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We extend coherent diffraction imaging (CDI) to a high numerical aperture reflection mode geometry for the first time. We derive a coordinate transform that allows us to rewrite the recorded far-field scatter pattern from a tilted object as a uniformly spaced Fourier transform. Using this approach, FFTs in standard iterative phase retrieval algorithms can be used to significantly speed up the image reconstruction times. Moreover, we avoid the isolated sample requirement by imaging a pinhole onto the specimen, in a technique termed apertured illumination CDI. By combining the new coordinate transformation with apertured illumination CDI, we demonstrate rapid high numerical aperture imaging of samples illuminated by visible laser light. Finally, we demonstrate future promise for this technique by using high harmonic beams for high numerical aperture reflection mode imaging.
© 2012 Optical Society of America
1. Introduction
The last decade has seen dramatic advances in the development of coherent diffraction imaging (CDI) techniques [1–6]. In CDI, a lens is essentially replaced by an iterative phase retrieval algorithm that, in theory, allows diffraction-limited imaging at the illuminating wavelength [7–12]. In traditional CDI, a coherent plane wave illuminates an isolated sample (i.e. where the entire finite-sized object is illuminated), and the intensity of the far field scatter pattern is measured by a detector. Because the phase of the scatter pattern is lost, an iterative phase retrieval algorithm must be used to reconstruct the exit surface wave directly after the sample. As a result, CDI can be categorized as a type of in-line, phase-contrast microscopy. The most prevalent iterative phase retrieval algorithms combine two constraints for image reconstruction: 1) the measured diffraction data, and 2) an oversampling condition on the detector that corresponds to an isolated object in the sample plane [13]. CDI has already been used to produce phase contrast images of magnetic materials [14,15], nano-scale strain [16], integrated circuits [17] and biological samples [18, 19].
Nearly coincident with the development of CDI, high harmonics of femtosecond lasers have advanced from interesting strong field science phenomena to a unique and robust light source used for a wide range of experiments in molecular, materials and energy sciences [20–22]. To produce bright, fully spatially and temporally coherent, laser-like high harmonic beams, this extreme nonlinear frequency up conversion process must be phase matched to ensure coherent buildup of the emission from many atoms. [20–24]. Phase matching of the high harmonic generation (HHG) process has now been demonstrated up to photon energies > 1.5 keV [20–22]. In fact, the only other available coherent EUV and soft x-ray light sources are large-scale synchrotrons and free electron lasers, and lab-scale laser-plasma-based soft x-ray lasers [25–27]. Tabletop HHG sources are complementary to large-scale synchrotron and x-ray free electron laser facilities: they can achieve sub-fs time resolution, are perfectly synchronized to the driving laser, can operate at multi-kHz repetition rates, and have a broad range of harmonics that can probe function at many different atomic sites simultaneously. Although the average x-ray flux and pulse energy is lower than what can be generated at the new, large, x-ray free electron laser facilities, nevertheless HHG sources have been successfully used for coherent diffractive imaging of isolated test samples, with record spatial resolution of 22 nm for a tabletop full field optical microscope [28].
Interestingly, to date the application of CDI using coherent light sources in any region of the spectrum has been limited almost exclusively to transmission mode imaging, with the exception of two recent proof-of-principle experiments that used synchrotron and helium-neon (HeNe) laser sources [29, 30]. This is due to the fact that reflection mode geometries are intrinsically more demanding than transmission mode samples since it is far more difficult to isolate the sample, and because of low reflectivities at EUV and x-ray wavelengths. In recent work by Roy et al. [29], isolation of the sample was accomplished by placing a pinhole between the sample and the detector, reconstructing the exit surface wave at the pinhole and back propagating the field to the sample. This severely limits the numerical aperture (NA) of the system. Reference [30] isolated the object by placing a pinhole directly onto the sample. However, this process will damage or alter sensitive samples.
