1. Introduction

Coherent X-ray diffractive imaging (CXDI) is used to numerically reconstruct the structure image of a noncrystalline specimen to nanometric scale resolution from its diffraction patterns. Being free of the resolution limit set by the fabrication of X-ray optics, CXDI has been promising the potential to reach atomic scale resolution with the advent of highly coherent sources, such as the third generation synchrotron radiation and X-ray free-electron laser (FEL). Since the first experimental demonstration of this microscopic technique in 1999 [1], impressive advances have been achieved in physics, material science, and the biology communities [2–4].

Although sophisticated synchrotrons and FELs have been used, CXDI has not reached atomic resolution, with the achieved highest resolution being about 10~20 nm for biological specimens [2]. There are some obstacles that CXDI has to overcome in practice.

First, the high dynamic range of diffraction patterns imposes great difficulties on dada collection. The pattern intensity scales with spatial frequency $f$ as $I(f)\propto f^{-4}$ [5]; so one usually gets an intensity range of $~{10}^6$ when measuring diffraction patterns. Conditions get more severe for biological specimens because of their weak photons scattering capability. Most X-ray CCDs’ (charge-coupled device) dynamic range is just $~{10}^4$ [6], which means one needs a dynamic range at least 100 times higher to record the whole pattern without saturation. Therefore a common practice is to block the central intense beam with a beamstop and to collect the scattered light with multiple exposures [7, 8]. Pixel array detectors of extreme photon-counting depth have been used to directly measure the diffraction patterns [9]; but presently they are not widely available. In ptychography, illumination diffusers, such as a pseudo-random mask [10], pinhole perturbation [11] or even a plastic window [12], have been used to reduce the dynamic range of the diffraction patterns up to ten times. Aiming to capture the entire X-ray diffraction pattern during one exposure, here we advocate a phase modulation method to reduce the dynamic range by three orders of magnitude.

A second practical problem revolves around the uniqueness and robustness of phase retrieval (PR) algorithms, which eventually determines the achievable resolution by CXDI. Classical PR algorithms [13] iteratively seek the signal estimate through alternating projections onto spatial constraints and Fourier modulus constraints. Although the relative uniqueness for real and non-negative objects has been mathematically proved (up to global phase, spatial shift and conjugate inversion), stagnation in the optimization process is still a significant problem for practical applications due to different sources of noise and artifacts in raw data. Moreover, the signals of interest in biological specimens are usually complex valued, for which classical phasing algorithms are less capable of successful reconstruction [14]. These challenges stem from the inherent non-uniqueness and non-convexity of the phase problem. In recent years, phase modulation of the signal of interest has been proposed to address these difficulties. In the literature, the modulation methods can be categorized into two classes: structured modulation [15–17], in which the values of modulator follow a specific analytic function; and random modulation [10, 18, 19], in which the modulator consists of realizations of some probabilistic distribution. There also has been significant work [20–22] exploring the sparsity constraint of the underlying signal for exact PR, which has the potential of breaking the resolution limit of the oversampling reconstruction technique [23].

A third challenge is the attained resolution for biological specimens that is limited by radiation damage: biomolecules are damaged by the intense X-ray beam before the dose reaches the requirement for an expected spatial resolution. The imaging dose $D$, scales proportionally with resolution $d$as $D\propto 1/d^{3~4}$. In the resolution range from 0.1 nm to 10 nm, $D$ becomes higher than the maximum tolerable dose of specimens [5, 24]. This explains why to date the achieved resolution in CXDI has been 10 nm without the cryogenic protection of biological specimens [25]. One way to bypass the radiation damage limit is to illuminate a stream of samples with ultrashort X-ray pulses and measure their diffraction patterns before each individual molecule is destroyed by the extremely brilliant laser beam [26, 27]. However, a single short pulse cannot generate a diffraction pattern with sufficient signal-to-noise ratio (SNR). This method thus has to collect millions of diffraction patterns and average identical copies to assemble up the final diffraction pattern for reconstruction. Nevertheless, data collection errors will inevitably affect the achievable resolution.

