Coherent modulation imaging using unknown modulators

Bingyang Wang; Zhenfei He; Fucai zhang

doi:10.1364/OE.434111

1. Introduction

Coherent diffraction imaging (CDI) uses diffraction intensity information to reconstruct the wavefield distribution through iterative calculation [1–4], and its theoretical resolution is solely diffraction-limited. CDI has become one important imaging technique in the field of X-ray imaging [5–7] and visible light imaging [8–10], as well as the high-resolution electron microscopy [11,12] because it does not require high-quality lenses or complicated optical elements like aberration correctors. The conventional CDI has some disadvantages though, namely, poor robustness to noise and missing data due to beam stop [13], high dynamic requirements of detectors [14], and inapplicability for extended objects [15].

Ptychography uses a localized illumination probe and records diffraction patterns of a sample at multiple lateral positions relative to the illumination probe. The overlapping of the illumination area between adjacent measurements provides a such strong constraint that both the complex amplitude of the probe and the sample function can be well retrieved [16–19]. However, due to its long data acquisition process, ptychography is not well suitable for fast dynamics study. Some attempts to implement ptychography from a single measurement have been reported, but at the cost of a complicated setup and low information acquisition efficiency [20–22]. When imaging biological samples like frozen and hydrated samples, cumulative radiation damage caused by x-rays or electrons [23,24] is inevitable. Therefore, for those radiation-sensitive samples, a single-shot technique that can collect information before the onset of sample damage is highly preferred.

Coherent modulation imaging (CMI) is a variant of the single-shot CDI method [25,8]. Combining the phase modulation and the support constraint in sample space, the ambiguous solutions can be effectively eliminated in the iterative process, leading to fast convergence. It has been successfully demonstrated in X-ray phase imaging of extended samples with inner structures [26]. Since only one diffraction pattern is required, CMI is suitable for studying dynamical samples and even with multiple wavelengths [27]. In addition, because the flexibility of the modulator designing, the weak cascade modulators [28] or other special structures can be used.

In the first reported implementation of CMI, the modulator function is required to be precisely known in advance. The modulator function is usually obtained by other measurement techniques, such as the ptychography. To make CMI a standalone technique, less dependent on other methods, an automatic modulator refinement algorithm has recently been proposed [29]. It works well when an initial guess of the modulator function that deviates from the actual function mainly in an additive manner can be provided. In this letter, we propose an improved algorithm that allows for the transmission function of a completely unknown modulator to be reconstructed along with the multiple object exit waves. The key aspects of the proposed algorithm lie in 1) multiple measurements with the same modulator, 2) an averaging operation on the estimates of the modulator function, and importantly 3) the use of a range constraint on the modulator amplitude. The proposed algorithm is based on the fact that the modulator function remains the same among all measurements, and the average of estimated modulator functions deduced from each measurement will provide a more accurate estimate than any individuals.

2. Methods

Figure 1(a) shows our visible light experimental layout of CMI. A plane wave emitted from a laser source illuminates the sample through a pinhole. The exiting waves of objects are modulated by a phase plate (modulator) and then reached the detector, where the diffraction pattern is recorded. In the proposed method, multiple diffraction patterns are recorded for different objects. The multiple objects can be formed by rotating one sample to different angles, or by switching to completely different samples.

Fig. 1. System layout. (a) schematic of optical setup arrangement; by rotating the sample holder or replacing the sample, a series of diffraction patterns are recorded at the detector. (b) illustrations of the light path and key planes.

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Figure 1(b) shows the key planes along the wave propagation path. We use the subscripts of the positional vector ${\mathbf r}$, namely, ${{\mathbf r}_\textrm{s}},{\; }{{\mathbf r}_\textrm{m}}$, ${{\mathbf r}_M}$ and ${{\mathbf r}_D}$, to represent the plane that the support constraint is to be applied, the front and rear surfaces of the modulator, and the detector, respectively. ${z_1}$represents the distance from the object to the modulator, and ${z_2}$ is the distance from the modulator to the detector.

Figure 2 shows the flowchart of the reconstruction algorithm. This algorithm iterates back and forth through three planes: the support plane, the modulator plane, and the detector plane. The reconstruction process starts from random initial guesses ${M_0}$ for the modulator function and ${\varphi _{n,0}}({{{\mathbf r}_\textrm{s}}} )$ for the exiting waves of multiple objects, $n = 1,2, \ldots ,{\; }N$. n is the measurement number. N is the total number of objects used. In the following, details of the key steps are described.