In this paper we demonstrate the most straightforward strategy for reflection mode CDI by simply illuminating the object with the image of a pinhole, an approach we term apertured illumination CDI (AICDI). We show for the first time that CDI can image and scan over aperiodic samples in a high numerical aperture (NA), off-axis reflection geometry, demonstrating a versatile reflection-mode microscope. To achieve this, we show that a coordinate transform can be used to rewrite the far-field scatter pattern from a tilted object as a uniformly spaced Fourier transform of the object. Using this approach, propagation between sample and detector planes can be accomplished using standard FFTs, making AICDI practical for rapid scanning and imaging of samples in reflection mode. Our non-contact technique images a pinhole onto the sample plane, resulting in no damage to the sample and no restriction of the NA of the imaging system, while our advances in the algorithms enable rapid image reconstruction and an ability to correct the diffraction pattern for sample tilt that is inherent to high-NA imaging. These advances are important because high-NA reflection mode CDI has many potential applications for example as a nanometrology tool for future generations of semiconductor patterning [31], for dynamic imaging of magnetic domains [14, 32] or catalytic surfaces, and for use when the sample is thicker than the absorption length of the illuminating light.
In the following, we first demonstrate AICDI using a HeNe laser in a simple, in-line transmission geometry. We then derive the coordinate transform that must be performed in the case of the off-axis geometry, in order to take advantage of the FFT to speed up the phase retrieval process and image reconstruction. Finally, we make use of this transform to demonstrate reflection-mode CDI using both a HeNe and a high harmonic extreme ultraviolet (EUV) source at a wavelength of 29 nm, with spatial resolutions of 1.4 μm and 100 nm, respectively.
2. Transmission mode AICDI
In order to implement and test the AICDI technique and data processing algorithms, we first developed a proof-of-concept system using a polarized 632.8 nm HeNe laser. A schematic diagram of the setup is shown in Fig. 1(a). First we spatially filter and collimate the beam to overfill a 300 μm wide circular aperture. The aperture is imaged to the sample plane using a one-to-one 4f imaging system. A positive lens placed directly after the sample sends the scattered light into the Fourier plane at the CMOS detector (Mightex Systems MCE-B013, 5.2 μm pixel size). In general the positive lens after the sample is unnecessary, however, the detectors used in this experiment were small enough that a demagnification of the far field was required in order to use a wavelength as large as 633nm. The aperture size is selected to satisfy the oversampling criterion [13], where the distance between the lens and the detector is 11.6 mm, corresponding to an NA of 0.22 for a 5.3 mm diameter detector.
Fig. 1 The apertured illumination CDI scheme isolates an extended transparent sample by imaging a aperture onto the sample plane. (a) A schematic of the setup, including a traditional bright-field microscope image of the sample. (b) The scatter pattern recorded by the detector in the Fourier plane scaled by the fourth root. (c) The exit surface wave reconstructed from the scatter pattern. The circle outlines the area illuminated by the aperture. (d) A reconstruction when the illumination is subtracted out. (e) Overlay of the images from many scan positions. The circles represent the outline of the aperture illumination at different scan positions.
In this first experiment, we illuminated a bundle of suspended, 26 μm diameter copper wires with an image of the aperture (Fig. 1(a) inset). An example of a scatter pattern obtained is shown in Fig. 1(b). The scatter pattern we obtain is proportional to the modulus of the Fourier transform of the illuminated portion of the sample. We use the RAAR algorithm as outlined in Ref. [12] (including the correct form of non-negativity, a fast ramp in the feedback parameter and a modified amplitude constraint) with shrinkwrap [11] to recover the phase. With the recovered phase, we are able to reconstruct the exit surface wave (Fig. 1(c)). This technique constitutes a bright-field imaging microscope. If instead we first record the scattered light with the sample removed (leaving the rest of the system unchanged) and reconstruct the electric field at the image plane of the aperture, then during the reconstruction process we can subtract the complex amplitude of the illumination at the detector plane per iteration. Subtracting out the illumination results in the wires appearing bright instead of dark (Fig. 1(d)); this constitutes a dark-field microscope. By scanning the sample in a plane perpendicular to the optical axis, we are able to reconstruct different areas of the extended sample independently. With overlap between the scan positions, we can register adjacent reconstructions to build up a large-area, high-resolution image. Figure 1(e) shows a number of reconstructions of this sample, which were overlaid in post-processing. Colored circles indicate the area that was illuminated by the imaged aperture for each individual reconstruction.
