High numerical aperture reflection mode coherent ...High numerical aperture reflection mode...

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High numerical aperture reflection mode coherent diffraction microscopy using off-axis apertured illumination Dennis F. Gardner, 1,Bosheng Zhang, 1 Matthew D. Seaberg, 1 Leigh S. Martin, 1 Daniel E. Adams, 1 Farhad Salmassi, 2 Eric Gullikson, 2 Henry Kapteyn, 1 and Margaret Murnane 1 1 JILA, University of Colorado, 440 UCB, Boulder, Colorado 80309-0440, USA 2 Center for X-ray Optics, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA [email protected] Abstract: 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 OCIS codes: (340.7460) X-ray microscopy; (100.5070) Phase retrieval; (340.7480) X-rays, soft x-rays, extreme ultraviolet (EUV); (190.2620) Harmonic generation and mixing. References and links 1. J. Miao, P. Charalambous, and J. Kirz, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999). 2. H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4, 833–839 (2010). 3. P. Thibault and E. Veit, “X-Ray Diffraction Microscopy,” Annu. Rev. Condens. Matter Phys. 1, 237–255 (2010). 4. K. S. Raines, S. Salha, R. L. Sandberg, H. Jiang, J. A. Rodr´ ıguez, B. P. Fahimian, H. C. Kapteyn, J. Du, and J. Miao, “Three-dimensional structure determination from a single view,” Nature 463, 214–217 (2010). 5. A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35, 2585–2587 (2010). 6. B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008). 7. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978). 8. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). 9. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003). 10. V. Elser, “Random projections and the optimization of an algorithm for phase retrieval,” J. Phys. A: Math. Gen. 36, 2995–3007 (2003). 11. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003). 12. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005). #171081 - $15.00 USD Received 21 Jun 2012; revised 22 Jul 2012; accepted 23 Jul 2012; published 3 Aug 2012 (C) 2012 OSA 13 August 2012 / Vol. 20, No. 17 / OPTICS EXPRESS 19050

Transcript of High numerical aperture reflection mode coherent ...High numerical aperture reflection mode...

Page 1: High numerical aperture reflection mode coherent ...High numerical aperture reflection mode coherent diffraction microscopy using off-axis apertured illumination Dennis F. Gardner,1,∗

High numerical aperture reflection modecoherent diffraction microscopy using

off-axis apertured illumination

Dennis F. Gardner,1,∗ BoshengZhang,1 Matthew D. Seaberg,1 Leigh S.Martin, 1 Daniel E. Adams,1 Farhad Salmassi,2 Eric Gullikson, 2 Henry

Kapteyn,1 and Margaret Murnane1

1JILA, University of Colorado, 440 UCB, Boulder, Colorado 80309-0440, USA2Center for X-ray Optics, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

[email protected]

Abstract: We extend coherent diffraction imaging (CDI) to a highnumerical aperture reflection mode geometry for the first time. We derive acoordinate transform that allows us to rewrite the recorded far-field scatterpattern from a tilted object as a uniformly spaced Fourier transform. Usingthis approach, FFTs in standard iterative phase retrieval algorithms can beused to significantly speed up the image reconstruction times. Moreover,we avoid the isolated sample requirement by imaging a pinhole onto thespecimen, in a technique termed apertured illumination CDI. By combiningthe new coordinate transformation with apertured illumination CDI, wedemonstrate rapid high numerical aperture imaging of samples illuminatedby visible laser light. Finally, we demonstrate future promise for thistechnique by using high harmonic beams for high numerical aperturereflection mode imaging.

© 2012 Optical Society of America

OCIS codes:(340.7460) X-ray microscopy; (100.5070) Phase retrieval; (340.7480) X-rays,soft x-rays, extreme ultraviolet (EUV); (190.2620) Harmonic generation and mixing.

References and links1. J. Miao, P. Charalambous, and J. Kirz, “Extending the methodology of X-ray crystallography to allow imaging

of micrometre-sized non-crystalline specimens,” Nature400, 342–344 (1999).2. H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics4, 833–839 (2010).3. P. Thibault and E. Veit, “X-Ray Diffraction Microscopy,” Annu. Rev. Condens. Matter Phys.1, 237–255 (2010).4. K. S. Raines, S. Salha, R. L. Sandberg, H. Jiang, J. A. Rodrıguez, B. P. Fahimian, H. C. Kapteyn, J. Du, and

J. Miao, “Three-dimensional structure determination from a single view,” Nature463, 214–217 (2010).5. A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with

useful resolution,” Opt. Lett.35, 2585–2587 (2010).6. B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty,

“Keyhole coherent diffractive imaging,” Nat. Phys.4, 394–398 (2008).7. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett.3, 27–29 (1978).8. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21, 2758–2769 (1982).9. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A20, 40–55 (2003).