Fig. 2. The flowchart of the algorithm with multiple objects. $\varphi $ wave fields; M modulator function; $\textrm{S}$ support; ${\textbf{P}_{{Z_{1,2}}}}$ propagation operator; ${\textbf P}_{{Z_{1,2}}}^{ - 1}$ inverse propagation operator; I diffraction intensity; The superscript ^ indicates the revised version after the modulus constraint has been applied; The subscript n indicates the measurement number; The subscript k is the current running iteration number; The subscript s, m, M, D indicate the support plane, the front and rear surface of the modulator, and the detector plane, respectively.

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2.1 Support constraint

A localized illumination probe is formed on the object by either an aperture or a focusing lens. The extent of the probe is used as the support $S({{{\mathbf r}_\textrm{s}}} )$, defined as

(1)$$S({{\bf r}_s}) = \left\{ {\begin{array}{cc} 1&{{{\bf r}_s} \in \textrm{support region}}\\ 0&{\textrm{otherwise }} \end{array}} \right.,$$

The support constraint is applied on the wavefield estimate ${\varphi _{\textrm{n},\textrm{k}}}({{{\mathbf r}_\textrm{s}}} )$ according to Eq. (2),

(2)$${\varphi _{n,k}}({{\bf r}_s}) = {\hat{\varphi }_{n,k - 1}}({{\bf r}_s}) \cdot S + \beta ({\hat{\varphi }_{n,k - 1}}({{\bf r}_s}) - {\varphi _{n,k - 1}}({{\bf r}_s})) \cdot (1 - S),$$

The constant $\beta $ is a control parameter, determining the rate that the pixel values outside the support are driven to zero [26]; $\beta $ usually takes value within [0, 1] and a value of 0.35 was used throughout our work. k is the current running iteration number. $\beta = 0.35$

2.2 Modulation

After propagation a distance of ${\textrm{z}_1}$, the wavefield arrives at the modulator. The resultant wavefield ${\varphi _{\textrm{n},\textrm{k}}}({{{\mathbf r}_m}} )$ is then modulated. For thin modulators, the wavefield exiting from the modulator is given by

(3)$${\varphi _{n,k}}({{\bf r}_M}) = {M_k} \cdot {\varphi _{n,k}}({{\bf r}_m}),$$

2.3 Modulus constraint

The wave ${\varphi _{\textrm{n},\textrm{k}}}({{{\mathbf r}_\textrm{M}}} )$ propagates further a distance ${\textrm{z}_2}$, reaches the detector. The modulus of calculated ${\varphi _{\textrm{n},\textrm{k}}}({{{\mathbf r}_D}} )$ is replaced by the square root of the measured intensity${\; }{I_\textrm{n}}({{{\mathbf r}_D}} ).{\; }$The revised wavefield is given as

(4)$${\hat{\varphi }_{n,k}}({{\bf r}_D}) = {{{\varphi _{n,k}}({{\bf r}_D}) \cdot \sqrt {{I_n}({{\bf r}_D})} } / {({{|{{\varphi_{n,k}}({{\bf r}_D})} |}_{ds}} + \varepsilon )}},$$

2.4 Modulator function updating

The updated wavefield ${\hat{\varphi }_{\textrm{n},\textrm{k}}}({{{\mathbf r}_D}} )$ is propagated backward through the distance ${\textrm{z}_2}$, yielding revised wavefield, ${\hat{\varphi }_{\textrm{n},\textrm{k}}}({{{\mathbf r}_M}} )$, on the rear surface of the modulator. The revised modulator function is obtained according to Eq. (5),

(5)$${\tilde{M}_k} = \sum\nolimits_{n = 1}^N {{{{{\hat{\varphi }}_{n,k}}({{\bf r}_M}) \cdot \varphi _{n,k}^ \ast ({{\bf r}_m})} / {(\sum\nolimits_{n = 1}^N {{\varphi _{n,k}}({{\bf r}_m}) \cdot \varphi _{n,k}^ \ast ({{\bf r}_m})} + \varepsilon )}}} ,$$

where we use the superscript * to indicate the conjugation operation.