Having shown the ability to isolate and image part of an extended sample in transmission, we next broaden the applicability of AICDI to image samples in a reflection geometry. In reflection mode, the sample must be at some non-zero angle with respect to the incident beam in order to achieve high NA imaging for most experimental setups. Scattering geometries for normal incidence and non-zero incidence angles are shown in Figs. 2(a) and 2(b), respectively. At non-zero incidence angles and at high NA, the far field scatter pattern is no longer proportional to a uniformly spaced Fourier transform of the specimen, as evidenced by the curvature of the pattern in Fig. 3(a). Below we present the numerical correction needed to reconstruct samples with high NA and/or at large incidence angles using standard iterative phase retrieval algorithms.
Fig. 2 Scattering geometry for (a) normally incident sample illumination and (b) obliquely incident illumination. The unit vectors n̂S and n̂D are normal to the sample and detector planes, respectively. The unprimed coordinate system refers to the detector coordinates (D), the primed coordinate system refers to the untilted sample coordinates (S), the double primed coordinate system refers to the tilted sample coordinates, and α is the angle between the tilted and untilted sample coordinate systems. q⃗ is the momentum transfer vector, defined as k⃗f − k⃗i, where k⃗i is the incident wavevector and k⃗f is the scattered wavevector. Note that the incident illumination vector, k⃗i, is anti-parallel to the detector-normal vector n̂D, in both (a) and (b). Also note that the origin of the unprimed coordinate system is located at the sample position.
Fig. 3 Mapping the diffraction from a tilted sample to a diffraction pattern linear in frequency space. (a) The raw data from light scattered by the sample at an angle of 30 degrees. (b) The data mapped onto a linear grid in frequency space. The dotted black lines are added to highlight the curvature seen in the tilted sample diffraction shown in (a). Both images have been scaled by the fourth root.
3. Tilted plane correction
We can write the far-field scatter pattern as the Fourier transform of a scattering potential following the first order Born approximation [33]
where μ is a parameter that specifies the strength of the interaction with the potential V(r′⃗) and f(q⃗) is the far-field scattering amplitude. We begin by examining the case where a ray is normally incident on the specimen and the sample and detector planes are parallel to each other. A schematic of this geometry is shown in Fig. 2(a) with the following relevant quantities: n̂S is the normal vector defining the sample plane, S, n̂D is the normal vector defining the detector plane, D, r′⃗ describes points on S, k⃗i is the incident wavevector, k⃗f is the final scattering vector, q⃗ is the momentum transfer vector (k⃗f − k⃗i), ϕ is the azimuthal angle in S, θ is the angle between k⃗i and k⃗f, and α is the angle between the tilted and untilted sample coordinate systems. We start by writing the momentum transfer vector in a coordinate system where k⃗i = k0ẑ: where k0 is the wavenumber. Our goal here is to identify the unit vectors associated with q⃗ (x̂, ŷ, ẑ in Eq. (2)) with those that are associated with r′⃗. In order to take advantage of the FFT algorithm, the real-space sampling grid must be linear in r′⃗ and the frequency-space sampling grid must be linear in q⃗. However, it is clear from Eq. (2) that the spatial frequencies qx and qy are linear in sin(θ) for a given ϕ. At this point, it is instructive to rewrite Eq. (2) in the Cartesian coordinates of the detector where x and y are the detector plane coordinates and R is the distance from the sample to the center of the detector. The values at each pixel of the components of q⃗ are clearly spaced non-uniformly on the detector due to the square root term in the denominator of each component. The square root in the denominator is a result of mapping the Ewald sphere onto a flat detector. This should be corrected by a two-dimensional interpolation onto a linearly spaced Cartesian grid. It is also worth noting that the light at the edges has propagated further than light at the center. Because the intensity falls off as 1/r2, we rescale the data by r2/R2 in order to correct for this, where r is the distance from the sample to a given pixel on the detector. Written in terms of detector coordinates we have: 1 + (x2 + y2)/R2. This rescaling should be done on the raw data, but after centering and before doing any transformation of coordinates.To summarize the above steps, we start with the measured diffraction pattern intensity, I(x,y,z), in terms of the detector coordinates. We then perform the intensity rescaling as
