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Coherent diffraction imaging (CDI) is an established lensless imaging method widely used at the x-ray regime applicable to the imaging of non-periodic materials. Conventional CDI can practically image isolated objects only, which hinders the broader application of the method. We present the imaging of non-isolated objects by employing recently proposed “non-scanning” apodized-illumination CDI at an optical wavelength. We realized isolated apodized illumination with a specially designed optical configuration and succeeded in imaging phase objects as well as amplitude objects. The non-scanning nature of the method is important particularly in imaging live cells and tissues, where fast imaging is required for non-isolated objects, and is an advantage over ptychography. We believe that our result of phase contrast imaging at an optical wavelength can be extended to the quantitative phase imaging of cells and tissues. The method also provides the feasibility of the lensless single-shot imaging of extended objects with x-ray free-electron lasers.
© 2015 Optical Society of America
1. Introduction
Many objects of biological interest such as cells and tissues are transparent in their natural state at the optical wavelengths. Despite being widely exploited tools in life sciences, common absorption-contrast optical microscopes cannot display natural contrast of these transparent specimens in the in-focus image plane. Early solutions provided to the problem by Zernike’s phase contrast microscopy [1,2] and differential interference contrast (DIC) microscopy [3] can transform the phase modulation of the sample into the intensity variations. These methods, however, have difficulty providing phase information quantitatively: Zernike’s phase contrast microscopy is known to suffer from characteristic halo artifacts; and DIC can only provide the gradient of the phase and the image contrast is usually low.
A broad variety of interferometric methods [4–6] have been used for the quantitative phase shift measurement of biological specimens. Precise phase determination in interferometric methods, however, usually requires measurement of multiple interferograms, which is not suitable for imaging of fast cellular dynamics, and also sensitive to environmental vibrations.
Non-interferometric quantitative phase imaging methods also often demand multiple wavefield intensity patterns. The transport of intensity equation needs the intensities at multiple planes [7]. Coherent diffraction imaging (CDI) is a quantitative phase imaging method [8,9] and has been used for high-contrast bioimaging in the X-ray regime [10–14]. Conventional CDI, however, can practically image isolated objects only. Ptychography, a scanning coherent imaging method, can be applied to extended objects and thus broadened the applicability of CDI [15–22].
To observe fast cellular dynamics, however, quantitative phase imaging from a single wavefield intensity pattern is preferred. In the present study, we employ recently proposed “non-scanning” apodized-illumination CDI to image non-isolated objects [23]. The use of focusing optics can also make CDI measurement efficient [24–27]. In the process of development of apodized-illumination CDI, we first demonstrate the imaging of extended objects by using an optical laser. We also show that the methods are useful in imaging of transparent phase objects at the optical wavelengths.
Apodized-illumination CDI is a recently proposed lensless imaging technique, which can image extended objects with spatially-confined focused illumination produced by specially designed optics. The spatially-confined coherent illumination interacts with the extended object and a Fraunhofer diffraction pattern is recorded by a two-dimensional imaging detector. The image of the object is then computationally reconstructed. The computational part of the microscopy is an iterative process, which in the most primitive form can be described as the back and forth propagation between the sample plane and the Fourier plane satisfying constraints in the both planes [28]. The constraint in the Fourier plane is called the modulus constraint and that in the sample plane the support constraint.
The reconstruction of the image using iterative phase retrieval methods requires that the diffraction intensities be sampled finely enough to satisfy the oversampling condition [29]. In conventional CDI, the oversampling condition is satisfied by illuminating an isolated sample, which precludes the imaging of extended objects.
CDI of extended objects at the optical wavelengths using an aperture to selectively image a region of extended objects has been reported earlier [30], but the aperture was fixed to the sample. A similar aperture assisted object area selection approach has also been used to image extended object by selected area coherent electron diffraction imaging [31]. Diffraction imaging of extended objects at the optical wavelengths has also been accomplished with a divergent wave [32]. The method assumes the prior knowledge of the incident wavefield. In apodized-illumination CDI, however, an isolated illumination with a constant phase defines the field of view and a part of the extended sample is selectively imaged. This would be particularly advantageous in the quantitative phase imaging of living cells and tissues.
2. Optical design
The isolated illumination for extended objects is the central idea of apodized-illumination CDI. Hence, we have designed an optical configuration to construct a spatially-confined constant-phase illumination. Before discussing the entire optical configuration, we briefly discuss the capability of lenses to generate a constant-phase main lobe at its back focal plane with a specific optical configuration.