10. V. Elser, “Random projections and the optimization of an algorithm for phase retrieval,” J. Phys. A: Math. Gen.36, 2995–3007 (2003).

11. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H.Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B68, 140101 (2003).

12. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl.21, 37–50 (2005).

#171081 - $15.00 USD Received 21 Jun 2012; revised 22 Jul 2012; accepted 23 Jul 2012; published 3 Aug 2012(C) 2012 OSA 13 August 2012 / Vol. 20, No. 17 / OPTICS EXPRESS 19050

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13. J. Miao and D. Sayre, “On possible extensions of X-ray crystallography through diffraction-pattern oversam-pling,” Acta. Crystallogr. A56, 596–605 (2000).

14. P. Fischer, “Studying nanoscale magnetism and its dynamics with soft X-ray microscopy,” IEEE Trans. Magn.44, 1900–1904 (2008).

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16. I. Robinson and R. Harder, “Coherent X-ray diffraction imaging of strain at the nanoscale,” Nat. Matter.8, 291–298 (2009).

17. B. Abbey, G. J. Williams, M. A. Pfeifer, J. N. Clark, C. T. Putkunz, A. Torrance, I. McNulty, T. M. Levin, A. G.Peele, and K. A. Nugent, “Quantitative coherent diffractive imaging of an integrated circuit at a spatial resolutionof 20 nm,” Appl. Phys. Lett.93, 214101 (2008).

18. J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending X-ray crystallography to allow the imaging of non-crystalline materials, cells, and single protein complexes,” Annu. Rev. Phys. Chem.59, 387–410 (2008).

19. J. Nelson, X. Huang, J. Steinbrener, D. Shapiro, J. Kirz, S. Marchesini, A. M. Neiman, J. J. Turner, and C. Ja-cobsen, “High-resolution x-ray diffraction microscopy of specifically labeled yeast cells,” Proc. Natl. Acad. Sci.U.S.A.107, 7235–7239 (2010).

20. T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics ofbright coherent X-ray generation,” Nat. Photonics4, 822–832 (2010).

21. M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. Murnane, andH. Kapteyn, “Bright, coherent, ultrafast soft x-ray harmonics spanning the water window from a tabletop lightsource,” Phys. Rev. Lett.105, 173901 (2010).

22. T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciu-nas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja,A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in thekeV x-ray regime from mid-infrared femtosecond lasers,” Science336, 1287–1291 (2012).

23. C. Durfee, A. Rundquist, S. Backus, C. Herne, M. Murnane, and H. Kapteyn, “Phase matching of high-orderharmonics in hollow waveguides,” Phys. Rev. Lett.83, 2187–2190 (1999).

24. X. Zhang, A. R. Libertun, A. Paul, E. Gagnon, S. Backus, I. P. Christov, M. M. Murnane, H. C. Kapteyn, R. A.Bartels, Y. Liu, and D. T. Attwood, “Highly coherent light at 13 nm generated by use of quasi-phase-matchedhigh-harmonic generation,” Opt. Lett.29, 1357–1399 (2004).

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27. D. Alessi, Y. Wang, B. Luther, L. Yin, D. Martz, M. Woolston, Y. Liu, M. Berrill, and J. Rocca, “EfficientExcitation of Gain-Saturated Sub-9-nm-Wavelength Tabletop Soft-X-Ray Lasers and Lasing Down to 7.36 nm,”Phys. Rev. X1, 021023 (2011).

28. M. D. Seaberg, D. E. Adams, E. L. Townsend, D. A. Raymondson, W. F. Schlotter, Y. Liu, C. S. Menoni, L. Rong,C.-C. Chen, J. Miao, H. C. Kapteyn, and M. M. Murnane, “Ultrahigh 22 nm resolution coherent diffractiveimaging using a desktop 13 nm high harmonic source,” Opt. Express19, 22470–22479 (2011).

