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Partially coherent image formation with x-ray microscopes L. Jochum and W. Meyer-Ilse Image formation with partially coherent radiation is evaluated with the Hopkins formula and then applied to x-ray microscopy. Image characteristics expected from instruments with circular and annular pupils in partially coherent conditions are considered for two-point objects and a knife-edge object. The theoretically expected values for image characteristics that are easy accessible by an experiment, such as the width of a knife edge, are given for various x-ray microscopes. Key words: Image formation, partially coherent imaging, x-ray microscopy, resolution limit. 1. Introduction X-ray microscopy is a unique technique that extends visible light microscopy to higher resolution and makes use of unique contrast mechanisms. It does not compete with techniques such as electron microscopy in terms of resolution but rather offers unique advantages including the ability to image samples in an aqueous environment. X-ray microscopy yields information with high resolution from thick samples that cannot be obtained by any other technique including methods such as atomic force microscopy or near-field microscopy. Applications of x-ray microscopy are in several fields including biology and material science. The interaction of x rays with matter provides unique elemental and chemical contrast. Construction of an x-ray microscopy resource center is planned at the Advanced Light Source 1ALS2 in Berkeley, Calif.1,2 that will provide a broad user community with the necessary instrumentation to use this technique optimally. There are two basic types of x-ray microscope, conventional x-ray microscopes (XM’s) and scanning x-ray microscopes (SXM’s). XM’s are similar to their visible light counterparts with critical illumination. Sometimes they are also called imaging or full-field x-ray microscopes. XM’s with zone plates were pioneered at the University of Gottingen.3 The most prominent instruments of each kind are the Göttingen XM at the Berlin electron storage ring4 The authors are with the Center for X-ray Optics, Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, California 94720. Received 14 July 1994; revised manuscript received 3 February 1995. 0003-6935@95@224944-07$06.00@0. r 1995 Optical Society of America. 4944 APPLIED OPTICS @ Vol. 34, No. 22 @ 1 August 1995 1BESSY2 and the STXM operated at the Brookhaven National Laboratory 1BNL2.5 Because these two microscope types are complementary, there are plans to build them at the ALS. An XM that uses bending magnet radiation is completed and has been operational since the Fall of 1994. The resolution of x-ray microscopes is currently not limited by the wavelength of the imaging light but rather by the ability to fabricate optical elements that provide high-resolution images. The highest resolution for both types of microscope is achieved with zone plate lenses. So far, the numerical aperture 1NA2 of the zone plates is typically below NA 5 0.1. For microscopes operating in the water window 12.4 nm # l # 4.5 nm2, a resolution approximately five times better than with visible light is commonly achieved. Unlike conventional refractive lenses, the focusing properties of zone plates are based on diffraction. This implies that the focal length of a zone plate lens strongly depends on the wavelength and that multiple diffraction orders exist. In most cases only the first diffraction order is used. The diffraction efficiency in the first diffraction order of objective zone plates is currently in the range of 10%, with improvements expected in the future. If operated with quasimonochromatic radiation, and if the number of zones is large, zone plate lenses obey the same laws as refractive lenses.6 In the scope of this paper we assume that the above-mentioned assumptions are fulfilled and treat zone plates as conventional lenses. Therefore the conclusions in this paper also hold for microscopes other than x-ray microscopes. 2. X-Ray Microscopes A schematic of the optical layout of an XM like the one at BESSY and the one at the ALS is shown in Fig. Fig. 1. 