The method introduced here, spread spectrum phase modulation (SSPM), can uniformly disperse the scattered photons on the diffraction plane, so as to mitigate the aforementioned problems and increase the achievable resolution limit. The idea builds on an appropriate phase modulator that can evenly diverge the scattered rays on all directions, which will significantly reduce the dynamic range of diffraction patterns, and substantially increase the SNR at high scattering angle positions. The phase retrieval algorithm can then be used to yield high-quality reconstructions due to two advantages. Higher SNR raw data is used on one hand, and on the other increased well-posedness for better convergence and uniqueness.

Although the motivation of phase modulation for CXDI is compelling, the fabrication of a phase modulator matching the desired spatial resolution of the probed specimen is challenging. The available fabrication resolution for X-ray optics is about 10~20 nm [28]. The low-resolution in the fabrication process inevitably inhibits the effectiveness of phase modulation in CXDI. The reason is that a desirable phase modulator should have a very sharp autocorrelation function in order to uniformly disperse the scattered photons onto diffraction plane. Previous coding methods, such as random modulation, rely on the abrupt transitions among coding elements to maintain a broad and flat PSD (power spectrum density).

The proposed method overcomes this limitation by use of a Fresnel zone plate (FZP) as a practical implementation of SSPM. The experimental parameters are derived for the fulfillment of SSPM via ray optics analysis. To the extent of our knowledge, this is the first time that SSPM is devised to overcome the resolution limit in CXDI. Numerical simulations demonstrate its potential to help address the aforementioned three problems in a single strobe.

2. Basics of modulated coherent X-ray diffraction imaging

In CXDI a highly coherent beam of X-rays is incident upon the sample and the scattered X-ray intensities are measured by a detector in the far-field; then the sample’s image is reconstructed via a phase retrieval algorithm since the phase information is lost during the measuring process. We introduce a modulator into the classical CXDI layout, where it can be situated either upstream [Fig. 1(a)] or downstream [Fig. 1(b)] from the specimen.

 

Fig. 1 Imaging geometry for modulated CXDI with modulator in the upstream (a) or downstream (b) from the object. In both schemes, the signal of our interest is the product of object’s transmission function $\psi$ and modulation function $a$, which can be related by wave propagation.

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Let the modulation signal be $a$ and the object’s signal of interest be $\psi$. Assuming the specimen and the aperture are thin objects, the modulation effect of $a$ on $\psi$ can be expressed as a multiplication $P\left(a,Z_1\right)\cdot \psi$ in Fig. 1(a) and $P\left(\psi ,Z_1\right)\cdot a$ in Fig. 1(b), where $P\left(s,Z\right)$ is the wave propagation operator predicting the complex amplitude of signal $s$ propagated over distance $Z$ . For conciseness, we omit the propagation operator $P\left(\cdot ,\cdot \right)$ hereinafter and express the modulated field at position $z=Z_1$ as $m\left(x,y\right)=a\cdot \psi \left(x,y\right)$ . Once one recovers the complex amplitude $m\left(x,y\right)$ from the intensity measurements ${\left|E\left(u,v\right)\right|}^2$, the underlying signal $\psi$ can be analytically solved given the known modulation $a$.

Under the paraxial propagation assumption, the complex amplitude $E\left(u,v\right)$ at the detector plane is represented by the Fresnel diffraction integral

It is obvious that the modulated signal $m\text{(}x,y\text{)}$ should have a very narrow autocorrelation function in order to reduce the dynamic range of the diffraction pattern. Most existing modulation methods resort to random modulation, while their performances rely on the sharpness of the modulation elements. We will demonstrate this point in the following section.

