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A microscope using Zernike’s phase contrast method and a hard xray Gabor hologram
Journal of the European Optical SocietyRapid Publicationsvolume 12, Article number: 25 (2016)
Abstract
Background
In hard Xray phase imaging using interferometry, the spatial resolution is limited by the pixel size of digital sensors, inhibiting its use in magnifying observation of a sample.
Methods
To solve this problem, we describe a digital phase contrast microscope that uses Zernike’s phase contrast method with a hard Xray Gabor holography associated with numerical processing and spatial frequency domain filtering techniques. The hologram is reconstructed by a collimated beam in a computer. The hologram intensity distributions itself become the reconstructed wavefronts. For this transformation, the Rayleigh Sommerfeld diffraction formula is used.
Results
The hard Xray wavelength 0.1259 nm (an energy of 9.85 keV) was employed at the SPring8 facility. We succeeded in obtaining highresolution images by a CCD sensor with a pixel size of 3.14 μm, even while bound by the need to satisfy the sampling theorem and by the CCD pixel size. The test samples used here were polystyrene beads of 8 μm, and human HeLa cells.
Conclusions
We thus proved that the resolution 0.951 μm smaller than the pixel size of CCD (3.14 μm) was achieved by the proposed reconstruction techniques and coherent image processing in the computer, suggesting even higher resolutions by adopting greater numerical apertures.
Background
Hard Xrays allow visualization of objects at high resolutions because of their inherent smaller wavelengths [1]; they are therefore highly desirable in various application areas, including biological studies [2, 3]. Even though optical imaging methods are nondestructive (and therefore useful), they are typically limited in spatial resolution because of the long wavelengths used [4] and transmission limitations. Imaging with electron beams depends on fixing the sample under vacuum conditions, and this is detrimental to the morphology of the biological specimens under observation. On the other hand, hard Xrays may suffer the effect of external vibrations, because of their short wavelengths.
Phase imaging using hard Xray makes it possible to visualize phase objects using interferometry or holography. Computed tomography using an Xray interferometer [5] and observation of biological soft tissues using interferometry [6] have been used for phase imaging. Methods using differential interference contrast (DIC) to form Talbot images with hard Xrays have been reported [6]. Given that samples are imaged on a digital sensor, the spatial resolution in such methods is limited, because of the pixel size.
Many reports of digital holography [7] and holographic lateral shear for DIC [8] using laser beam holography have been made. High resolution is possible in holography, because the images are obtained at a position apart from the digital sensor. The twopoint resolution in coherent illumination hard Xrays can be obtained by 0.82λ/NA [9]—if the relative phase between the two points can be considered as zero—where λ is the wavelength and NA is the numerical aperture. It should be pointed out that the sensor pixel size does not appear in the equation for resolution. Accordingly, hard Xray digital holographic systems may provide high resolution. However, the conjugate images and reconstructed images overlap, a problem that must be solved. Hard Xrays have been used in Gabor holography [10], to obtain specklefree coherent illumination [1–11], and in phase contrast microscopy [12]. Hard Xrays have also been used in coherent diffractive imaging [13] with high sensitivity, lensless imaging with an extended source [14], and phase contrast imaging [15]—using polychromatic hard Xrays [16].
This paper describes a digital holographic microscope using Zernike’s phase contrast observation method [4] using a hard Xray Gabor hologram, which is recorded in a computer. In Gabor holograms, the most important aspect is to reduce the effect of conjugate images. Good quality highresolution images were obtained after some numerical processing.
Methods
Description of the used hard Xray Gabor hologram setup
Figure 1 shows the Gabor hologram process using a common path interferometer. In our optical arrangement, the maximum optical path difference between the reference and object beams in the hologram plane is very short. The synchrotron source (the beamline 20XU of SPring8, Japan) used in this study produces quasimonochromatic^{4} Xrays with a monochromator, as shown in Fig. 1. The central part of the expanded Xray beam is filtered by aperture CS_{1}, with a size of 50 μm, and is focused by a Fresnel zone plate (FZP) with a diameter of 104 μm and a focal length of 498 mm. Another cross slit aperture CS_{2}, with a size of 2 μm, is used to filter the beam and acts as the point source for hologram recording. The beam illuminates an object placed at a distance of Z _{ 0 } from CS_{2}. The size of the hologram is determined by the size of the reference beam at the hologram plane (diameter D = 1.483 mm at a distance of Z _{1} from CS_{2}). The distance between the object and the hologram is Z. A 16bit Hamamatsu chargecoupled device (CCD) sensor (C47429824) with a 3.14 μm pixel size which consists of 1344 × 1024 pixels was used to record the holograms. Interference fringes with a narrowest spacing of 15.3 μm were produced at the edges of the sensor. The minimum width of the fringe was about 4.9 times the pixel size, thus satisfying the sampling theorem. The Xray energy was 9.85 keV, with a corresponding wavelength of 0.1259 nm. The test samples used in this study were 8 μm polystyrene sphere beads and human HeLa cells. The spatial resolution of the reconstructed image in our setup was calculated at 0.951 μm [1], which is less than the sensor pixel size.
