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Holography with extended reference by autocorrelation linear differential operation Manuel Guizar-Sicairos and James R. Fienup The Institute of Optics, University of Rochester, Rochester, N.Y. 14627, USA [email protected], [email protected]

Abstract: We introduce a generalization of Fourier transform holography that allows the use of the boundary waves of an extended object to act as a holographic-like reference. By applying a linear differential operator on the field autocorrelation, we use a sharp feature on the extended reference to reconstruct a complex-valued image of the object of interest in a single-step computation. We generalize the approach of Podorov et al. [Opt. Express 15, 9954 (2007)] to a much wider class of extended reference objects. Effects of apertures in Fourier domain and imperfections in the reference object are analyzed. Realistic numerical simulations show the feasibility of our approach and its robustness against noise. © 2007 Optical Society of America OCIS codes: (090.1995) Digital holography; (070.0070) Fourier optics and signal processing; (110.3010) Image reconstruction techniques; (110.7440) X-ray imaging; (100.5070) Phase retrieval

References and links 1. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962). 2. J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (Roberts & Company, Englewood, 2005). 3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). 4. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987). 5. P. S. Idell, J. R. Fienup and R. S. Goodman, “Image synthesis from nonimaged laser-speckle patterns,” Opt. Lett. 12, 858–860 (1987). 6. J. N. Cederquist, J. R. Fienup, J. C. Marron and R. G. Paxman, “Phase retrieval from experimental far-field speckle data,” Opt. Lett. 13, 619–621 (1988). 7. J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14, 498–508 (2006). 8. J. Miao, P. Charalambous, J. Kirz and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999). 9. 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). 10. S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004). 11. D. Shapiro et al., “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102, 15343–15346 (2005). 12. H. N. Chapman et al., “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys. 2, 839–843 (2006). 13. L.M. Stadler, R. Harder, I. K. Robinson, C. Rentenberger, H. P. Karnthaler, B. Sepiol and G. Vogl, “Coherent x-ray diffraction imaging of grown-in antiphase boundaries in Fe65 Al35 ,” Phys. Rev. B 76, 014204 (2007). 14. H. N. Chapman et al., “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).

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Received 5 Oct 2007; revised 29 Nov 2007; accepted 1 Dec 2007; published 11 Dec 2007

24 December 2007 / Vol. 15, No. 26 / OPTICS EXPRESS 17592

15. J. R. Fienup, T. R. Crimmins and W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,” J. Opt. Soc. Am. 72, 610–624 (1982). 16. T. R. Crimmins, J. R. Fienup and B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–13 (1990). 17. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986). 18. H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini and H. N. Chapman, “Use of extended and prepared reference objects in experimental Fourier transform x-ray holography,” Appl. Phys. Lett. 85, 2454–2456 (2004). 19. O. Hellwig, S. Eisebitt, W. Eberhardt, W. F. Schlotter, J. Lüning and J. Stöhr, “Magnetic imaging with soft x-ray spectroholography,” J. Appl. Phys. 99, 08H307 (2006). 20. I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells and D. P. Kern, “High-Resolution Imaging by Fourier Transform X-ray Holography,” Science 256, 1009–1012 (1992). 21. W. F. Schlotter et al., “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. 89, 163112 (2006). 22. S. G. Podorov, K. M. Pavlov and D. M. Paganin, “A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging,” Opt. Express 15, 9954–9962 (2007). 23. G. W. Stroke, R. Restrick, A. Funkhouser and D. Brumm, “Resolution-retrieving compensation of source effects by correlative reconstruction in high-resolution holography,” Phys. Lett. 18, 274–275 (1965). 24. M. R. Howells, C. J. Jacobsen, S. Marchesini, S. Miller, J. C. H. Spence and U. Weirstall, “Toward a practical X-ray Fourier holography at high resolution,” Nucl. Instrum. Methods Phys. Res. A 467, 864–867 (2001). 25. R. N. Bracewell, K.-Y. Chang, A. K. Jha and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304 (1993). 26. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd Ed. (McGraw-Hill, New York, 1978). 27. J. D. Gaskill and J. W. Goodman, “Use of multiple reference sources to increase the effective field of view in lensless Fourier-transform holography,” Proc. IEEE 57, 823–825 (1969).