3. Spread spectrum phase modulation

We start from a chirp modulation phase coding scheme, $a\left(x,y\right)=\exp \left[i\pi \beta \text{(}{x}^2+y^2\text{)}\right]$, in which the phase $\phi \text{(}x,y\text{)=}\pi \beta \left(x^2+y^2\right)$ is band limited while the modulator $a\text{(}x,y\text{)}$ is capable of evenly dispersing the modulated signal in the frequency domain. This treatment in nature is similar to the spread spectrum technique widely used in digital communications, and we borrow this concept to name the proposed modulation scheme for CXDI: spread spectrum phase modulation.

3.1 Quadratic phase modulation

The phase modulation is of the form of a finite chirp function:

 

Fig. 2 Example of 2-D QPF and its normalized MTF, where $L=256,d=2,\beta =1.01\times 2/\left(dL\right)$. (a) Modulator’s phase distribution. (b) Phase picture $\mathrm{mod}\left[\phi \left(x,y\right),2\pi \right]$. (c) $\left|A\left(f_X,f_Y\right)\right|$, MTF of (a). (d) Cross section $A\left(f_X,0\right)$.

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Next we proceed to relate the values of $\beta$ and the aperture dimension $L$ (assuming $L_X=L_Y$) with the expected object resolution $d$(the smallest non-negligible lateral length scale of the object). The single sided bandwidth of object is $B{W}_{\psi }=1/d$. Requiring $B{W}_a\ge B{W}_{\psi }$ for a full coverage of the object’s spectrum by the modulator, one gets the following spread spectrum condition:

3.2 Beam optics analysis of SSPM

A practical implementation of the previous QPM may take advantage of a defocused Gaussian beam to introduce a quadratic phase wavefront to the sample, as shown in Fig. 3. The incident X-ray beam is focused by a FZP, and an order sorting aperture (OSA) is planted onto the focus point to allow for the transmission of only the first order diffracted wave. This geometry was first proposed by K. A. Nugent et al. to address the ambiguity problem in phase recovery using a curved wavefront illumination [15, 30], and in recent years has developed into a new X-ray diffractive imaging method, Fresnel Coherent Diffractive Imaging (FCDI) [31], although it works in the Fraunhofer region. Now we derive the optimal defocus distance $Z^{\prime }$ to fulfill SSPM to the maximum extent.

 

Fig. 3 SSPM implementation via astigmatic illumination: a parabolic phase curvature is introduced to the sample at an out-of-focus position $Z^{\prime }$ ; the diffraction pattern is measured from the far-field.

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When the sample is placed on a defocused position $Z^{\prime }$, it is illuminated by a Gaussian beam with the complex amplitude [32]:

Since our interest is the spread spectrum modulation effect of $a\left(x,y\right)$ on the far-field measured patterns, we can safely neglect the amplitude variation $C\left(Z^{\prime }\right)\cdot \exp \left[-\pi \left(x^2+y^2\right)/2\lambda Z^{\prime }\right]$. The justice of such a simplification stems from the fact that $ℱ\left\{\exp \left[-\pi \left(x^2+y^2\right)/2\lambda Z^{\prime }\right]\right\}$ is still a real Gaussian, and much weaker in terms of spectrum modulation than $ℱ\left\{\exp \left[i\pi \left(x^2+y^2\right)/2\lambda Z^{\prime }\right]\right\}$, which is almost a flat spectrum as shown in Fig. 2. Therefore, Eq. (6) reduces to

Starting from such a quadratic phase form, we can shape the phase distribution in Eq. (9) to meet the spread spectrum condition by requiring

When one uses a Gaussian beam to illuminate the sample, there also exists an implicit constraint that the beam’s diameter must be greater than the aperture size,

Equation (7) and (11) tell that $L\ge 2\lambda Z^{\prime }/d$; and similarly Eq. (8), (12) and (13) show that $L\le 2W_0{Z}^{\prime }/Z_0$. Therefore, taking Eq. (8) into account, one derives the following spatial resolution limit for SSPM:

