Contrast transfer functions for Zernike phase contrast in full-field transmission hard X-ray microscopy

Full-field transmission hard X-ray microscopy (TXM) has been widely applied to study morphology and structures with high spatial precision and to dynamic processes. Zernike phase contrast (ZPC) in hard X-ray TXM is often utilized to get an in-line phase contrast enhancement for weak-absorbing materials with little contrast differences. Here, following forward image formation, we derive and simplify the contrast transfer functions (CTFs) of the Zernike phase imaging system in TXM based on a linear space-shift-invariant imaging mode under certain approximations. The CTFs in ZPC in their simplified forms show a high similarity to the one in free-space propagation X-ray imaging systems. © 2016 Optical Society of America OCIS codes: (340.7460) X-ray microscopy; (340.7440) X-ray imaging; (110.4850) Optical transfer functions; (110.2990) Image formation theory. References and links 1. W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. 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Introduction
X-ray microscopy (XRM) in both soft and hard X-ray regimes has been utilized to probe with higher precision into small spatial and even temporal scales [1,2] to access the morphology of macro-and micro-3D structures [3,4], as well as chemical information regarding elemental distribution and concentration [5], chemical states [6], etc.
Hard X-ray microscopy offers a non-invasive and versatile probe with varying spatial resolutions covering a wide range from a few hundred nm down to tens of nm [7,8]. Bridging the gap between light microscopy and electron microscopy, it is becoming more and more crucial in the mesoscopic regime in three dimensions [9]. Compared to soft XRM, hard XRM is more suited for investigations under more complex and flexible sample environments such as in-situ and in-vivo observation [10,11], probing non-invasively into three dimensions when combining with computed-tomography.
One of the major challenges for hard X-ray microscopy is related to the low sensitivity for distinguishing various structural components of light materials in biological specimens or composites of similar densities. Especially in life science, soft tissues or cells usually produce poor absorption contrast, i.e. small differences in image intensities that are merely observable. This is due to the fact that elemental compositions of low atomic numbers and low densities in soft tissues or cells are very similar to that of water [12]. Therefore in hard X-ray (>10 keV) regime, especially for light materials (low-Z elements) it becomes more interesting for imaging techniques to explore the phase shifts induced from the wave-matter interaction rather than modulations in amplitude. The sensitivity gain can reach up to an order of three magnitudes [12]. The phase ring is placed in the back focal plane of the objective lens (Fresnel zone plate). U i is the wavefield behind the sample. U b is the wavefield behind the phase ring (in the back focal plane of FZP). U o is the wavefield arriving at the detector plane. The P wave (red) represents the zeroth-order diffraction; the S wave (green) is the higher-order diffraction that passes the phase ring untouched. The interference between the phase altered P wave and the S wave will lead to contrast enhancement in the intensity recorded at the imaging plane.
Zernike phase contrast (ZPC) in full-field hard X-ray microscopy [13] integrates a phase ring in the back focal plane of the objective as shown in Fig. 1. By phase shifting the zerothorder diffraction beam, namely, the primary beam, the recorded intensity of the interference between the primary wave and the diffracted wave can attain increased sensitivity particularly when applied to weak-absorbing objects. This enhanced contrast can be observed in-line and is approximately linearly proportional to the optical path difference (OPD), therefore implying the object-induced phase variation [14], which is the prominent advantage of ZPC.
However, due to its intrinsic low frequency artifacts from interference, to approach quantitative analysis remains difficult. Discussion began to emerge recently on how to tackle this problem. In general, one way to approach the quantification is off-line phase map retrieval using multiple radiographs under various conditions [15][16][17]. The other way is to modify the optics, such as the condensor and the coupled phase ring in order to suppress the ZPC artifacts [18].
Here, we would like to introduce and center on the contrast transfer functions (CTFs) of ZPC in full-field transmission hard X-ray microscopy in a simplified way under coherence assumptions. They have been derived as an extension of the formulation in electron microscopy [19], however they can be further simplified due to the properties of X-ray optics. Similarity can be found in CTFs of free-space propagation X-ray imaging system, which have been developed more than one decade ago [20] and have been extensively applied to make considerable achievements in varying research fields [21,22]. Our derivation of simplified CTFs will be useful in deepening the understanding of this imaging system, with potential to improve the quantification of the image information.

