannot be adopted for real-time or single-shot imaging.

Recently, it was demonstrated that single-shot ptychography [16,35,36] or Fourier ptychography [35,37,38] could be realized with pinhole array [35,36], microlens array [38] or a Dammann grating [37]. Here, we propose a single-shot aperture-scanning Fourier ptychography by inserting a Dammann grating at a certain distance before the aperture plane (Fourier plane of the 4f system). Multiple shifted focal spots are formed at the aperture plane simultaneously, which is equivalent to the scanning of the Fourier spectrum with respect to the aperture. The intensity of an image array generated at the detector plane is recorded within a single camera shot and a standard FP algorithm [5] is used to convert the recorded single intensity pattern into a complex wavefront. The proposed method is demonstrated by proof-of concept experiments.

2. Methods

The original configuration of the aperture-scanning Fourier ptychography [32] is shown in Fig. 1(a). A plane wave illuminates a sample located upstream of the 4f imaging system. A circular aperture fixed on a 2D mechanical scanning stage is placed at the Fourier plane of the 4f system. For each scanning position of the aperture, the detector located at the image plane records the corresponding intensity image. Each measured intensity image is proportional to the magnitude square of the Fourier transform of the selected part of the Fourier spectrum, defined as

I_{j} (r) = {| F^{-} [O (u) A (u - u_{j})] |}^{2} .

Here, u and r are the coordinates in the Fourier and detector planes, respectively, O represents the Fourier spectrum of the complex wavefront exiting the thick sample, A(u−u_j) is the function of the employed scanning aperture, u_j is the center position of the aperture in scanning step j, and

F^{-}

stands for the 2D inverse Fourier transform operator. Then, the complex wavefront is reconstructed from the set of measurements by a standard FP algorithm. An alternative implementation of the aperture-scanning Fourier ptychography is to replace the mechanical scanning aperture with a programmable SLM in the Fourier plane [33,34], as shown in Fig. 1(b), and a series of intensity images of the complex wavefront is recorded with different parts of the SLM aperture opened.

Fig. 1 Schematics of the setups for the (a) original aperture-scanning Fourier ptychography, (b) SLM based aperture-scanning Fourier ptychography, and (c) proposed single-shot aperture-scanning Fourier ptychography. (d) Ray tracing in the proposed single-shot aperture-scanning Fourier ptychography.

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Figures 1(c) and 1(d) show the setup for the proposed single-shot aperture-scanning Fourier ptychography scheme. A plane wave collimated from a coherent light source illuminates a sample located upstream of the 4f imaging system. A Dammann grating is inserted downstream of lens L1 to simultaneously generate multiple shifted focal spots at the Fourier plane, and every shifted version of the focal spot is equivalent to the shifting of the Fourier spectrum with respect to the aperture. The distribution of the multiple shifted Fourier spectrum is $\sum_{m, n} O (u - u_{m, n}) \exp (i k_{m, n} u)$ , where $u_{m, n}$ is the (m,n)th order shift of the grating in the Fourier plane and k_m_,_n is the transverse k vector of (m,n)th order.

The measured intensity in the proposed single-shot aperture-scanning Fourier ptychography is given by:

I (r) = {| F^{-} [\sum_{m, n} O (u - u_{m, n}) \exp (i k_{m, n} u) A (u)] |}^{2} .

As the role of k_m,n is to shift the low-resolution image laterally in the detector plane, the measured intensity is an image array consisting of a series of sub-images corresponding to different parts of the Fourier spectrum. The complex distribution of the exiting wavefront can be recovered from the measured intensity, provided that all sub-images are isolated and not overlapped with each other. This requires that the size of the individual sub-image need to be smaller than the separation between adjacent sub-images. As shown in Fig. 2, S_m is the distance between the central position of the mth image and the optical axis, and f is the focal length of lens L2. Thus, the separation between adjacent sub-images can be calculated by:

L = S_{m} - S_{m - 1} = f \cdot \tan θ_{m} - f \cdot \tan θ_{m - 1}

where θ_m and θ_m_-1 are the mth and (m−1)th diffraction angle of the grating, respectively. As the magnification of this 4f system is one, the width of the object W is also less than L, which is determined by the diffraction angle of the grating and the focal length of the lenses used in the system. The shift of the Fourier spectrum in the Fourier plane,

(d \cdot \tan θ_{m} - d \cdot \tan θ_{m - 1})

, where d is the distance between the grating and the Fourier plane [Fig. 1(c)], results in an overlap ratio in the frequency domain,

R = 1 - \frac{d \cdot \tan θ_{m} - d \cdot \tan θ_{m - 1}}{D},

where D is the diameter of the aperture. The overlap is determined by the distance between the grating and the Fourier plane, and the change of it doesn’t alert the field of view. Thus, compared with previous work [37], the difficulty in optics alignment is remarkably reduced in this method. By appropriately selecting the period of the grating and its distance to the Fourier plane, the setup in Fig. 1(c) meets all of the requirements of the aperture-scanning Fourier ptychography. A standard FP algorithm can be employed to convert the single recorded image array to a complex wavefront.

Fig. 2 Distance between the central position of the mth image and the optical axis.

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After the recorded image array is split into separated sub-images according to Eq. (3), the reconstruction process begins with an initial guess for the Fourier spectrum function of the wavefront, O_g. The initial guess of O_g can be obtained by Fourier transforming the square root of the (0,0)th order image of the recorded image array. An iterative reconstruction process is performed for the k^th iteration via the following steps.

(1). The filtered spectrum is established immediately behind the aperture by multiplying $O_{g}^{k}$ at the current position $u_{m}_{, n}$ , $O_{g}^{k} (u - u_{m}_{, n})$ , by the aperture function A(u): $ψ_{m, n}^{k} = O_{g}^{k} (u - u_{m}_{, n}) A^{k} (u) .$
The diameter of A(u) is determined by the cut-off frequency of the imaging system, $D / (2 λ f)$ , where D is the diameter of the aperture and λ is the wavelength of the illumination.
(2). The established wavefront distribution at the detector plane can be obtained by: $Ψ_{m, n}^{k} = F^{-} (ψ_{m, n}^{k}) .$
(3). The amplitude of $Ψ_{m, n}$ is replaced by the square root of $I_{m, n}$ which is divided by the recorded image array: ${\hat{Ψ}}_{m, n}^{k} = \sqrt{I_{m, n}} \cdot \frac{Ψ_{m, n}^{k}}{| Ψ_{m, n}^{k} |} .$
(4). By transforming ${\hat{Ψ}}_{m, n}^{k}$ back to the aperture plane, an improved spectrum behind the aperture can be obtained: ${\hat{ψ}}_{m, n}^{k} = F ({\hat{Ψ}}_{m, n}^{k}) .$
(5). The spectrum and the aperture function is updated as: $\begin{array}{l} O_{g}^{k + 1} (u - u_{m}_{, n}) = O_{g}^{k} (u - u_{m}_{, n}) + \frac{A^{k}^{*} (u)}{{| A^{k} (u) |}_{\max}} \frac{| A^{k} (u) |}{{| A^{k} (u) |}^{2} + α} \cdot ({\hat{ψ}}_{m, n}^{k} - ψ_{m, n}^{k}) \\ A^{k + 1} (u) = A^{k} (u) + \frac{O_{g}^{k *} (u - u_{m}_{, n})}{{| O_{g}^{k} (u - u_{m}_{, n}) |}_{\max}} \frac{| O_{g}^{k} (u - u_{m}_{, n}) |}{{| O_{g}^{k} (u - u_{m}_{, n}) |}^{2} + β} \cdot ({\hat{ψ}}_{m, n}^{k} - ψ_{m, n}^{k}) . \end{array}$
respectively, where constants α and β, controlling the step size, are set to unity in this paper.
(6). The next position $u_{m}_{+ 1, n}$ is reached and steps (1)–(5) are repeated until all positions are reached. The process is iterated several cycles until O(u) and A(u) are converged, and then the complex wavefront exiting the thick sample can be obtained by the inverse Fourier transformation of O(u).