Another important step is to enforce a range constraint $[{{c_{\textrm{min}}},{c_{\textrm{max}}}} ]$ onto the resultant modulator function to form a new estimate of ${\hat{M}_\textrm{k}}$, as shown in Eqs. (6a–6b). Note that there is no need to know precisely the range of modulator amplitude variation in advance. It is usually sufficient to set to [0, 1], to eliminate the unrealistic values for any passive objects [30]. A narrow range could help to increase the convergence rate though.

(6a)$$\begin{array}{cc} {{{\hat{M}}_k} = \frac{{{c_{\max }} \cdot {{\tilde{M}}_k}}}{{|{{{\tilde{M}}_k}} |+ \varepsilon }},\begin{array}{cc} {}&{\textrm{if}} \end{array}}&{|{{{\tilde{M}}_k}} |} \end{array} > {c_{\max }},$$

(6b)$$\begin{array}{cc} {{{\hat{M}}_k} = \frac{{{c_{\min }} \cdot {{\tilde{M}}_k}}}{{|{{{\tilde{M}}_k}} |+ \varepsilon }},\begin{array}{cc} {}&{\textrm{if}} \end{array}}&{|{{{\tilde{M}}_k}} |} \end{array} \le {c_{\min }},$$

In Eqs. (4-6), $\varepsilon $ is a small number to prevent from dividing by zeros. A value of 0.001 was used in the paper.

2.5 demodulation

To demodulate the revised modulator function of ${\hat{M}_\textrm{k}}$ to obtain the wave fields ${\hat{\varphi }_{\textrm{n},\textrm{k}}}({{{\mathbf r}_m}} )$ just in front of the modulator, a formula from the ePIE algorithm [19] is used, as shown in Eq. (7).

(7)$${\hat{\varphi }_{n,k}}({{\bf r}_m}) = {\varphi _{n,k}}({{\bf r}_m}) + \frac{{\hat{M}_k^ \ast }}{{|{{{\hat{M}}_k}} |_{\max }^2}} \cdot ({\hat{\varphi }_{n,k}}({{\bf r}_M}) - {\varphi _{n,k}}({{\bf r}_M})),$$

Back-propagating ${\hat{\varphi }_{\textrm{n},\textrm{k}}}({{{\mathbf r}_m}} )$ by the distance ${\textrm{z}_1}$gives the revised object exiting waves ${\hat{\varphi }_{\textrm{n},\textrm{k}}}({{{\mathbf r}_s}} )$. The steps (i-v) are repeated until a fixed number of iterations is conducted or an error metric criterion is satisfied [26].

3. Simulations

To verify the effectiveness of the proposed method, we have first conducted simulations in the near-field geometry. The laser wavelength was λ = 520 nm. The illumination probe was about 1.2 mm in diameter. The distances ${\textrm{z}_1} = 40$ mm, and ${\textrm{z}_2} = 20$ mm. The array size of diffraction patterns was $512 \times 512$ pixels, and the detector pixel size was $5.4{\; }\mathrm{\mu }\textrm{m}$. Angular spectrum algorithm was used for the wave propagation along ${\textrm{z}_1}$ and ${\textrm{z}_2}$. The calculated field of view was 2.76 mm. The illumination probe was set to have 10¹⁰ photons and the Poisson noise was added in the calculated diffraction intensities.

In the simulations, the modulator was a binary phase plate with a randomly distributed phase of 0 or $0.5\mathrm{\pi }$, whose distribution is shown in Fig. 3(a1). A total of 10 object functions were used in our simulation, and one of them is shown in Figs. 3(b1-b2), wherein the yellow circle indicates the extent of the illuminated area.

Fig. 3. Simulation results of the proposed method. (a1) is the phase of the simulated modulator, (b1) is the amplitude of the first simulated object. Reconstruction results (a2, b2) after 8000 iterations, (a2) is the recovered phase of the modulator, and (b2) is the amplitude reconstruction for (b1). The red square inset in (a1) and (a2) show the zoom-in of the central 20×20 region. (a3) is the difference in the phase value between the simulated modulator and the reconstructed modulator. The color bars in (a1), (a2) are in radians. Phase profile (b3) takes across the red line in (b1) and (b2).

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In the reconstruction process, an all-one matrix was used for the initial guesses of the modulator function and the object exiting waves. We set the amplitude constraints $[{{c_{\textrm{min}}},{c_{\textrm{max}}}} ]$ to $[{1,1} ]$, assuming a pure phase plate modulator. This tight amplitude range constraint will be relaxed later.