1a2 Schematic optical layout of an XM. 1b2 Schematic optical layout of an SXM. 11a2. Polychromatic synchrotron radiation is focused onto the sample by a condenser zone plate, and an objective zone plate forms an enlarged high-resolution image of the object. This image is detected with an x-ray CCD camera. Because zone plates are diffraction optics, they have a strong chromatic error, and the image quality depends on the monochromaticity of the image-forming light. To provide the required monochromaticity, a pinhole is put in close proximity to the sample plane. This pinhole together with the condenser zone plate acts as a linear monochromator. With D as the diameter of the zone plate and d as the pinhole diameter, the spectral bandwidth of the illuminating light is given by7; Dl@l 5 2d@D. The center wavelength can be tuned by changing the distance between the condenser zone plate and the pinhole. A schematic of an SXM like the one at the BNL is shown in Fig. 11b2. A high-resolution objective zone plate is spatially coherent illuminated and forms a diffraction-limited x-ray spot. This spot is scanned over the sample area while the transmitted x-ray intensity is recorded by a detector. Because only the spatially coherent fraction of the x-ray source can be used, such an instrument would preferably be built at an undulator light source to achieve reasonably short exposure times. A central stop on the objective zone plate and an order-sorting aperture between the objective zone plate and the sample are used to block out zero-order radiation. The main advantage of an SXM compared with an XM is the absence of an objective zone plate downstream of the sample. The low efficiency of the objective zone plate contributes to the transmission losses in an XM. Therefore an SXM requires a smaller radiation dose on the sample to obtain the same signal-to-noise ratio as an XM. To take advantage of the increased resolution compared with visible light microscopy, microscopy with x-ray-excited visible light luminescence8,9 requires an SXM. 3. Image Formation in X-Ray Microscopes Microscopes use a probe to determine the distribution of one or more physical quantities in the sample with high spatial resolution. In x-ray microscopy the probe consists of x rays. For low-energy x rays as used in x-ray microscopy, the main interactions are photoelectric absorption and elastic scattering, whereas inelastic scattering 1Compton scattering2 may be neglected.10 In a single-atom model, the interaction of any material with x rays can be described by the complex atomic scattering factor f 5 f1 1 if2; values of f1 and f2 are listed in Ref. 10. On a macroscopic scale the interaction of x rays with matter is characterized by the complex index of refraction n 5 1 2 1l@2p2h 2 i1l@4p2µ, where µ is the linear absorption coefficient and h is the phase shift coefficient of the wave with respect to its propagation in vacuum. The atomic scattering factors are related to µ and h as µ 5 2r0lnA f 2 and h 5 r0lnAf1 with r0 being the classical electron radius and nA the number of atoms per unit volume.10 In x-ray microscopes the sample is imaged in transmission. This image contains the information about the spatial distribution of the complex index of refraction n1x, y, z2 in the sample. For the low-energy x rays used in x-ray microscopy, where inelastic scattering processes can be neglected, the image formation for both types of x-ray microscope 1SXM and XM2 may be described by the same formalism.11,12 The equivalence of scanning and conventional microscopes is valid even for imaging thick objects. The differences between the two instruments are purely technical. Therefore the conclusions and results in this paper apply for both types of x-ray microscope, although the 1 August 1995 @ Vol. 34, No. 22 @ APPLIED OPTICS 4945 parameters below have different meanings, depending on the type of instrument. The microscopes considered here detect the intensity distribution in the detector plane, so that only one two-dimensional 12D2 section of the sample can be seen at a time. To record the complex index of refraction n1x, y, z2 in all three spatial dimensions with such an instrument, one needs to take several images with different views of the sample. To gather three-dimensional 13D2 information in one exposure by using only one detector plane, the recording of the amplitude and the phase of the wave transmitted by the sample is required, as in holography. With only one image the intensity recorded in the detector plane of a microscope from a 3D object cannot be distinguished from a 2D object that produces the identical wave field in a plane downstream of the object. Below we briefly review the theory of partially coherent image formation and with that background describe the properties of a 2D view of different samples in an x-ray microscope. 