Substituting Eq. (11) and (14) into $\beta \le 2/L{d}_0$ derived from Eq. (15), one arrives at

For any defocus distance $Z^{\prime }>{Z^{\prime }}_{\min }$, the parameter $\beta$ will decrease and thus the band width of the modulator $B{W}_a$ decreases linearly with the increase of $Z^{\prime }$. If the desirable resolution of samples is thinner than $d_0$, say $d=d_0/M,M>1$, then $B{W}_a$ will be a fraction of the object’s band width, $B{W}_a=B{W}_{\psi }/M$. One may argue that the resolution limit of available FZPs shall inhibit the complete fulfillment of SSPM: since a widely recognized spot size ($2W_0$) of FZPs is about $20nm$, $d_0$ will be more than $30nm$, which is far from a desirable CXDI resolution. However, our numerical results in Section 4 demonstrate that even if the QPM’s band width is only 1/8 of the sample’s spectrum, it still can exhibit rather useful performance. Furthermore, different phase distributions will be introduced onto the sample when relative movements occur between the illumination beam and the sample, either transversely or longitudinally. This feature allows for a convenient implementation of variant phase modulators, which is of great significance when multiple measurements are employed to improve the reconstruction performance.

4. Numerical simulations

In this section we conduct numerical experiments to verify the performance gain attained in overcoming the aforementioned limitations of CXDI. The signal of interest in CXDI is the transmission function of the specimen, which can be regarded as a complex image $\text{o}\left(x,y\right)=M\left(x,y\right)\cdot exp\left[\phi \left(x,y\right)\right]$, where the magnitude $M\left(x,y\right)$, weakly less than 1, represents the absorption by the specimen, and $\phi \left(x,y\right)\in \left[0,2\pi \right]$ is the phase variation caused by the electrons in the specimen atoms. With such a configuration, we can simulate the transmission function of a thin biological specimen and investigate the performance of SSPM.

4.1 Dynamic range suppression

Recall that a phase modulator should be able to suppress the dynamic range of diffraction patterns beyond two orders of magnitude so that whole pattern data can be collected in a single shot. We examine this point by comparing the PSDs of a complex image with and without phase modulation. Two schemes in this paper are tested: QPM in Eq. (3) and random phase modulation (RPM) in the form$\exp \left(i\phi \left(x,y\right)\right)$, where $\phi \left(x,y\right)~U\left[0,2\pi \right]$. First, we assume the synthetic complex image is undergoing high resolution (HR) phase modulation, by which we mean the precision of optical devices is high enough that all frequency components of samples can be transferred. Figure 4 shows the simulated diffraction patterns in logarithmic scale and the color bar on their right side indicate the orders of magnitude of dynamic range.

 

Fig. 4 Dynamic range suppression of modulated diffraction patterns. (a) Original complex image. (b) ${\log }_{10}\left(PSD\right)$ of (a). (c) ${\log }_{10}\left(PSD\right)$ of (a) after RPM. (d) ${\log }_{10}\left(PSD\right)$ of (a) after QPM. Data within the white circles in (b)~(d) are used to plot Fig. 5.