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1. Introduction

The last decade has seen dramatic advances in the development of coherent diffraction imag-ing (CDI) techniques [1–6]. In CDI, a lens is essentially replaced by an iterative phase re-trieval algorithm that, in theory, allows diffraction-limited imaging at the illuminating wave-length [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 pat-tern is measured by a detector. Because the phase of the scatter pattern is lost, an iterative phaseretrieval algorithm must be used to reconstruct the exit surface wave directly after the sam-

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ple. As a result, CDI can be categorized as a type of in-line, phase-contrast microscopy. Themostprevalent iterative phase retrieval algorithms combine two constraints for image recon-struction: 1) the measured diffraction data, and 2) an oversampling condition on the detectorthat corresponds to an isolated object in the sample plane [13]. CDI has already been used toproduce phase contrast images of magnetic materials [14,15], nano-scale strain [16], integratedcircuits [17] and biological samples [18,19].

Nearly coincident with the development of CDI, high harmonics of femtosecond lasers haveadvanced from interesting strong field science phenomena to a unique and robust light sourceused for a wide range of experiments in molecular, materials and energy sciences [20–22]. Toproduce bright, fully spatially and temporally coherent, laser-like high harmonic beams, thisextreme nonlinear frequency up conversion process must be phase matched to ensure coherentbuildup of the emission from many atoms. [20–24]. Phase matching of the high harmonic gen-eration (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 syn-chrotrons 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 electronlaser facilities: they can achieve sub-fs time resolution, are perfectly synchronized to the driv-ing laser, can operate at multi-kHz repetition rates, and have a broad range of harmonics thatcan probe function at many different atomic sites simultaneously. Although the average x-rayflux and pulse energy is lower than what can be generated at the new, large, x-ray free electronlaser facilities, nevertheless HHG sources have been successfully used for coherent diffractiveimaging of isolated test samples, with record spatial resolution of 22 nm for a tabletop full fieldoptical microscope [28].

Interestingly, to date the application of CDI using coherent light sources in any region of thespectrum has been limited almost exclusively to transmission mode imaging, with the exceptionof 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 intrinsicallymore demanding than transmission mode samples since it is far more difficult to isolate thesample, and because of low reflectivities at EUV and x-ray wavelengths. In recent work by Royet al. [29], isolation of the sample was accomplished by placing a pinhole between the sampleand the detector, reconstructing the exit surface wave at the pinhole and back propagating thefield 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 processwill damage or alter sensitive samples.

In this paper we demonstrate the most straightforward strategy for reflection mode CDI bysimply illuminating the object with the image of a pinhole, an approach we term apertured illu-mination CDI (AICDI). We show for the first time that CDI can image and scan over aperiodicsamples in a high numerical aperture (NA), off-axis reflection geometry, demonstrating a ver-satile reflection-mode microscope. To achieve this, we show that a coordinate transform can beused to rewrite the far-field scatter pattern from a tilted object as a uniformly spaced Fouriertransform of the object. Using this approach, propagation between sample and detector planescan be accomplished using standard FFTs, making AICDI practical for rapid scanning andimaging of samples in reflection mode. Our non-contact technique images a pinhole onto thesample plane, resulting in no damage to the sample and no restriction of the NA of the imagingsystem, while our advances in the algorithms enable rapid image reconstruction and an abil-ity to correct the diffraction pattern for sample tilt that is inherent to high-NA imaging. Theseadvances are important because high-NA reflection mode CDI has many potential applicationsfor 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

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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 trans-

mission geometry. We then derive the coordinate transform that must be performed in thecase of the off-axis geometry, in order to take advantage of the FFT to speed up the phaseretrieval process and image reconstruction. Finally, we make use of this transform to demon-strate 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 1.4µm and 100 nm, respectively.

Fig. 1. The apertured illumination CDI scheme isolates an extended transparent sample byimaginga aperture onto the sample plane. (a) A schematic of the setup, including a tra-ditional bright-field microscope image of the sample. (b) The scatter pattern recorded bythe detector in the Fourier plane scaled by the fourth root. (c) The exit surface wave recon-structed 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 imagesfrom many scan positions. The circles represent the outline of the aperture illumination atdifferent scan positions.