4. Partially Coherent Imaging The theory of partially coherent image formation was first developed by Hopkins.13,14 If 1x0, y02 and 1x1, y12 are Cartesian coordinates in the object plane and in the image plane, respectively, the intensity profile I1x1, y12 in the image plane of an isoplanatic microscope is I1x1, y12 5 ` ` ` ` 2` 2` 2` 2` eeee J1x0 2 x08, y0 2 y082 3 F1x0, y02F*1x08, y082K1x1 2 x0, y1 2 y02 3 K*1x1 2 x08, y1 2 y082dx0dy0dx08dy08. 112 In Eq. 112 J1x, y2 is the mutual intensity in the object plane, F1x, y2 is the complex transmission of the object, and K1x, y2 is the Fourier transform of the complex transmission of the objective lens. Any objective lens aberrations or effects of defocusing can be included in K1x, y2. The physical meaning of the mutual intensity is that it characterizes the phase correlation between any two points in the object plane. If the illuminating light is perfectly incoherent, the phases vary statistically so that the mutual intensity equals zero unless the two points are identical. Mathematically, this behavior is described by the Dirac d function, and Eq. 112 reduces to the well-known linear relation between the object and the image intensity with the intensity point-spread function as the linear filter. For totally coherent illumination the phase correlation and thus the mutual intensity is constant for any two object points. In this case Eq. 112 describes a linear transformation of the complex amplitude from the object to the image plane with the amplitude point-spread function as the linear filter. To make use of this linearity, the detector must record the amplitude and phase in the image plane as it is done in holography. 4946 APPLIED OPTICS @ Vol. 34, No. 22 @ 1 August 1995 The results of investigations of the resolution in x-ray microscopy for the limiting cases of coherent and incoherent imaging have been published in Ref. 15. In general, the image formation in a microscope is partially coherent. As we show below, the effects of partial coherence are most important for feature sizes near the limit of resolution. When imaging with a circular symmetric aberrationfree objective lens that might have a central stop of relative radius 0 # ek , 1, the objective lens function K1x, y2 is given by K1x, y2 5 J11n2 2 23 n 2 11 2 ek 2 ekJ11ekn2 n 4 with n 5 2p1x2 1 y221@2. 122 J1 is the Bessel function of first order and must not be mistaken for the mutual intensity J1x, y2. The spatial dimensions of x and y are measured in units of l@NAo, where l is the wavelength and NAo is the numerical aperture of the objective lens. In these normalized units and if ek 5 0, the first minimum of K1x, y2 is at a distance of 0.61 from the central maximum. In an XM there is normally no central stop on the objective zone plate. That the innermost zone of a zone plate may act as a central stop can be neglected, since it results in an effective ek of 1@Œn with n being the total number of zones. The n is of the order of 102 for objective zone plates and 105 for condenser zone plates. In an SXM, however, a central stop on the objective zone plate is normally used to remove zero-order radiation 1ek . 02. Assuming that the mutual intensity in the object plane is caused by a circular symmetric aberrationfree aperture that might have a central stop, we obtain J1x, y2 5 J11mn2 2 11 2 e 2 3 mn 2 2 j ejJ11ejmn2 4 mn with n 5 2p1x2 1 y221@2, 132 with ej describing the relative radius of the central stop and m the width of the central maximum of J1x, y2 relative to the one of K1x, y2. Equation 132 includes the limiting cases of coherent and incoherent image formation as discussed above. If m = 0, J1x, y2 is constant over the sample area leading to a coherent image formation, and if m = `, J1x, y2 can be approximated by a d function that corresponds to an incoherent image formation. In an XM, ej would be the radius of the central stop of the condenser zone plate divided by the radius of the condenser zone plate. A central stop is necessary in an XM to remove zero-order radiation, so that normally ej . 