The magnitude of the wavefield at the back focal plane of a lens is the Fourier transform of the wavefield placed either in front of the lens or immediately behind the lens [33]. The associated phase factor of the wavefield depends on the position of the input plane relative to the lens. Let U(x,y) be the input wavefield at a distance d in front of a lens with a pupil function P. The wavefield at the back focal plane of the lens with a focal length f is given by
Provided that the pupil of the lens projected on the input plane is large compared to the dimensions of the input object, we can neglect the effect of the pupil function.In the cases when the quadratic phase factor in Eq. (1) should be avoided, the input is placed at the front focal plane of the lens. In such case, i.e. when d = f, the wavefield at the back focal plane of the lens is given by
The expression in Eq. (2) is the simple Fourier transform of the input wavefield. Hence, a constant-phase main lobe can be realized at the back focal plane of the lens for centrosymmetric aperture located at d = f illuminated with a plane wave, because the Fourier transform of any real centrosymmetric function is real and centrosymmetric. Our optical configuration stands on this property of lenses.The schematic representation of the optical configuration is shown in Fig. 1(a). Coherent light from a laser is filtered and collimated by the spatial filter and the lens L1, respectively. The rectangular slit placed after the lens L1 defines the illumination shape and size. By controlling the slit aperture size, the illumination size at the sample plane can be adjusted. An assumption made at this point is that the Gaussian beam diameter incident on the slit is much larger than the slit aperture, and hence the wavefield at the slit plane can be regarded as a plane wave. The optical components in between the light source and the beam-defining slit can be avoided, provided that the beam size at the beam-defining slit is broad enough to satisfy this approximation. The lens L2 generates a Fraunhofer diffraction pattern of the beam-defining slit at its back focal plane. The distance between the slit and lens L2 is equal to the focal length of the lens. Hence, by Eq. (2), the wavefield at the back focal plane of the lens has a constant phase within the main lobe. The apodizing slit placed at the back focal plane of the lens L2 functions to remove the side lobes of the Fraunhofer diffraction pattern of the beam-defining slit. The lenses L3 and L4 constitute the so-called 4f imaging system, which forms the image of the apodized wavefield at the back focal plane of the lens L4. Finally, the isolated plane wave illumination is incident on a non-isolated object. The scattered radiation from the sample is collected by the lens L5 placed immediately behind the sample. The lens produces the Fraunhofer diffraction pattern at its back focal plane, where an imaging detector records the pattern.
3. Experimental results and discussions
3.1 Optical setup
An experimental setup based on the optical design was realized. A He-Ne laser (LASOS Lasertechnik GmbH LGR 7634) emitting light at a wavelength λ of 632 nm with a power of 2 mW was used as a coherent light source. A neutral density filter (not shown in the experimental configuration in Fig. 1) was placed immediately after the laser system to adjust the light intensity. The parameters of all the lenses used in our experiments are summarized in Table 1. A mirror after the lens L1 was used to change the course of the light by 90° to make the setup compact. The size of the collimated Gaussian beam produced by the spatial filter and the lens L1 was about 15 mm. The aperture size of the beam-defining slit was 1 mm in the both directions with an alignment precision of ± 10 μm. Since the Gaussian beam size is substantially larger than the slit aperture size, the wavefield at the plane of the beam-defining slit can be regarded as a plane wave. We used a Photon Inc. 12-bit USBeamPro charge-coupled device (CCD) Model 2312 with 1360 × 1024 pixels, each with a size δ of 4.65 μm, for both recording diffraction patterns and taking bright-field images of the sample.
Table 1. Parameters of the lenses used in the experiment
In CDI measurement, the lens L5 was placed physically as close as possible to the sample, and was located at ~15 mm from the sample. The lens-detector distance was equal to the focal length of the lens L5 and was 72 mm. In all the results presented in this article, reconstruction was performed using the central 980(N) × 980(N) pixels of the CCD detector. Hence, the pixel size in the real space for all the reconstructed images presented here is λl/δN ≈13 μm, where l is the sample-detector distance and l = 94.2 mm.
By translating the lens L5 and the detector to the positions 110 mm and 322 mm from the sample, respectively, bright-field images of the sample were directly obtained. We refer to the setup as the direct imaging mode and the image thus obtained as the direct image. The magnification of the direct images was 1.92. Defocused images were taken by locating the detector 3 mm downstream the in-focus image plane.
3.2 Apodization of illuminating wavefield
The intensity distributions at the sample plane with and without the apodizing slit were measured by the CCD detector in the direct imaging mode. The measurement at various exposure times of 0.1, 1, 10 and 100 ms was made to increase the dynamic range, and were merged into a single intensity distribution.