Figure 3(a2) shows the reconstructed modulator function. It can be seen from the middle $20 \times 20$ pixels area that the restored values are well consistent with the original one shown in Fig. 3(a1). Furthermore, Fig. 3(a3) shows the difference between the simulated and reconstructed phases of the modulator, which varies within [-0.001, 0.001] for the central $200 \times 200$ pixels region. The binary phase of the modulator has been recovered precisely. Figure 3(b2) shows the reconstructed object exiting wave, in comparison to the original one in Fig. 3(b1). Figure3(b3) shows the phase difference details of the corresponding red cross-sections in Fig. 3(b1) and Fig. 3(b2). There is a trivial constant deviation of 0.0358rad, and the fluctuation is less than $8 \times {10^{ - 4}}\; $rad.

To evaluate the quality of the reconstructed modulator function and object exiting waves, we use the RMS error metric [31], defined in Eq. (8).

{\gamma _{N,O}} = {\bf F}\{ {\varphi _N}({f_x},{f_y})\} \cdot {\bf F}{\{ {\varphi _O}({f_x},{f_y})\} ^\ast }

{\gamma _{N,N}} = {\bf F}\{ {\varphi _N}({f_x},{f_y})\} \cdot {\bf F}{\{ {\varphi _N}({f_x},{f_y})\} ^\ast }

{\gamma _{O,O}} = {\bf F}\{ {\varphi _O}({f_x},{f_y})\} \cdot {\bf F}{\{ {\varphi _O}({f_x},{f_y})\} ^\ast }

(8)$$\textrm{RMS} = \sqrt {1 - {{{{\left|{\sum\limits_{{f_x},{f_y}} {{\gamma_{N,Org}}} } \right|}^2}} / {\left( {\sum\limits_{{f_x},{f_y}} {{\gamma_{N,N}}} \sum\limits_{{f_x},{f_y}} {{\gamma_{Org,Org}}} } \right)}}} ,$$

where $\textrm{F}\{{\; } \}$ denotes the Fourier transform, and ${\varphi _O}({{\textrm{f}_\textrm{x}},{\textrm{f}_\textrm{y}}} )$ is the original wave field or modulator function, ${\varphi _\textrm{N}}({{\textrm{f}_\textrm{x}},{\textrm{f}_\textrm{y}}} )$ is the calculated wavefield or modulator function.

Figure 4 shows the RMS of the reconstructed object wave and the modulator function. The relationship between the RMS and the number of measured objects is shown in Fig. 4(a). The comparison between the extended algorithm and the original algorithm in Ref. [26] is shown in Fig. 4(b). Here 6 objects were used in this simulation.

Fig. 4. Reconstruction RMS error metric. (a) RMS evolution and its dependence on the number of measurements. (b) The comparison of different algorithms in terms of RMS: original CMI algorithm (red lines), extended algorithm (blue lines). The solid lines are for the object, and the dashed lines are for the modulator.

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It can be seen in Fig. 4(a) that the RMS values for the modulator decrease as the iteration progressed. When the number of measurements is greater than 3, the proposed algorithm starts to work. The reconstruction quality improves as the number of measurements increases. When the number of measurements is greater than 6, the quality of reconstruction tends to be stable, with an RMS reaching the order of magnitude of 10⁻³ to 10⁻⁴. More simulations show that there is no clear quantitative limit on the required number of measurements, which depends on the size of the illumination area and the complexity of the object waves. Figure 4(b) shows the reconstruction results of different algorithms. The traditional CMI algorithm cannot work entirely for an unknown modulator, but the proposed algorithm can provide results of good quality. In the presented results, a range of [1,1] has been used. When the range was relaxed to [0, 1], we found that a rerun of the algorithm the second time using the retrieved modulator function will give a final RMS error of 0.006.