2. Transmission mode AICDI

In order to implement and test the AICDI technique and data processing algorithms, we firstdeveloped a proof-of-concept system using a polarized 632.8 nm HeNe laser. A schematicdiagram of the setup is shown in Fig. 1(a). First we spatially filter and collimate the beam tooverfill a 300µm wide circular aperture. The aperture is imaged to the sample plane usinga one-to-one 4f imaging system. A positive lens placed directly after the sample sends thescattered 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, thedetectors used in this experiment were small enough that a demagnification of the far field wasrequired in order to use a wavelength as large as 633nm. The aperture size is selected to satisfythe oversampling criterion [13], where the distance between the lens and the detector is 11.6mm, corresponding to an NA of 0.22 for a 5.3 mm diameter detector.

In this first experiment, we illuminated a bundle of suspended, 26µm diameter copper wireswith an image of the aperture (Fig. 1(a) inset). An example of a scatter pattern obtained isshown in Fig. 1(b). The scatter pattern we obtain is proportional to the modulus of the Fouriertransform of the illuminated portion of the sample. We use the RAAR algorithm as outlinedin Ref. [12] (including the correct form of non-negativity, a fast ramp in the feedback param-eter and a modified amplitude constraint) with shrinkwrap [11] to recover the phase. With the

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recovered phase, we are able to reconstruct the exit surface wave (Fig. 1(c)). This techniqueconstitutesa bright-field imaging microscope. If instead we first record the scattered light withthe sample removed (leaving the rest of the system unchanged) and reconstruct the electric fieldat the image plane of the aperture, then during the reconstruction process we can subtract thecomplex amplitude of the illumination at the detector plane per iteration. Subtracting out theillumination results in the wires appearing bright instead of dark (Fig. 1(d)); this constitutesa 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 overlapbetween 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, whichwere overlaid in post-processing. Colored circles indicate the area that was illuminated by theimaged 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 re-flection mode, the sample must be at some non-zero angle with respect to the incident beamin order to achieve high NA imaging for most experimental setups. Scattering geometries fornormal 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 propor-tional to a uniformly spaced Fourier transform of the specimen, as evidenced by the curvatureof the pattern in Fig. 3(a). Below we present the numerical correction needed to reconstructsamples with high NA and/or at large incidence angles using standard iterative phase retrievalalgorithms.

3. Tilted plane correction

We can write the far-field scatter pattern as the Fourier transform of a scattering potential fol-lowing the first order Born approximation [33]

f (~q) =µ4π

V (~r′)exp−2πi~q·~r′ d~r′ (1)

whereµ is a parameter that specifies the strength of the interaction with the potentialV (~r′)and f (~q) is the far-field scattering amplitude. We begin by examining the case where a ray isnormally 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: ˆnS isthe normal vector defining the sample plane, S, ˆnD is the normal vector defining the detectorplane, D,~r′ describes points on S,~ki is the incident wavevector,~k f is the final scattering vector,~qis the momentum transfer vector(~k f −~ki), φ is the azimuthal angle in S,θ is the angle between~ki and~k f , andα is the angle between the tilted and untilted sample coordinate systems. We startby writing the momentum transfer vector in a coordinate system where~ki = k0z:

~q = ~k f −~ki = k0[sinθ cosφ x+sinθ sinφ y+(cosθ −1)z] (2)

wherek0 is the wavenumber. Our goal here is to identify the unit vectors associated with~q(x, y, z in Eq. (2)) with those that are associated with~r′. In order to take advantage of the FFTalgorithm, the real-space sampling grid must be linear in~r′ and the frequency-space samplinggrid must be linear in~q. However, it is clear from Eq. (2) that the spatial frequenciesqx andqy

are linear in sin(θ) for a givenφ . At this point, it is instructive to rewrite Eq. (2) in the Cartesiancoordinates of the detector

~q = k0

[

x√

x2 + y2 +R2x+

y√

x2 + y2 +R2y+

(

R√

x2 + y2 +R2−1

)

z

]

(3)

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Fig. 2. Scattering geometry for (a) normally incident sample illumination and (b) obliquelyincident illumination. The unit vectors ˆnS and nD are normal to the sample and detectorplanes, respectively. The unprimed coordinate system refers to the detector coordinates(D), the primed coordinate system refers to the untilted sample coordinates (S), the doubleprimed coordinate system refers to the tilted sample coordinates, andα is the angle betweenthe tilted and untilted sample coordinate systems.~q is the momentum transfer vector, de-fined as~k f −~ki, where~ki is the incident wavevector and~k f is the scattered wavevector. Note

that the incident illumination vector,~ki, is anti-parallel to the detector-normal vector ˆnD, inboth (a) and (b). Also note that the origin of the unprimed coordinate system is located atthe sample position.