Figure 2 shows a comparison of the intensity profiles at the sample plane with and without the apodizing slit. In the measurement with the apodizing slit, the aperture size and position of the apodizing slit were adjusted while observing the intensity distribution at the sample plane, and were optimized to minimize the side lobe intensities. The precision of apodizing slit adjustment was ± 5 µm. The logarithmic plot in Fig. 2 distinctly displays the suppression of the side lobes of the illumination, when the apodizing slit is used at the optimized aperture size and position. The magnitude of the side lobes in the intensity profile with the apodizing slit has been reduced by approximately three orders of magnitude compared to that without the apodizing slit.
Fig. 2 Logarithmic plot of the intensity profiles at the sample plane with and without the apodizing slit.
This dramatic decrement in the side lobe intensities of the illumination allows us to perform apodized-illumination CDI. The apodized illumination size a determined from Fig. 2 is approximately 538 μm in both dimensions. The oversampling ratio, given by σ = λl/δa, is 23.82.
3.3 Imaging of amplitude object
The imaging of an amplitude test pattern was performed in the CDI mode with and without the apodizing slit. A USAF positive resolution target bought from Sigma Koki as shown in Fig. 3(a) was used as a test sample. The diffraction patterns at exposure times of 0.05, 1 and 10 ms were recorded, and a diffraction pattern with a higher dynamic range was tailored by merging them. In merging the diffraction patterns, the intensities at saturated pixels at longer exposures were substituted by the time-scaled intensities of the corresponding pixels with shorter exposures. The diffraction pattern at the shortest exposure time was devoid of saturated pixels. The dark charge noise of the detector at each exposure time was subtracted from the corresponding diffraction pattern before the merging. The center of the diffraction pattern was set to the pixel with the maximum intensity in the diffraction pattern. We centro-symmetrized the merged diffraction pattern based on the fact that the diffraction pattern is centro-symmetric in three-dimensional reciprocal space for amplitude objects and that the curvature of the Ewald sphere can be neglected in the present case. The maximum error due to the curvature of the Ewald sphere is given by Q(1-cos θmax), where Q is the wavevector and θmax is the maximum diffraction angle. In our CDI experiment, the maximum error corresponds to ~12 pixels for a maximum diffraction angle of ~2° at the corner of the CCD detector. This value is smaller than an oversampling ratio of ~24, thus we can neglect the curvature of the Ewald sphere. In addition, for thin samples as in the present case, each speckle is elongated in the direction of the incident beam, so we can safely neglect the curvature of the Ewald sphere.
Fig. 3 (a) A USAF resolution target. (b)The direct image. Merged diffraction patterns after centro-symmetrization (c) with apodizing slit at the optimized aperture size and position and (d) without the apodizing slit. (e) and (f) Reconstructed images from (c) and (d), respectively. (g) and (h) Sets of the direct image and the reconstructed image from other parts of the resolution target.
The reconstruction of the image from the merged diffraction pattern was carried out by using the shrink-wrap hybrid input-output (HIO) algorithm [28,34]. The algorithm uses a dynamic support constraint unlike the fixed support constraint in the conventional HIO algorithm. The phasing procedure started with a rectangular support with a size slightly larger than the illumination size, and the support was updated by convoluting the reconstructed image with a Gaussian function. The width of the Gaussian function was initially set to 3 pixels, and was reduced by 1% every 10 iterations until the 500th iteration. The threshold for the dynamic update of the support was taken in the range from 8% to 12% of the maximum amplitude of the reconstructed object. After the 500th iteration, the support was held fixed till the 5000th iteration, after which the phasing procedure was ceased. Figure 3(b) represents the direct image of a part of the sample illuminated by the apodized focused beam. The diffraction patterns with and without the apodizing slit are shown in Fig. 3(c) and 3(d), respectively, and the corresponding reconstructed images in Fig. 3(e) and 3(f), respectively. The dramatic improvement in the quality of the reconstructed image with the apodizing slit can be distinctly observed. Figure 3(g) and 3(h) are sets of the direct image and the reconstructed image from other parts of the sample. The results prove that the apodization of the illumination is crucial in non-scanning CDI of extended objects. The brighter region at the center of the reconstructed and direct images can be attributed to the amplitude and intensity distribution of the apodized illumination, respectively.
3.4 Imaging of phase object
We also applied apodized-illumination CDI in imaging a phase object. A binary rectangular-groove phase grating was fabricated on the central 3.5 × 3.5 mm2 area of a 12 × 12 mm2 cover glass using optical lithography. The procedure of optical lithography involves spin coating the photoresist (Tokyo Ohka Kogyo Co., Ltd. OFPR5000LB) on the cover glass, followed by direct laser writing (NEOARK Corp. DDB-201-200). The non-protected region of the cover glass was etched by Reactive Ion Etching (SAMCO Inc. RIE-10NRV). Finally, the photoresist was removed by wet etching with acetone. The lower stripes were designed to be 140 nm deep from the upper stripes. The extinction coefficient of the cover glass at the optical wavelengths is zero. The refractive index of the cover glass was 1.5 and hence the rectangular groove phase grating is expected to produce a phase shift of ~0.22π rad.