4. Experiment demonstration

Visible light experiments using the setup as shown in Fig. 1(a) have been carried out. The optical source used was a Thorlabs 4-channel pigtailed diode laser with a wavelength of 517.2 nm. A 1.4 mm diameter pinhole was attached to the sample, i.e., $\textrm{d} = 0$. A fabricated phase plate with randomly distributed pillars was placed between the object and the detector. The transmission function of the phase plate was unknown in this experiment. The distance z1 was measured to be 45.4 mm, and the distance z2 was 22.1 mm. The detector used was an Andor Zyla 5.5 with a pixel size of $6.5\; \mathrm{\mu }\textrm{m}$. The central part of $512 \times 512\; $pixels of the recorded diffraction pattern was cropped out for reconstruction. The object was a tissue paper and was randomly rotated to different angles to record 30 diffraction patterns. The proposed algorithm was used to reconstruct the recorded patterns. The amplitude range $[{{c_{\textrm{min}}},{c_{\textrm{max}}}} ]$ is set to $[{0.4,0.5} ]$. The reconstructed results of the objects and modulator after 5000 iterations are shown in Fig. 5 and Fig. 6.

Fig. 5. The reconstruction results by the extended CMI algorithm. (a1) and (a2) show the 3^rd and the 5^th diffraction pattern recorded by the detector. The amplitude (b) and the phase (c) of the object reconstruction correspond to (a), respectively. In (b-c) only the area of 256×256 pixels in the middle of the reconstruction is shown.

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Fig. 6. The reconstruction of unknown modulators by the extended CMI algorithm. (a1) and (b1) represent the amplitude and phase respectively; (a2) and (b2) are the zoom-in views of the central 100×100 region in the red boxes, and (b3) show the phase profile along the blue dashed line in (b2).

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Figures 5(a1) and 5(a2) show two typical recorded diffraction patterns. Figure 5(b) and 5(c) are the reconstructed amplitude, and phase, corresponding to (a), respectively. From the results, the fabric structure of the tissue paper can be distinguished, and the diameter of a tissue fibre in the red box is about 6.6${\; }\mathrm{\mu }$m wide, which is close to the imaging resolution of the system that equals to the detector pixel size. The amplitude and the phase of the reconstructed modulator are shown in Fig. 6(a1) and Fig. 6(b1), respectively. It can be seen that the reconstructed region is circular with a diameter of 1.6 mm. The size of the illuminated area is related to the size of the pinhole and the angle of light scattered from the sample. Figures 6(a2) and 6(b2) show the zoom-in view of the red box regions. It can be seen that the modulator details have been well reconstructed. Figure 6(b3) shows the phase profile along the blue line in Fig. 6(b2).

To further verify the accuracy of the reconstructed modulator, we conducted an experiment using an onion epidermis as the sample and then performed the conventional CMI reconstruction using the retrieved modulator function as shown in Fig. 6.

Figure 7(a) shows the recorded diffraction intensity of the onion sample. Figure 7(b1) shows the reconstructed sample amplitude. The zoom-in view of the box region is shown in Fig. 7(b2). It can be seen that the nucleus, chloroplast plasma membrane, and cell wall of onion epidermal cells can be observed. This result proves the validity of the extended algorithm for the measurement of the modulator.

Fig. 7. The reconstruction of the onion epidermal sample from a single measurement using the retrieved modulator as a priori information. (a) the recorded diffraction pattern, (b1) the reconstructed amplitude, (b2) the zoom-in of red boxed area in (b1) with 6 times magnification.

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5. Conclusion

In summary, we report an extended algorithm for coherent modulation imaging. The algorithm allows the use of an unknown modulator in CMI. The core of the algorithm is to reconstruct the modulator function by averaging the multiple estimates derived from each measurement and a range constraint is set on the modulator amplitude. As the multiple object exit-waves are reconstructed, the modulator function can be retrieved. This method saves the need to calibrate the modulator in advance when conducting CMI experiments. The performance of the proposed method has been demonstrated and discussed in detail in both simulations and experiments. We believe the extended algorithm would make CMI a much more independent technique and would find wide use for materials science and biology.

Funding

National Natural Science Foundation of China (11775105, 12074167); Science and Technology Planning Project of Shenzhen Municipality (KQTD20170810110313773); Centers for Mechanical Engineering Research and Education at MIT and SUSTech (MechERE Centers at MIT and SUSTech) (6941806).

Disclosures

The authors declare no conflicts of interest. This work is original and has not been published elsewhere.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Coherent modulation imaging using unknown modulators

Abstract

1. Introduction

2. Methods

2.1 Support constraint

2.2 Modulation

2.3 Modulus constraint

2.4 Modulator function updating

2.5 demodulation

3. Simulations

4. Experiment demonstration

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (12)

Optics Express