where Ir is now properly scaled as the Fourier transform of the object. The second step is to perform the interpolation in order that the pattern is sampled on a grid that is linear in q⃗, which can be written as where in the case of a planar sample, Ir = 0 for qz ≠ 0. This allows us to simply use a two-dimensional rather than three-dimensional interpolation. In the case of a three-dimensional sample, a three-dimensional interpolation must be performed [4].We now turn to the case (shown in Fig. 2(b)) where the specimen plane has been rotated by some angle α about the ŷ′ axis, as would be the case in a reflection geometry (or a tilted sample in transmission). In this case Eq. (2) is still valid in the unprimed coordinate system, however as expected, the coordinate axes associated with q⃗ are no longer aligned with the coordinate axes associated with r̂′. To fix this, we simply rewrite q⃗ in the sample coordinate system. In practice, this corresponds to rotating q⃗ by −α, given by
where Ry is the three-dimensional (gimbal-like) rotation matrix about ŷ′ and q⃗(x″, y′, z″) is the momentum transfer in the sample coordinate system, in terms of the components of q⃗ in the unprimed coordinates. A more general case can be obtained by applying a further rotation Rx(−β), allowing for the sample plane to be in any orientation relative to the detector plane. As in the untilted case, a two-dimensional interpolation should be performed (Eq. (5) using q⃗ in the sample coordinates) to resample the nonlinearly spaced components of q⃗ onto a linearly spaced Cartesian grid. With both rotation matrices applied, the final form of q⃗ is:The procedure for the most general case, where q⃗i × n̂D ≠ 0, can be summarized by the following algorithm: First, write q⃗ in a plane normal to k⃗i then perform rotations on q⃗ in order to represent it in 1) the sample plane and then 2) the detector plane. This transform allows us to write the detected scatter pattern as the Fourier transform of the sample potential. Finally, in order to use the FFT algorithm, an interpolation must be performed to resample the resulting spatial frequencies onto a uniformly spaced grid, once again as in Eq. (5). It is worth noting that the interpolation must be performed with care so that the oversampling ratio does not fall below the minimum requirement. This can be achieved by interpolating onto a grid with a larger number of pixels than the original grid.
4. Reflection mode AICDI
To demonstrate our tilted sample correction as well as reflection mode imaging we modified the transmission mode setup so that the sample is at an angle α = 30 degrees (Fig. 4(a)). The same detector and Fourier transform lens were used as in the transmission setup, but were repositioned such that they were aligned along the specular reflection from the sample (Fig. 4(a)). Thus the NA was kept at 0.22 and the resolution at 1.4 μm. The sample used was a positive 1951 USAF Resolution Target. Figure 3(a) shows a scatter pattern form the vertical bars of group 5 (element 1) of the resolution target. The black dashed lines are overlaid to illustrate the curvature in the diffraction resulting from a tilted sample. In Fig. 3(b) we show the scatter pattern after mapping the diffraction onto a grid that is linear in spatial frequency, as discussed in section 3. After interpolation, the scatter pattern is proportional to the modulus of Fourier transform of the sample. Using the same iterative phase retrieval algorithm as mentioned above, we are able to reconstruct any arbitrary position of the target. These reconstructions are overlaid and shown in Fig. 4(b). An objective based bright-field microscopy image is also shown in Fig. 4(c) for comparison.
Fig. 4 Visible laser apertured illumination and tilted sample correction by reconstructing a 1951 USAF resolution target in a reflection mode geometry. (a) A schematic of the setup. Note that a negative USAF pattern is shown, but a positive USAF was used in the experiment. (b) Several reconstructions with different scan positions are overlaid to show the AICDI reconstruction. (c) A traditional bright-field microscopy image of the sample.