The Gabor hologram is reconstructed by numerical processing in the computer. In the conventional reconstruction method, the object image and its twin image appear close to each other, as shown in Fig. 2, and removal of this twin image is therefore crucial to obtain the final image [17]. To reduce the effects of both the dc noise and the twin image, defocus is introduced in the image plane. For reconstruction, a collimated beam parallel to the optical axis is used to illuminate the hologram, so that the complex amplitude of the reconstructed image is obtained by simply carrying out Fresnel back propagation of the hologram to the image plane. In this way, the twin image is produced far from the object image, as shown in Fig. 3. The convolution of the twin image with an impulse response having a diverging curvature produces a defocus, thereby reducing the twin image effects.
Mathematical description of Zernike’s method
The Zernike’s phase contrast method can be mathematically derived using the complex amplitudes reconstructed from the Gabor hologram, as will be shown. If the complex amplitudes of the reference beam and the phase object beam in the hologram plane at coordinates (x _{1}, y _{1}) are denoted by r(x _{1}, y _{1}) and g(x _{1}, y _{1}) respectively, the intensity distribution I _{h}(x _{1}, y _{1}) of hologram is given by
The complex amplitude of r(x _{1}, y _{1}) is given by
where Z _{1} is the distance between the point source and the hologram, and λ is the Xray wavelength, and j is an imaginary unit, \( \mathrm{j} = \sqrt{1} \). If the phase object is illuminated by the hard Xrays originating from a point source, the complex amplitude g(x _{1}, y _{1}) of the phase object in the hologram plane is given by
where Z _{0} is the distance between the point source and the object, Z is the distance between the object and the hologram, and Φ(x, y) is the phase distribution of the object. It is noted that if the phase object is thicker, the wavefront is further advanced. The intensity distribution I _{h}(x _{1}, y _{1}) of the hologram is obtained by substituting Eq. (2) and Eq. (3) into Eq. (1).
For reconstruction the Gabor hologram is illuminated by a collimated beam of unity amplitude to separate the object image and its conjugate image. The complex amplitudes reconstructed in the image plane are obtained by carrying out a Fresnel Transform of I _{h}(x _{1}, y _{1}) from the hologram plane to the image plane. The complex amplitude of image u _{i}(x_{i}, y_{i}) = Fr[r ^{*}(x _{1}, y _{1})g(x _{1}, y _{1})] is given by
The image is reconstructed at position Z _{i0} where the condition 1/Z – 1/Z _{1} +1/Z _{i0} = 0 is satisfied; the image plane position is therefore given by Z _{i0} = −ZZ _{1} /Z _{0} (using Z _{1} = Z _{0} + Z). The complex amplitude of the reconstructed image is given by
where Φ(x _{i0}/M, y _{i0}/M) is a real periodic function (with period d) and M = Z _{i0} /Z is the system magnification. We assume that the magnitude of Φ(x _{i0}/M, y _{i0}/M) is small compared to unity, so that we may write [4]
where the amplitude k _{i} = 1/(λ ^{2} Z _{1} ^{2}) is constant, and Fr[] stands for the Fresnel transform. Z _{i0} = −ZZ _{1} /Z _{0} is thus obtained. It is also noted that exp[jФ] = 1 + jФ – Ф^{2}/2! + • • • .