Derivation of contrast transfer under Zernike phase contrast mode for full-field transmission X-ray microscopy
The TXM system can be modeled as a linear shift-invariant system composed of four parts, the input U i , the lens, the Zernike phase ring U b and the output U o , as specified in Fig. 1. A coherent illumination is assumed here while in practice it is generally taken as a mutually incoherent emission. Readers are referred to [23,24] for the detailed discussion of coherence conditions. The complex transmission function of the sample can be linearized as follows Here r is the spatial coordinates in the transverse plane of the object, φ (r) is the relative phase variation in contrast to the amount of absolute phase shift δ n is the real part of the complex refractive index of the medium. This linearization between the phase variation φ (r) and the transmission function of the object T (r) is achieved by taking the assumptions of a weak-absorbing object with slowly varying phase shifts The incident wavefield U i (r) is then forward propagated to the exit plane of the phase ring, which is also the back focal plane of the objective lens, where it equals [25] U (3) f is the spatial frequency in Fourier domain corresponding to r.
is the pupil function of the phase ring. The cut-off spatial frequency f p is defined as f p = r p d f λ , where d f is the focal length of the objective lens, and r p corresponds to the radius of the hole in the phase ring. The frequency f p should be infinitely close to zero, since phase modification should only be applied to the zeroth-order diffraction beam. P(m, λ ) is the attenuation factor of the phase ring, depending on the materials (m) and the X-ray wavelength (λ ).
H(f) is the function characterizing optical deformation induced by the objective lens, including the pupil function, optical aberrations, and defocusing by shifting the lens, expressed in the form of [26] H Here ∆z denotes the defocusing distance, and g(f) is the lens aberration factor, attributed to the optical aberrations inherent in the objective lens. In XRM systems with ZPC, the objective lens is mostly realized by a Fresnel zone plate (FZP). The FZPs in hard X-ray systems tend to have lower aberrations compared to lenses in electron microscopy or for soft X-rays because of their lower numerical apertures. Hence under the current diffraction resolution limit for hard X-rays [27], the spherical aberration as well as other aberrations can be safely neglected. In general, the pupil function of the FZP acts as a low-pass filter, limiting the achievable spatial resolution. As we focus our discussion on the transfer of the low frequencies, which carry the overall image contrast, this effect is not taken into account. Hence this pupil function is also neglected for simplicity.
By substituting Eq. (1) into Eq. (3), we get from which we can observe that the ring functions as a high-pass phase shifter here since it in principle only modulates the phase of the Dirac delta function δ (f). In the end U i (r) is the wavefield modified by the phase plate. HereB(f) andφ (f) means B(r) and φ (r) in Fourier space. The image intensity in Fourier space at the imaging plane after far-field propagation can be written as an autocorrelation of the real space wavefield U i (r) in the exit-plane of the objective lens [28] F in which () * means conjugation. If we consider the wavefield U i (r) in Fourier domain F {U i (r)} = δ (f) −B(f) + iφ (f) without the phase ring, i.e. in bright-field mode, and substitute Eq. (1) to Eq. (6) we get [29] I Here, the factor sin(πλ ∆zf 2 ) in front of the relative phase shiftsφ (f) is defined as the phase contrast transfer function (phase CTF). And the factor cos(πλ ∆zf 2 ) in front of the absorption modulationB(f) is the amplitude contrast transfer function (amplitude CTF). Eq. (7) indicates a linear relation between the Fourier transform of the image intensityĨ o (f) and the Fourier transform of the phase-amplitude modulationφ (f) andB(f), with a small defocus distance ∆z. This turns out to be one of the most important formulas in electron microscopy optics, and has been adopted to X-ray fields in the last decade [20,30]. The phase and amplitude CTFs in the equation each describe a spectral filter of the object information to pass through, and the resolution limit lies in the first zero crossings for a single record.