where x and y are the detector plane coordinates andR is the distance from the sample to thecenter of the detector. The values at each pixel of the components of~q 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 Cartesiangrid. It is also worth noting that the light at the edges has propagated further than light at thecenter. Because the intensity falls off as 1/r2, we rescale the data byr2/R2 in order to correctfor this, wherer is the distance from the sample to a given pixel on the detector. Written interms of detector coordinates we have: 1+(x2 + y2)/R2. This rescaling should be done on theraw 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

Ir(x,y,z) =

(

1+x2 + y2

R2

)

I(x,y,z), (4)

whereIr is now properly scaled as the Fourier transform of the object. The second step is toperform the interpolation in order that the pattern is sampled on a grid that is linear in~q, whichcan be written as

Ir(x,y,z) 7→ Ir(qx,qy,qz), (5)

where in the case of a planar sample,Ir = 0 for qz 6= 0. This allows us to simply use a two-dimensional rather than three-dimensional interpolation. In the case of a three-dimensionalsample, 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 bysome angleα about they′ axis, as would be the case in a reflection geometry (or a tilted samplein transmission). In this case Eq. (2) is still valid in the unprimed coordinate system, however asexpected, the coordinate axes associated with~q are no longer aligned with the coordinate axesassociated withr′. To fix this, we simply rewrite~q in the sample coordinate system. In practice,

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Fig. 3. Mapping the diffraction from a tilted sample to a diffraction pattern linear in fre-quency 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 addedto highlight the curvature seen in the tilted sample diffraction shown in (a). Both imageshave been scaled by the fourth root.

this corresponds to rotating~q by−α, given by

~q(x′′,y′,z′′) = Ry(−α)~q(x,y,z) (6)

whereRy is the three-dimensional (gimbal-like) rotation matrix abouty′ and~q(x′′,y′,z′′) is themomentum transfer in the sample coordinate system, in terms of the components of~q in theunprimed coordinates. A more general case can be obtained by applying a further rotationRx(−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~qin the sample coordinates) to resample the nonlinearly spaced components of~q onto a linearlyspaced Cartesian grid. With both rotation matrices applied, the final form of~q is:

~q(x′′,y′,z′′) = Rx(−β )Ry(−α)~q(x,y,z) =

qx cosα −qz sinαqx sinα sinβ +qy cosβ +qz cosα sinβqx sinα sinβ −qy sinβ +qz cosα cosβ

(7)

The procedure for the most general case, where~qi × nD 6= 0, can be summarized by thefollowing algorithm: First, write~q in a plane normal to~ki then perform rotations on~q in orderto represent it in 1) the sample plane and then 2) the detector plane. This transform allows usto write the detected scatter pattern as the Fourier transform of the sample potential. Finally, inorder to use the FFT algorithm, an interpolation must be performed to resample the resultingspatial frequencies onto a uniformly spaced grid, once again as in Eq. (5). It is worth notingthat the interpolation must be performed with care so that the oversampling ratio does not fallbelow the minimum requirement. This can be achieved by interpolating onto a grid with a largernumber of pixels than the original grid.

4. Reflection mode AICDI

To demonstrate our tilted sample correction as well as reflection mode imaging we modifiedthe transmission mode setup so that the sample is at an angleα = 30 degrees (Fig. 4(a)). Thesame detector and Fourier transform lens were used as in the transmission setup, but were repo-sitioned 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 1.4µm. The sample used was a positive1951 USAF Resolution Target. Figure 3(a) shows a scatter pattern form the vertical bars ofgroup 5 (element 1) of the resolution target. The black dashed lines are overlaid to illustratethe curvature in the diffraction resulting from a tilted sample. In Fig. 3(b) we show the scatter

#171081 - $15.00 USD Received 21 Jun 2012; revised 22 Jul 2012; accepted 23 Jul 2012; published 3 Aug 2012(C) 2012 OSA 13 August 2012 / Vol. 20, No. 17 / OPTICS EXPRESS 19056

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pattern after mapping the diffraction onto a grid that is linear in spatial frequency, as discussedin section 3. After interpolation, the scatter pattern is proportional to the modulus of Fouriertransform 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 over-laid and shown in Fig. 4(b). An objective based bright-field microscopy image is also shown inFig. 4(c) for comparison.