0 with ej 5 1@3 as a typical number. In unnormalized coordinates of the XM, the coherence parameter m would be m5 lo@NAo lc@NAc , 142 with lc and lo being the wavelengths that pass the condenser and objective lens and NAc and NAo being the numerical apertures of the condenser and objective lens, respectively. As for soft x rays, where inelastic scattering can be neglected, lc equals lo, the coherence parameter m is given by the ratio of the numerical apertures of the condenser and the objective lens. Equation 142 also holds for an SXM if NAc is replaced by the sine of the half angle of the acceptance cone of a circular detector. If either the object itself or its Fourier spectrum is given by the sum of a finite number of Dirac d functions, an analytical solution for the image intensity in Eq. 112 can be found, but for arbitrary objects a numerical evaluation is necessary. We have chosen a two-point object and a knife edge as an object to investigate the effects of partial coherence and central stops on the properties of the optical system. 5. Partially Coherent Imaging of a Two-Point Object A two-point amplitude object with a point distance of d is described by F1x, y2 5 d121⁄2 d 2 x, y2 1 d11⁄2 d 2 x, y2. 152 With the above object function the four integrations in Eq. 112 have been evaluated. The result is the image intensity of the two-point object. We calculated the image contrast of a two-point object as a function of the parameters m, ej, and ek for point distances of 0 # d # 1.22. As mentioned above, the linear dimensions are measured in units of l@NAo. The figures show examples of how the variation of the parameters m, ej, and ek affects the image contrast. Figure 2 shows the image contrast of a two-point object as a function of the inverse point distance 1@d for different relative numerical apertures m, indicating different degrees of coherence in the object plane. In the range of 0 # m # 1 the image contrast at a fixed object point distance generally increases with increasing m. If we compare the image contrast for an incoherent imaging system 1m = `2 with the case of matching numerical apertures 1m 5 12, the image and thus the contrast is identical only for a point separation of d 5 0.61. Two points, which are unresolved in a coherent imaging system 1m 5 02, can Fig. 2. Image contrast of a two-point object imaged with ej 5 ek 5 0 and coherence parameters m of 0, 0.5, 1, and infinity; d is in units of l@NAo. Fig. 3. Influence of m on the image contrast of a two-point object for four selected point distances d imaged with circular pupils 1ej 5 ek 5 02; d is in units of l@NAo. be imaged with a contrast of more than 40% with a partially coherent imaging system and an appropriate coherence parameter m. For a fixed point distance and increasing m, the image contrast steadily increases to a maximum between m 5 1 and m 5 1.5, which is shown in Fig. 3. As m approaches infinity, the contrast is slightly oscillating around the contrast obtained with an incoherent imaging system 1m = `2. Figure 4 and 5 show the influence of different central stops on the image contrast. The coherence parameter is set to m 5 1 for these examples. The mutual intensity J1x, y2 in the object plane varies with the parameter ej, whereas ek describes a central stop on the objective lens and thus modifies K1x, y2. For a two-point object a central stop on the objective lens 1ek . 02 has a greater effect on the image contrast than a change of the mutual intensity from a similar stop on the condenser lens 1ej . 02. Fig. 4. Image contrast of a two-point object versus the inverse of the point distance d 1in units of l@NA2 for m 5 1 and 1a2 ek 5 0 for different mutual intensities 10 # ej # 0.92, 1b2 ej 5 0 for different central stops on the objective lens 10 # ek # 0.92. 1 August 1995 @ Vol. 34, No. 22 @ APPLIED OPTICS 4947 Fig. 5. Image contrast of a two-point object for four selected point distances d imaged with m 5 1: 1a2 ek 5 0, different mutual intensities 10 # ej # 0.92; 1b2 ej 5 0, different central stops on the objective lens 10 # ek # 0.92. Fig. 6. Distance L between two points that form an image with 15.3% contrast as a function of the coherence parameter m at 1a2 ek 5 0, different mutual intensities 10 # ej # 0.92, 1b2 ej 5 0, different central stops on the objective lens 10 # ek # 0.92. For any point distance d an increase in ek results in a higher image contrast. The contrast enhancement by changing ek is less significant for larger point distances, which can be seen as