A traditional bright-field microscopy image (KEYENCE VK-X200) of the phase grating pattern shown in Fig. 4(a) does not provide any contrast as it should be. The inset in Fig. 4(a) is an image of a part of the phase grating pattern taken by a scanning white-light interferometer (Zygo NewView 7300). The widths of the upper and lower stripes were 35 and 45 μm, respectively, and the depth of the groove was ~138 nm as determined by the scanning white-light interferometer. Figure 4(b) and 4(c) are the defocused and focused images, respectively, taken using our optical setup in the direct imaging mode. The diffraction patterns at exposure times of 0.05, 1 and 10 ms were recorded and merged by patching, as explained earlier, to obtain a higher dynamic range diffraction pattern without saturated pixels. We centro-symmetrized the merged diffraction pattern, since the diffraction pattern is centro-symmetric at small angles for pure phase objects (negligible absorption) with weak phase. Figure 4(d) shows the merged diffraction pattern after centro-symmetrization. The diffraction pattern consists of periodic maxima along the x-axis as expected from the phase grating pattern. The image was reconstructed from the merged diffraction pattern by using the shrink-wrap HIO algorithm. The threshold for the shrink-wrap algorithm was set to 10% of the maximum of the reconstructed amplitude. The reconstructed image shown in Fig. 4(e) was obtained after the 5000th iterations. The reconstructed image obtained in the CDI mode and the defocused image in the direct imaging mode was found to be in good agreement. Figure 4(f) is the line profile along the dashed line in the reconstructed image. The full width at half maximum (FWHM) of the upper and lower stripes as determined from the reconstructed images was 36 μm and 45μm respectively. The line profile agrees well with the fabricated pattern proving the reliability of the method in imaging phase objects. The brighter region at the center of the reconstructed image can be attributed to the amplitude distribution of the apodized illumination as in the case of the amplitude object. For pure phase objects with weak phase, the reconstructed image is proportional to both the illumination amplitude and the phase shift [35]. The phase shift distribution can be quantitatively obtained by appropriately normalizing the reconstructed image by the amplitude distribution of the illumination. For the apodized illumination with a constant phase, the amplitude distribution is just given by the square root of the intensity distribution, which we did not measure at the CDI experiment.
Fig. 4 (a) Optical microscope image of a binary phase grating pattern; the inset is the image from a scanning white-light interferometer. (b) Defocused and (c) focused images in the direct imaging mode. (d) Merged diffraction pattern after centro-symmetrization. (e) Reconstructed image from (d). (f) Line profile along the dashed line in the reconstructed image (e).
4. Conclusions
To summarize, we have demonstrated a non-scanning CDI of extended objects with apodized illumination. In the process, we have realized isolated apodized illumination with a constant phase by using a specially designed optical configuration. The optical system is composed of a beam defining slit illuminated by a plane wave at the front focal plane of a lens, an apodizing slit at the back focal plane of the lens, and a subsequent 4f imaging system. The apodizing slit successfully reduced the side lobe intensities approximately by three orders of magnitude as compared to the case without the apodizing slit.
In apodized-illumination CDI experiment, we used a USAF positive resolution target as an amplitude object and a binary rectangular-groove grating as a phase object. The grating pattern was fabricated by optical lithography. In recording coherent diffraction patterns, we merged CCD data with different exposure times to increase the dynamic range. The shrink-wrap HIO algorithm was used for image reconstruction. For the both samples, the dramatic improvement in the quality of the reconstructed image with the apodizing slit was observed.
We believe that the method can be expanded to quantitative phase imaging of cells and tissues. The non-scanning nature of the method will be useful in the quantitative study of fast cellular activities, such as membrane fluctuations [36]. Our results also proved the feasibility of the imaging of extended objects with x-ray free-electron lasers [37–39], where other methods fall short. The concept of the spatially localized illumination in apodized-illumination CDI can also be transferred to the ptychography community to realize aperture-less ptychography.
Acknowledgments
This study was supported by CREST (JST); KAKENHI (JSPS) Grant Numbers 15H05737, 26870006, 25400438, 26286077, and 23226004; the X-ray Free-Electron Laser Priority Strategy Program (MEXT); and the Cooperative Research Program of “Network Joint Research Center for Materials and Devices”.
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