5. Reflection mode CDI using short wavelength high harmonic beams
We also demonstrated the utility of tilted plane correction for reflection mode CDI in the EUV by using a fully spatially coherent high harmonic beam with a center wavelength of 29 nm. The sample was a two-dimensional array of identical square, nickel nano-pillars, each ∼ 2μm in width and 20nm high, patterned on a sapphire substrate. Rather than using AICDI, a slightly simpler geometry was used where the beam was loosely focused directly onto the object, with a spot size of approximately 25 μm, so that many pillars were illuminated (Fig. 5(a)). The incident angle of the HHG beam on the sample was 45 deg, and as a result the scatter pattern in Fig. 5(b) displays a high degree of asymmetry, making this specimen a good demonstration of the need for tilted plane correction. Figures 5(b) and 5(c) show uncorrected and corrected scatter patterns respectively. The 27.6 mm square detector with 13.5 μm pixels (Andor iKon) was placed 4.5 cm past the object, resulting in a NA of 0.29. An integration time of 20 minutes was required in order to obtain the diffraction pattern in Fig. 5(b). The missing center in the diffraction pattern is the result of a beam stop used to prevent saturation of the bright zero order peak. The beam stop was placed as close to the detector as possible (≈ 2mm) in order to minimize edge diffraction effects.
Fig. 5 EUV microscope image in reflection mode of a 2D array of nickel nano-pillars.(a) A schematic of the setup. Uncorrected (b) and corrected (c) diffraction patterns. The data in (c) was resampled onto a coarser grid, shown in (d), containing only the peak intensity points. (e) Reconstructed image using the diffraction pattern in (d). The missing data shown in (b), (c) and (d) was left as a free parameter and is a result of a beam stop used to prevent saturation of the bright zero order peak on the camera.
This specimen can be thought of as a convolution between a Dirac comb function with a single nickel nano-pillar of 1/4 duty cycle. With this in mind, and using the convolution theorem, we can consider the Fourier transform of this sample to be the product of the individual Fourier transforms of the Dirac comb and a single (averaged) nano pillar. This means that in the diffraction plane, a Dirac comb samples a sinc function, which is the Fourier transform of a single pillar. Using this idea we can increase the signal-to-noise ratio, after applying the tilted plane correction, by extracting the peak values (spaced by the period of the Dirac comb) of the scatter pattern and placing them on a new, coarser grid, shown in Fig. 5(d). This new grid was used in the averaged pillar reconstruction shown in Fig. 5(e), producing an image with ∼ 100 nm theoretical resolution. The reconstructions were carried out in the same manner and with the same algorithm as in the case of the 632.8 nm illumination. The missing data shown in Figs. 5(b)–5(d) is a result of a beam stop used to block the zero order diffraction, allowing us to measure the high-angle scatter while preventing saturation of the detector at the center. Clearly this method of resampling the data onto a separate grid by extracting the peaks in the diffraction plane is only applicable for arrays of identical objects. However, provided a contrast mechanism exists, the same apertured illumination technique discussed above can be implemented more generally for full field imaging of nanostructures in the EUV.
6. Conclusion
We have demonstrated the ability to image aperiodic, non-isolated samples at high numerical aperture using coherent diffractive imaging. We overcome the isolated object constraint of standard CDI by isolating the illumination using a technique we call apertured illumination CDI. Because many applications of reflection mode CDI will require an off-axis geometry, we derived a coordinate transform that allows the use of FFTs in standard iterative phase retrieval algorithms, which increases the speed of the reconstruction on a per-iteration basis. In the future, AICDI should achieve spatial reolution in the sub-10 nm range using shorter wavelength HHG sources and higher NA imaging.
Acknowledgments
The authors acknowledge support from a NSSEFF award, and used facilities supported by the National Science Foundation Engineering Research Center in EUV Science and Technology. M. S. acknowledges support from the NSF IGERT program. D.G. acknowledges support from a Ford Foundation Fellowship.
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