different slopes of the curves in Fig. 51b2. If m 5 1 and ej 5 ek 5 0, a point separation of d 5 0.61 corresponds to the resolution limit according to the Rayleigh criterion 1Ref. 14, p. 3332. Applying the Rayleigh criterion to an incoherent imaging system with circular pupils leads to an image contrast of 15.3% for a two-point object. By changing ek from 0 to 0.9 at a point distance of d 5 0.61, the image contrast increases from 15.3% to almost 70% 3see Fig. 51b24. Let L be the distance of two object points at which the image contrast is 15.3%. In unnormalized coordinates this distance is dx 5 Ll@NAo. Figure 6 shows L as a function of m for various ej and ek. In the range of 0 # m # 1, which is relevant for x-ray microscopy, an increase of ej as well as ek results in a decreasing L. This can be interpreted as an improved resolution, but one must remember that these results refer only to the imaging of a two-point object. Figure 7 shows two examples of an edge response and the differentiated response. The imaging parameters in Fig. 71a2 represent a typical layout for the XM-1 at the ALS, and in Fig. 61b2 are typical parameters for an SXM. The full width at half-maximum 1FWHM2 of the differentiated image from an edge is often used to determine the resolution of a system experimentally. 6. Partially Coherent Imaging of a Knife Edge After examining the behavior of a partially coherent imaging system in the case of a two-point object, we investigated the same system with a 2D knife edge as an object. Analyzing the image of a knife edge is a very common and convenient method to check the image quality experimentally. To simulate the partially coherent imaging of a knife edge, Eq. 112 was evaluated numerically with the program SPLAT,16 developed by Toh at the Department of Electrical Engineering and Computer Sciences, University of California. 4948 APPLIED OPTICS @ Vol. 34, No. 22 @ 1 August 1995 Fig. 7. Response from imaging an edge with partially coherent light and the differentiated response at 1a2 m 5 0.54, ej 5 0.33, and ek 5 0 1same as XM-1, 30 nm in Table 12; 1b2 m 5 1, ej 5 0, and ek 5 0.45 1same as SXM in Table 12. Table 1. Optical Setup Incoherent SXM-lum SXM XM-2 XM-1, 30 nm XM-1, 25 nm Coherent Image Characteristics of a Knife-Edge Object and a Two-Point Object for Different Optical Setups m ej ek Different Edge Response FWHM 1l@NA2 ` ` 0 0 0 0 0.33 0.33 0 0 0.45 0.45 0 0 0 0 0.49 0.47 0.45 0.47 0.45 0.45 0.42 1 1 0.54 0.45 0 Edge Response 25–75% 1l@NA2 Edge Response 10–90% 1l@NA2 L 5 Two-Point Distance for 15.3% Contrast Two-Point Contrast at d 5 0.61 1%2 0.31 0.42 0.40 0.29 0.22 0.21 0.19 0.62 1.32 1.31 0.59 0.39 0.37 0.34 0.61 0.55 0.57 0.61 0.70 0.73 0.82 15.3 29.5 29.5 15.3 0 0 0 The indicated value for the FWHM in Fig. 7 is what can be theoretically expected from that particular instrumental configuration. FWHM values of the differentiated edge response for other instrument configurations are tabulated in Table 1. Although the intensity profiles of the edge response in Fig. 7 look quite different, they result in the same FWHM of the differentiated response. This is because the FWHM, like the image contrast of a two-point object, is mainly characterized by the width of the central maximum of K1x, y2. By introducing a central stop in the objective lens, the central maximum of K1x, y2 becomes narrower, but the higher-order fringes around the maximum increase in intensity, as is shown in Fig. 8. As previously shown, the narrower central maximum results in a higher image contrast in the case of a two-point object, especially for small point distances. When a knife edge is imaged with the same optical system, the increased intensity in the higher-order fringes blur the image, as seen in Fig. 71b2, and tend to reduce the overall contrast of an arbitrary object. When looking at the distance of a 90–10% intensity drop in the edge response, which is the corresponding quantity to the Rayleigh resolution limit of an incoherent imaging system, we see that this distance is significantly larger in Fig. 71b2 than in Fig. 71a2. Other then the FWHM, which is the same for both cases, this distance is sensitive to the changed parameters of the optical system and thus is a more likely candidate to characterize the quality and performance of a particular imaging system. Fig. 8. Point images for the parameters in Fig. 71a2 1m 5 0.54, ej 5 0.33, and ek 5 0, continuous curve2 and Fig. 71b2 1m 5 1, ej 5 0, and ek 5 0.45, dashed curve2. 