Fig. 4. Visible laser apertured illumination and tilted sample correction by reconstructing a1951USAF 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 exper-iment. (b) Several reconstructions with different scan positions are overlaid to show theAICDI 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 EUVby using a fully spatially coherent high harmonic beam with a center wavelength of 29 nm. Thesample was a two-dimensional array of identical square, nickel nano-pillars, each∼ 2µm inwidth and 20nm high, patterned on a sapphire substrate. Rather than using AICDI, a slightlysimpler geometry was used where the beam was loosely focused directly onto the object, witha spot size of approximately 25µm, so that many pillars were illuminated (Fig. 5(a)). Theincident angle of the HHG beam on the sample was 45 deg, and as a result the scatter patternin Fig. 5(b) displays a high degree of asymmetry, making this specimen a good demonstrationof the need for tilted plane correction. Figures 5(b) and 5(c) show uncorrected and correctedscatter patterns respectively. The 27.6 mm square detector with 13.5µm pixels (Andor iKon)was placed 4.5 cm past the object, resulting in a NA of 0.29. An integration time of 20 minuteswas required in order to obtain the diffraction pattern in Fig. 5(b). The missing center in thediffraction pattern is the result of a beam stop used to prevent saturation of the bright zeroorder peak. The beam stop was placed as close to the detector as possible (≈2mm) in order tominimize edge diffraction effects.

This specimen can be thought of as a convolution between a Dirac comb function with asingle nickel nano-pillar of 1/4 duty cycle. With this in mind, and using the convolution theo-

#171081 - $15.00 USD Received 21 Jun 2012; revised 22 Jul 2012; accepted 23 Jul 2012; published 3 Aug 2012(C) 2012 OSA 13 August 2012 / Vol. 20, No. 17 / OPTICS EXPRESS 19057

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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 datain (c) was resampled onto a coarser grid, shown in (d), containing only the peak intensitypoints. (e) Reconstructed image using the diffraction pattern in (d). The missing data shownin (b), (c) and (d) was left as a free parameter and is a result of a beam stop used to preventsaturation of the bright zero order peak on the camera.

rem, we can consider the Fourier transform of this sample to be the product of the individualFourier transforms of the Dirac comb and a single (averaged) nano pillar. This means that inthe diffraction plane, a Dirac comb samples a sinc function, which is the Fourier transform of asingle pillar. Using this idea we can increase the signal-to-noise ratio, after applying the tiltedplane correction, by extracting the peak values (spaced by the period of the Dirac comb) of thescatter pattern and placing them on a new, coarser grid, shown in Fig. 5(d). This new grid wasused in the averaged pillar reconstruction shown in Fig. 5(e), producing an image with∼ 100nm theoretical resolution. The reconstructions were carried out in the same manner and withthe same algorithm as in the case of the 632.8 nm illumination. The missing data shown inFigs. 5(b)-5(d) is a result of a beam stop used to block the zero order diffraction, allowing us tomeasure the high-angle scatter while preventing saturation of the detector at the center. Clearlythis method of resampling the data onto a separate grid by extracting the peaks in the diffractionplane is only applicable for arrays of identical objects. However, provided a contrast mecha-nism exists, the same apertured illumination technique discussed above can be implementedmore 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 numericalaperture using coherent diffractive imaging. We overcome the isolated object constraint of stan-dard 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 de-rived a coordinate transform that allows the use of FFTs in standard iterative phase retrievalalgorithms, which increases the speed of the reconstruction on a per-iteration basis. In the fu-ture, AICDI should achieve spatial reolution in the sub-10 nm range using shorter wavelengthHHG sources and higher NA imaging.

#171081 - $15.00 USD Received 21 Jun 2012; revised 22 Jul 2012; accepted 23 Jul 2012; published 3 Aug 2012(C) 2012 OSA 13 August 2012 / Vol. 20, No. 17 / OPTICS EXPRESS 19058

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Acknowledgments

Theauthors acknowledge support from a NSSEFF award, and used facilities supported by theNational Science Foundation Engineering Research Center in EUV Science and Technology.M. S. acknowledges support from the NSF IGERT program. D.G. acknowledges support froma Ford Foundation Fellowship.

#171081 - $15.00 USD Received 21 Jun 2012; revised 22 Jul 2012; accepted 23 Jul 2012; published 3 Aug 2012(C) 2012 OSA 13 August 2012 / Vol. 20, No. 17 / OPTICS EXPRESS 19059