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Compared with the unmodulated pattern in Fig. 4(b), the dynamic range of RPM in Fig. 4(c) drops about seven orders of magnitude, and eight orders of magnitude by QPM in Fig. 4(d). For a better insight of the distribution trend of pattern intensity with respect to frequency, we plotted the radially averaged spectrum curves in Fig. 5. This treatment reveals the overall relationship between intensity and frequency, while the direct pattern data range may be misleading because of occasionally emerged pattern pixels with extremely low values. One of the advantages of QPM over RPM is that QPM is less vulnerable to imaging blur, introduced by the limit of nanoscale fabrication. We simulated the blurring effect for QPM by setting the spot size of FZP multiple times of the HR QPM requirement,${\text{W}}_0=M\left(d/\pi \right)$, with M = 2, 4 and 8. While for RPM, we introduce a Gaussian blur to the phase distribution, with the blur width equal to $2d,4d,8d$, respectively. QPM and RPM have comparable performances under HR configuration. However, the dynamic range of RPM extends rapidly with the blurring effect: when the blur width reaches eight times the desirable sample’s resolution, RPM yields an intensity distribution almost the same with the unmodulated pattern. On contrary, low resolution (LR) QPM maintains an acceptable performance through the whole frequency range: even at the highest frequency, corresponding to the finest resolution $d$, QPM offers a dynamic range suppression more than three orders of magnitude. Although the absolute dynamic range varies with different test images, the suppression ability remains at about three orders of magnitude, which will greatly facilitate the data collection efforts in real applications.

 

Fig. 5 Normalized radially averaged power spectrums in logarithmic scale, calculated against different extents of imaging blur effect. For QPF, it depends on the spot size of FZP; while for a random phase plate, it depends on the sharpness between neighboring phase plate elements.

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4.2 Phase retrieval performance

The uniqueness of phase retrieval by QPM [15, 33] and RMP [19] have been established. The experiments in this part focus on the practicability of the two schemes under the constraint of imaging blur described in the previous section. 25 complex images are generated from 5 natural images of size $256\times 256$, Lena, Boat, Mandrill, Peppers and Barbara, which act as the magnitude and phase of a complex signal. These complex test images were respectively modulated by QPM and RPM, and then were fed into a HIO + ER based phase retrieval algorithm [19]. For complex images, this algorithm needs two pattern measurements to recover the phase. Since we are more concerned with the modulation effects when fabrication accuracy limits the achievable imaging resolution, we go further to invest the reconstruction performance of LR modulations. Both QPM and RPM with 1/8 bandwidth of the test image were tested and their RMSE were listed in Table 1, where the results of HR modulation were provided for comparison. For a signal $X$ and its estimate $\widehat{X}$, the relative error is defined in terms of Frobenius norm as $RMSE\triangleq {{\Vert X-\widehat{X}\Vert }_F/\Vert \widehat{X}\Vert }_F$. One can see QPM and RMP exhibit almost identical reconstruction accuracy with HR modulation, while with LR modulation QPM brings around 3~7dB improvement over RPM when the SNR ranges from 20 to 50dB.

Table 1. Reconstruction Error RMSE (dB) vs. SNR (dB) ^a

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In Fig. 6 we present the recovery images of magnitude and phase, where Poisson noise (SNR = 30dB) was added to the original images and the total number of algorithm iterations was set to 1300. The results in Table 1 and Fig. 6 verify that for HR modulation QPM has the comparable numerical property with RPM, while for LR modulation QPM works better than RPM. From the perspective of optimization theory, this validates QPM contributes more to improve the well-posedness of the phase retrieval problem.

 

Fig. 6 Reconstruction results of phasing algorithm with HR and LR phase modulations. Top row, the magnitude (a) and phase (b) of the synthesized complex test signal. Middle row: results when modulation bandwidth matches the signal bandwidth. Bottom row: results when modulation bandwidth is only 1/8 of the signal bandwidth.

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4.3 Lower imaging dose requirement

Now we show why SSPM is a promising treatment for low dose CXDI. Under a fixed experimental setup, the coherently diffracted intensity of a 3-D specimen scales with diffraction resolution as [5]: $I\left(f_d\right)\propto I_0{d}^3$, where $f_d=1/d$ and $I_0$ is the intensity of incident X-ray beam. For an expected resolution $d$, the minimum intensity $I_0$ has to be high enough so that the refracted intensity is detectable relative to the imaging noise: $I\left(f_d\right)\ge SN{R}_0\sqrt{{\sigma }_P^2+{\sigma }_G^2}$, where $SN{R}_0$ is the minimum SNR, and ${\sigma }_P^2$ and ${\sigma }_G^2$ are the variances of Poisson and Gaussian distributions, which represent the photon fluctuation and detector noise, respectively. Corresponding to $I_0$, we refer to the dose during the integration time of CCD as ${\text{D}}_0$. Therefore, it is reasonable to conjecture that the required imaging dose for an expected resolution will decrease substantially via SSPM; or in other words, the achievable resolution will increase since the intensities at high frequencies are lifted up by SSPM.