Similarly, we can derive the complex amplitude of the conjugate image u _{c}(x _{i}, y _{i}) in the plane where the conjugate image is reconstructed; u _{c}(x _{i}, y _{i}) is then given by
If the conjugate image is reconstructed at position Z _{i} = Z _{ic}, the condition 1/Z – 1/Z _{1} – 1/Z _{ic} = 0 must be satisfied. The position in the conjugate image plane (x _{ic}, y _{ic}) is Z _{ic} = ZZ _{1}/Z _{0}, and the complex amplitude of conjugate image is given by
If the distance between the reconstructed image and its conjugate image is very long, a wavefront coming from the conjugate image plane may be considered as being emanated from a point source. This condition will be met if the radius MD/2 of the aperture in the image plane is smaller than the Airy disc caused by its aperture at the conjugate image plane; that is, the coherent illumination condition is given by
It is noted that the magnitudes at the image plane and conjugate image plane are equal. We shall now calculate the complex amplitude of the conjugate image in the image plane caused by a point in the conjugate image plane. In Eq. (8), x _{ic} = y _{ic} = 0 is substituted and given by
In the image plane (x _{i0}, y _{i0}), the complex amplitude of conjugate image is given by
When exp[jΦ(0, 0}] ≑ 1 + jΦ(0, 0) and the magnitude of Φ(0, 0) is small compared to unity,
The magnification of (jπ/2λZ _{ic}) (x _{i0} ^{2} + y _{i0} ^{2}) is small compared to unity and we may also write
Moreover, the complex amplitude of the zeroorder term is expressed by u _{0}(x _{i}, y _{i}) = Fr[r(x _{1}, y _{1})^{2} + g(x _{1}, y _{1})^{2}] and is given by
In these equations, k _{0} = (1/λ ^{2} Z _{1} ^{2} + 1/λ ^{2} Z _{0} ^{2}) is a constant. Three wavefronts of u _{i}(x _{i0}, y _{i0}), u _{c}(x _{i0}, y _{i0}), and u _{0}(x _{i0}, y _{i0}) exist in the image plane. To remove the zeroorder term, all complex amplitudes in the image plane are Fourier transformed, and the zero frequency is excluded; this means that u _{0}(x _{i}, y _{i}) expressed by Eq. (14) is also removed. These complex amplitudes are then inverse Fourier transformed to the image plane. If we expand exp[−jΦ(x _{i0}/M, y _{i0}/M)] with a Fourier series
then c _{0} = 1 and c _{m} = −c _{m} ^{*}, (m ≠ 0). We Fourier transform the complex amplitudes of u _{i}(x _{i}, y _{i}) and u _{c}(x _{i}, y _{i}) again, and we add exp(±jπ/2) = ±j at the zero frequency. Eq. (7) and Eq. (14) then become:
Using Z _{ic} = −Z _{i0} = ZZ _{1} /Z _{0}, the intensity distribution I _{i}(x _{i0}, y _{i0}) in the image plane is obtained as
In Eq. (19), if λZic ≫ 1 + Φ(0, 0), the complex amplitude of the conjugate image can be neglected. The phase term of the complex amplitude of the reconstructed image is cancelled by calculation of the intensity, so that Eq. (19) becomes
where k = {1/(λ ^{2} Z _{1} ^{2})}^{2} is a constant, and the term Ф ^{2} has been neglected because of its smallness. We should note that Eq. (20) shows that if a Gabor hologram is recorded in the computer, it is possible to use Zernike’s phase contrast observation method.
Results
The hard Xray Gabor hologram is reconstructed by a collimated beam parallel to the axis, so the hologram intensity distribution is multiplied by unity. This means that the intensity distribution itself can be regarded as the complex amplitude and transformed from the hologram plane to the object plane. For this transformation, the RayleighSommerfeld diffraction formula was used. The image wavefront is produced at a long distance from the hologram.
The correct position determination in the image plane is important. A method to obtain an autofocused image has been proposed [18]. However, we will present a method to elegantly find the position in the image plane of simple objects recorded in the hologram, such as the polystyrene sphere beads. In our method, we use the diffraction effect that results from the distance between the image plane and the hologram being very long. The principle of the method is that if the object is located at the image plane, no diffraction pattern appears around the image; however, if the image is located at a position away from the image plane, a ringshaped diffraction pattern appears around the image. Therefore, the position of the image plane can be elegantly determined by observing the diffraction patterns (the image plane Z _{i0} was 174.5 m). The magnification M of the optical system [19] was calculated to be M = Z _{i0} /Z = 25.55, given that the distance between object and hologram was Z = 6.830 m.
Figure 4a and b show the hard Xray Gabor holograms of the polystyrene beads and the HeLa cells, respectively. Since the wavefront produced by the Xrays transmitted through the phase object is advanced in comparison with the wavefront traveling in the air, the hologram becomes dark. Figures 5 and 6 show the results of Zernike’s phase contrast method. Figure 5 shows the results obtained with the 8μm polystyrene sphere bead samples. Figure 5a shows I _{i}(x _{i}, y _{i}) = k{1 − 2Φ(x _{i0}/M, y _{i0}/M)}, and Fig. 5b shows k{1 + 2Φ(x _{i0}/M, y _{i0}/M)} as noted in Eq. (20). The reconstructed phase object is the bright sphere in Fig. 5a and the dark sphere in Fig. 5b. Figure 6 shows the results obtained for the dried HeLa cell samples. Figure 6a shows I _{i}(x _{i}, y _{i}) = k{1 − 2Φ(x _{i0}/M, y _{i0}/M)}, and Fig. 6b shows k{1 + 2Φ(x _{i0}/M, y _{i0}/M)}. It should be pointed out that Φ(x _{i0}/M, y _{i0}/M)} is negative, because with hard Xrays the object refractive index is smaller than that of air.