In Zernike phase contrast mode, for U i (r) being phase modulated by θ p and hence F {U i (r)} = exp(iθ p )δ (f) +B(f) − iφ (f), the same amount of phase shift is induced in the transfer function and it becomes When applying a θ p = 3π/2 phase ring, Eq. (8) can be reduced tõ which has the simplest form in the extreme case when ∆z = 0, thenĨ o (f) = δ (f)P 2 − 2φ (f)P [25], where the linearity between the phase modulation and the intensity can be achieved. Despite that this linearity can be inevitably corrupted by the inherent ZPC artifacts, the absorption contrast image can compliment to retrieve the phase map [16]. For concision, P(m, λ ) is noted as P in the equations. One can also extend Eq. (8) to the dark field mode. Consider a beam stop with very high attenuation instead of the phase ring which blocks the direct beam, then we get This imaging mode can be applied to observe e.g. scattering signals from ultra-structures [31]. It is obvious that the distinct difference of Eq. (9) from Eq. (7) is that the phase CTF is shifted from a sine function to a negative cosine function, while for the amplitude CTF an inverse change from a cosine to a sine function occurs.
This difference is a significant improvement in the final image intensity contrast in Zernike phase contrast mode, for the change of a sine function to a cosine function in phase contrast produces a much higher contrast factor for the low spatial frequencies, thus leading to much better preservation of the large features in the object. At the same time, a more smooth spectral distribution is obtained compared to the summed up defocusing CTFs [32], where multiple distances are often required in the regularization of the zero crossings in the summed CTFs.

Discussion on coherence condition and its influence in CTF
In experimental conditions, the illumination in TXM is usually assumed to be a quasimonochromatic Gaussian source distribution with a finite beam divergence and a finite energy spread. Regarding the aspect of temporal coherence corresponding to the energy spread, the transfer function will be the product of the coherent CTF with an envelope function characterizing the energy spread [30]. This damping effect in the CTFs will mainly contribute to the degrading of amplitude of CTFs and also to an elimination in the oscillations, leading to contrast depression and loss of fine details.
Concerning the spatial coherence, which can be characterized by the van Cittert-Zernike theorem with a complex coherence factor, its influence has been analyzed by K. A. Nugent in [23,24]. In this study the impact of partial coherence on the contrast in microscopic systems is quantitatively analyzed. From the simulation, it is shown that under the same defocus distance the oscillations of both the amplitude and phase optical transfer functions (OTF) are rapidly suppressed as the coherence is reduced. It is more complicated than a simple envelope damping factor as in the temporal coherence case, but works in a similar manner in influencing the CTFs.
While the damping effect on CTFs may pose a trouble for defocusing algorithms which rely on a high degree of coherence, it will have less influence on Zernike phase contrast mode when phase retrieval is not required. Hence in this case the linearity of the exit wavefield does not need to be preserved, but it will become a linearity of the recorded image intensity instead [27].

Simulation of contrast transfer functions in TXM
To analyze the degree of contrast enhancement according to the CTF, the image of Lena is taken as the object for simulation. Image pixel size is set to be 30 nm, with image dimensions 20 µm by 20 µm. As a weak phase object with phase-attenuation duality, the phase shift ranges from 0.0029 to 0.039, and the amplitude attenuation is set to be even weaker by two orders of magnitude, what corresponds to typical values in biological applications. A small defocusing distance ∆z of 150 µm is introduced here to both absorption and Zernike phase contrast modes. The amplitude and phase CTFs are first simulated independently of specific objects, and are only scaled by the defocusing distance ∆z. So the further it is defocused, the narrower oscillations and zero crossings we get from the CTFs.
As shown in Fig. 2(a), in the bright-field (absorption) mode, the sine function of the phase CTF (green) stays below 0.5 for frequencies below below 2.7 µm −1 (corresponding to 0.37 µm in real space). Hence it acts as a high-pass filter for the phase modulation factorφ (f), which is the dominating contrast in our object. This suppression of the low frequencies results in the poor visibility of the features larger than 0.37 µm. On the other hand, high frequencies are more prominent and manifested as sharply defined edges, corresponding to the edge-enhancement effect we obtained from defocusing propagation. This is the reason why weak absorbing objects usually can not be imaged truly but just the small variations and edges appear enhanced. In contrast, in Zernike phase contrast mode the phase CTF (green in Fig. 2(b)) is shifted by 3π/2 to a cosine function, which is analogous to change a high-pass filter to a low-pass one. This change gives a boost to the final image contrast of the whole objects as the low