7. Summary The properties of the image formation with partially coherent light have been described for x-ray microscopes. The results are not restricted to x-ray microscopy; they also apply to other microscopes including visible light and transmission electron microscopes. The results of our evaluations are summarized in Table 1. It shows the image characteristics of a knife-edge object and a two-point object for different combinations of m, ej, and ek. Each row represents a particular optical setup. The first and the last rows show the theoretical cases of a totally incoherent and a totally coherent system, respectively. All the other optical setups shown in Table 1 are likely candidates for the microscopes planned at the ALS. SXM-lum stands for a scanning x-ray microscope operating with the x-ray excitation of visible light luminescence. Because the luminescence light is incoherent, the coherence parameter m is set to infinity. An unobstructed detector and a central stop on the objective zone plate define the parameters ej and ek, respectively. The incoherent image formation implies that the image formation is linear in intensity, even beyond the diffraction limit of resolution. This enables techniques that use inverse filtering to overcome the diffraction limit 1Ref. 17, p. 133ff 2. The SXM in Table 1 is the same instrument but not in the luminescence mode, so that m is given by the numerical aperture of the objective zone plate and half of the acceptance angle of the detector. XM-2 is a conventional microscope. The SXM and the XM-2 are to operate at a short-period undulator source. The optical design of XM-2 will be such that it has a circular entrance and exit pupils and matching numerical apertures of the illuminating and imaging optics. XM-1 is the conventional microscope that operates with bending magnet radiation. In Table 1 its imaging properties are shown for two different objective zone plates, one with a 30-nm and one with a 25-nm outermost zone width. In both cases it has a large diameter condenser zone plate with a 55-nm outermost zone width, so that these two setups have a different coherence parameter m. Comparing the first two rows in Table 1 or the SXM and the XM-2 cases, one can see how an annular exit pupil improves the two-point resolution but increases 1 August 1995 @ Vol. 34, No. 22 @ APPLIED OPTICS 4949 the width of the edge response at the same time. For circular exit pupils 1ek 5 02, the edge width decreases so that it appears sharper with a decreasing coherence parameter m, whereas the two-point resolution is reduced at the same time. This object dependence of the image formation is caused by the nonlinearity of Eq. 112. The fact that knife-edge images produced with partially coherent radiation appear sharper compared with the incoherent case is important for the characterization of instruments near their diffraction limit. Looking at the numbers in the first three rows of Table 1, one can see that a central stop on the objective zone plate, which is necessary in a SXM, affects the image characteristics significantly compared with the influence of the coherence parameter. In a knife-edge test an instrument such as the SXM in Table 1 should produce a 75–25% image intensity drop at a distance of 0.4 l@NAo or a FWHM of 0.45 l@NAo, to be considered as diffraction limited. We thank Derek Leed from the Department of Electrical Engineering and Computer Sciences, University of California, for helpful support with the program SPLAT, which was used to calculate the edge response data. This research is supported by the United States Department of Energy, Office of Basic Energy Sciences, and the Office of Health and Environmental Research under contract DE-AC0376SF00098. References 1. W. Meyer-Ilse, D. Attwood, and M. Koike, ‘‘The X-ray Microscopy Resource Center at the Advanced Light Source,’’ in Proceedings of the Fourth International Conference on Biophysics and Synchrotron Radiation, B. Chance, J. Deisenhofer, S. Ebashi, D. T. Goodhead, J. R. Heliwell, H. E. Huxley, T. Iizuka, J. Kirz, T. Mitsui, E. Rubenstein, N. Sakabe, T. Sasaki, G. Schmahl, H. B. Stuhrmann, K. Wüthrich, and G. Zaccai, eds. 1Clarendon, London, 19942, pp. 625–636. 2. W. Meyer-Ilse, M. Koike, R. Beguiristan, J. Maser, and D. 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