Discussion
It should be noted that the refractive index related to the phase difference in the hard Xray regime is less than unity, contrary to what happens with visible light; the wellknown refractive index equation is approximately given by n = 1–1.35 × 10 ^{−6} ρλ ^{2}, where ρ (g/cm^{3}) is the density, λ (Å) is the wavelength [12], and the terms representing absorption and scattering are ignored for simplicity. The maximum value of the optical path difference in the polystyrene sphere is about δ = 0.017 nm, if the density is taken as ρ ≑ 1. The condition needed to derive Eq. (20) is Φ(x _{i}/M, y _{i}/M) < 1; the magnitude of Φ(x _{i}/M, y _{i}/M) = 2πδ/λ recorded in the hologram is about 0.848. Since this value is smaller than unity, the condition is satisfied [4]. In Eq. (10), a value of (4(0.82λZ _{ ic})/M)^{1/2} = 53 μm in the object plane was obtained, so that the wavefront of the conjugate image in the image plane can be regarded as a wavefront caused from a point source. Samples of polystyrene sphere beads with 8μm diameter and dried HeLa cells are used as phase objects; in the used hologram, the aperture diameter magnitude can be considered to produce a wavefront originating from a point source. The numerical value of (1 +Φ(0, 0))/(2λZ _{ic}) in Eq. (20) is calculated, to check whether this term can be neglected or not, resulting in Φ(0, 0) < 1, λ = 0.1259 ×10 ^{−3}μm, and Z _{ic} = 174.5 × 10 ^{6} μm; therefore, (1 +Φ(0, 0))/(2λZ _{ic}) ~ 4.2 × 10 ^{−5} ≪ 1, and we can safely neglect this term.
Conclusions
We proposed a microscope using Zernike’s phase contrast observation method and a hard Xray Gabor hologram recorded in a computer. Two different sample types were used for demonstration purposes: polystyrene sphere beads with an 8μm diameter, and dried HeLa cells. Recording the hologram in a computer makes it possible to perform computational experiments on coherent Xray processing. This is very important, because Xray resources are limited. Moreover, coherent numerical processing becomes easy and simple; creating a perfect phase delay of π/2, for example, is trivially produced numerically. Even though a small 0.1259 nm wavelength was used, there is no influence of the external vibrations in image reconstruction. It is also pointed out that the use of holography with the exception of an image hologram can produce images with high resolution, since the pixel size of the CCD detector does not have influence on resolution, but its diameter has.
Abbreviations
 CCD:

Charge coupled device
 DIC:

Differential interference contrast
 FZP:

Fresnel zone plate
 NA :

Numerical aperture
 λ :

Wavelength
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Acknowledgements
The authors would like to thank Dr. Y. Koseki, a group leader, and Dr. K. Chinzei, a subdirector of AIST, Japan, for providing research facilities. Technical advices from Dr. K. Hibino, Dr. T. Eijyu and Dr. M. Yamauchi were highly appreciated.
Funding
Consejo Nacional de Ciencia y Tecnología(CONACYT), Mexico, has awarded to Dr. Juan C. Aguilar with a scholarship to make a postdoctoral research at National Institute of Advanced Industrial Science and Technology (AIST), Japan (250204). A part of this study was supported by 2016 Saga Prefecture Leading Industry Incubation Program, Japan.
Availability of data and materials
The datasets supporting the conclusions of this article are included within the present article.
Authors’ contributions
KM: He engaged in the idea of the method, mathematical analysis, simulation of laser and Xray holography and design of computer algorithm. JCA: He engaged in a computer code for hard Xray holography. MM: He engaged in the management of this study, discussion and advice of Xray measurement method. MY: He engaged in Xray optical design, production of hard Xray hologram and advice for simulation of laser holography. SR: He engaged in a part of the idea of the method, simulation of laser holography and computer algorithm. YS: He engaged in production of hard Xray hologram. AT: He engaged in production of hard Xray hologram. IN: He engaged in discussion of hard Xray optics and simulation of laser holography. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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This study does not need any ethical approval along with evidence. The samples used in this study are commercially distributed.
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Keywords
 Xray microscopy
 Distributedfeedback
 Digital holography
 Xray imaging
 Interference microscopy
 Xray interferometry