We still set the modulation bandwidth to 1/8 image bandwidth and simulate the effect of reduced illumination intensity upon the reconstruction results. The pixel values of a resolution test chart in Fig. 7(a) were scaled by $1/\sqrt{K}$, where $K=1,10,100$ and 1000; such a manipulation corresponds to the measured diffraction intensities that are $1/K$ weak. Apparently, in Fig. 7(g) QPM can yield better result with only one tenth dose required by non-SSPM illumination in Fig. 7(b). Even when the dose is reduced by one thousand times, QPM still yields acceptable reconstruction [Fig. 7(i)]. The results of RPM are slightly worse than QPM and omitted for space limitation. In effect, the role of QPM is the maximization of SNR around high frequencies, which facilitates the practice of CXDI under low-dose conditions.

 

Fig. 7 Effect of dose on reconstruction with and without SSPM. (a) Original resolution test chart. (b)-(e) Reconstructions without SSPM. (f)-(i) Reconstructions with QPM. The bandwidth of QPM is 1/8 the test image’s bandwidth. The relative errors Err are listed under each panel.

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4.4 Improvement via random coded apertures

The experiments in Section 4.2 require two modulated patterns to feed into the phase retrieval algorithm for a successful reconstruction: one pattern is taken with the defocus distance $Z^{\prime }={Z^{\prime }}_{\min }$ in Eq. (17); while we set $Z^{\prime }=10\times {Z^{\prime }}_{\min }$ for the other. In order to take full advantage of SSPM $(Z^{\prime }={Z^{\prime }}_{\min })$, we sequentially replaced the aperture in Fig. 3 with two complementary random coded binary apertures. Their “open ratio” is 0.5 and the element’s size is five times the object resolution $d$, which is feasible to fabricate. The resulted random support constraint allows for the reconstruction of an occluded version of the signal using a single diffraction pattern. Two complementary reconstructions were then merged to a full image. Figure 8 presents the simulation results with the same settings as in Fig. 6. Compared with Fig. 6, the improvement for HR modulation is minute while significant for LR modulation. However, the experiments using only two random masks without SSPM failed to reconstruct the complex image and we omitted their reconstruction results.

 

Fig. 8 Phase retrieval results combining SSPM with two complementary random coded apertures. (a) and (b) The pair of complementary random coded apertures. (c) Results of HR SSPM. (d) Results of LR SSPM. The LR SSPM result is comparable to the HR SSPM with the assistance of random coded apertures, acting as the enhanced spatial constraints with small correlation.

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7. Conclusion

Coherent diffraction imaging is characterized by its imaging formation model that relates the unknown signal with its measurements via the Fourier transform. Such a Fourier imaging model hints us to use spread spectrum technique, a frequency modulation method, to address some challenging problems in CXDI. Diffraction patterns with flat intensity distribution throughout the signal’s frequency range are desirable to reduce pattern’s dynamic range, which in turn is favorable to the well-posedness of PR algorithms as well as lower dose requirement for imaging. We proposed the SSPM with quadratic phase as a solution to these issues, and presented a practical implementation method using a FZP to generate the phase modulator required by SSPM. Numerical experiments demonstrate the privilege of SSPM over RPM under the limitation of nanoscale optics fabrication. A further research direction is a compressive sensing SSPM PR algorithm that recovers the signal from a single modulated pattern. We look forward to seeing real applications of SSPM in the CXDI community.