3. Experimental results

3.1 Setup

The experiments were carried out using a He–Ne laser (λ = 632.8 nm) to verify the feasibility of the proposed method. In the experimental setup [Fig. 1(c)], lenses L1 and L2 are the same with a diameter of 30 mm and a focal length of 100 mm. The diameter of the circular aperture [shown in Fig. 3(a)] is set to 2.5 mm providing a system numerical aperture (NA) of 0.0125. A Dammann grating [which is usually used to generate diffraction orders of uniform intensity [shown in Fig. 3(b)] of orders of 7 × 7 with $\tan θ_{m, n} = \pm 0.02 \sqrt{m^{2} + n^{2}}$ (m, n = 0, ± 1, ± 2, ± 3) was inserted in a distance of d = 50 mm upstream of the confocal plane, resulting in a 60% overlap in the Fourier spectral domain, which is equal to a synthetic NA of 0.042. A monochrome CCD camera (AVT, Pike F421B), with a pixel size of 7.4 μm and dynamic range of 14 bit, was inserted at 100 mm downstream of lens L2 to record the image array. In order to obtain the separations between sub-images and compensate the intensity differences between different orders of the grating which are used in the recovery process, we calibrated the system by removing the aperture from the 4f system and recording the high resolution image array formed on the detector. Figure 3(c) shows the record image array which consists of clearly separated sub-images, the separations and intensity differences of sub-images can be extracted from the figure.

Fig. 3 (a) The aperture used in the experiment; (b) an image the grating captured by conventional microscope; (c) image array recorded by the detector when the aperture is removed from the 4f system.

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3.2 Imaging performance

First, a USAF 1951 resolution test target was used as a sample to test the proposed single-shot imaging system. Figure 4(a) shows the recorded raw image array with an exposure time of 30 ms. The third element in group 4 can be resolved from the (0,0)th order image of the recorded image array, shown in Fig. 4(b). Figure 4(c) shows the reconstructed image of the proposed method, and from this figure element 1 in group 6 can be resolved. Figure 4(d) shows the recorded image when the Dammann grating and the aperture are removed from the 4f system. In the configuration when the aperture is removed from the system, the NA of the system is 0.15, providing a resolution of 4.2 μm, indicating that, according to the Nyquist sampling, a CCD camera with a pixel size of 2.1 μm [smaller than the pixel size of CCD used in the experiments] needs to be used to fully resolve the details of the image. Unlike other Fourier ptychography techniques which are used to improve the resolution of an imaging system, the resolution of proposed single-shot system does not exceed the resolution of the 4f system without the grating and aperture because the grating used to generate multiple images is placed after the first lens in the setup. However, in the case that the pixel size of the detector used to capture image is large, the image quality recovered by the proposed single-shot method is better than that directly captured by the 4f system because the latter image is not well sampled. Figures 4(e) and 4(f) show magnified images of the red box in Fig. 4(c) and the green box in Fig. 4(d), respectively. Obviously, the image in Fig. 4(f) is not well-sampled.

Fig. 4 Imaging performance of the proposed method: (a) recorded raw image array and (b) magnified image of (0,0)th order of the image array, (c) reconstructed image by the proposed method, (d) directly recorded image when the aperture and the grating are removed from the 4f imaging system, (e) magnified image of the red box in (c) and (f) magnified image of green box in (d).

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In addition, we compared the results with and without pupil recovery. Figure 5(a) shows the reconstructed image without pupil recovery. Figure 5(b) shows the magnified image of the red box in Fig. 5(a). Figure 5(c) shows the reconstructed image with the pupil recovery and the inset in Fig. 5(c) shows the recovered pupil phase. Figure 5(d) shows the magnified image of the red box in Fig. 5(c). Compared with Fig. 5(b), the image quality of Fig. 5(d) is improved.

Fig. 5 (a) Recovered image without pupil recovery; (b) magnified image of red box in (a); (c) recovered image with pupil recovery, the inset shows the pupil phase; (d) magnified image of red box in (c).

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3.3 Quantitative phase imaging

To verify the quantitative phase imaging capability of the method, we inspect a step-height sample with a height (silicon dioxide film thickness) of 240nm. Figure 6(a) shows the image of the sample captured by a conventional white light microscope. The recorded low resolution raw image array is shown in Fig. 6(b) at log scale. The central image of the recorded image array is shown in Fig. 6(c). Figures 6(d) and 6(e) show the reconstructed amplitude and phase of the sample, respectively. The phase through the blue line of Fig. 6(e) is converted to the height of the sample by $h = λ Δ φ / [2 π (n - 1)],$ where $λ$ = 632.8nm is the wavelength, $Δ φ$ is the measured phase of the sample, $n$ = 1.46 is the refractive index of the silicon dioxide film at 632.8nm. In Fig. 6(f), the blue curve (measured height) is in good agreement with the red curve (theoretical height).

Fig. 6 (a) Image of the step-height sample captured by a white light microscope; (b) recorded raw image array by the proposed system (at log scale); (c) central image of (b); recovered (d) amplitude and (e) phase of the sample; (f) line plots of measured height by the proposed method and theoretical height through the blue line of Fig. 6(e).

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3.4 Digital refocusing

Two microscope slides [one is a wing of a butterfly and the other is a wing of a bee] were stacked together to serve as the sample to demonstrate the 3D refocusing capability of the proposed method. The recorded raw image array is shown in Fig. 7(a) at log scale. The (0,0)th order image of the recorded intensity is magnified and shown in Fig. 7(b). Figures 7(c) and 7(d) show the reconstructed amplitude and phase of the complex wavefront exiting the sample. Figures 7(e1) and 7(e2) show complex images of two planes after digitally propagating the reconstructed complex wavefront to distances of −1200 μm and 900 μm. The red and yellow arrows in Fig. 7(e1) and Fig. 7(e2) indicate that two layers of the sample are focused, respectively.

Fig. 7 Experimental results of digital refocusing using the proposed method: (a) recorded raw image array (at log scale) and (b) magnified image of the (0,0)th order of the recorded intensity, reconstructed (c) amplitude and (d) phase of the complex wavefront exiting the sample. The complex images of two planes after digitally propagating the reconstructed complex wavefront to distances of (e1) −1200 μm and (e2) 900 μm.

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4. Summary

In summary, we proposed and experimentally demonstrated a single-shot aperture-scanning Fourier ptychography method for pixel super-resolution and complex wavefront reconstruction. In our proposed method, the complex wavefront exiting a thick sample can be reconstructed in a single camera shot by a standard Fourier ptychographic algorithm. Compared with the method based on mechanically scanning and the method based on SLM-scanning, the proposed method exhibits high temporal resolution and the acquisition time of the method is in the order of 10 ms; thus, this method a very useful tool for conditions where ultra-high imaging speed is required.

In this method multiple low-resolution images are formed on a single detector simultaneously. Compared with scanning based methods [32–34], the available pixel count for each sub-image is limited in this proposed method, and to avoid overlapping between sub-images the field of view cannot be very large. It should be noted that this issue can be addressed if a lens array and a detector array [39–41] are used to replace lens L2 and the detector.

The proposed method can be applied in wavefront measurement, label-free phase imaging for biomedical specimens, or further applications requiring rapid imaging speed.

Funding

National Natural Science Foundation of China (NSFC) (61675215); Shanghai Sailing Program (18YF1426600).

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Single-shot aperture-scanning Fourier ptychography

Abstract

1. Introduction

2. Methods

3. Experimental results

3.1 Setup

3.2 Imaging performance

3.3 Quantitative phase imaging

3.4 Digital refocusing

4. Summary

Funding

References

Cited By

Figures (7)

Equations (9)

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