1. Introduction Techniques that are able to form images without the use of lenses or mirrors have been developed and have found a wide range of applications in the optical regime. These techniques exploit the coherence of the illuminating beam to obtain a complex-valued diffraction-limited image from a measurement of the intensity of a diffraction pattern. Among the lensless imaging techniques are Fourier transform (FT) holography [1, 2] and image reconstruction by phase retrieval [3–9]. The limited capability for x-ray imaging with lenses and mirrors has produced an increasing interest in further developing lensless imaging modalities. The ability to form images by illuminating a non-crystalline sample with a coherent x-ray source offers unique imaging capabilities of thick samples due to the short wavelength. Furthermore, the advent of short-time coherent x-ray pulses by free-electron lasers promises high-penetration imaging of fast phenomena with very fine resolution. Imaging at these wavelengths has a very diverse range of applications both for the fields of biology and solid-state physics [8–13]. Image reconstruction by phase retrieval, originally developed and demonstrated experimentally for optical wavelengths [6], has recently shown a substantial success in the x-ray regime [9–14]. In this approach the intensity of the far-field diffraction pattern of the object is collected with a detector array and iterative transform algorithms [3] are used to computationally retrieve the far-field phase by using the measured intensity pattern and some knowledge of the object support. The support constraint may be known a priori or estimated from the object’s autocorrelation function [15, 16] which can be computed from the Fourier intensity. In this context, phase retrieval algorithms substitute for optics in the image formation process. The resolution of the reconstructed images depends on the maximum scattered angle that is captured with a moderately good signal-to-noise ratio (SNR) and the wavelength of the illuminating beam. Since several reconstructions with different random starting guesses are often required to ensure that the algorithm converges to the correct solution, phase retrieval is a technique that is time consuming and may suffer from stagnation problems, specially for cases where the object #88310 - $15.00 USD

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is complex-valued or has a support that is symmetric about the origin [4, 17]. Fourier transform holography is a non-iterative lensless imaging technique in which a complex-valued image may be reconstructed in a single-step deterministic computation [1, 2]. This is achieved by placing a coherent point source at an appropriate distance from the object and having the object field interfere with the reference wave produced by this point source at the detector plane. For x-ray applications, the resolution for holography is typically limited by the size and quality of the point-like source that needs to be placed in the vicinity of the object [10, 18, 19]. This approach also requires a higher transverse coherence on the illuminating beam than phase retrieval, since the object field and the point-like source have to satisfy the holographic separation condition and still interfere [2]. Typically the final resolution of the image will be worse than what can be achieved by phase retrieval because manufacturing an unresolved point source at these wavelengths and getting sufficiently bright radiation through it is difficult to achieve. Nevertheless the fast, unambiguous and direct reconstruction of FT holography is attractive for many applications [10, 20, 21]. Recently Podorov et al. described a novel technique for holographic-like non-iterative reconstruction in which an extended rectangular reference is used [22]. Upon application of a second-order differential operator on the field autocorrelation, which can be computed from the far-field intensity, their approach can unambiguously reconstruct a complex-valued image using the boundary waves of a rectangular aperture as a reference. This is analogous to FT holography in that it is capable of reconstructing a complex-valued object field in a non-iterative fashion. As the authors mentioned, in this approach the resolution is limited by the quality of the corners of the rectangular aperture. This approach is fundamentally different from previously reported methods to enhance the reconstruction resolution by means of a deconvolution, when an extended reference is used [18, 23, 24]. The approach described in this paper, holography with extended reference by autocorrelation linear differential operation (HERALDO), is a more general approach to FT holographiclike reconstruction based on linear differential operators applied to the field autocorrelation. It constitutes a generalization of the approach of Podorov et al. (it includes conventional FT holography and Podorov’s approach as special cases). HERALDO permits direct retrieval of a complex-valued field using boundary waves of more general extended objects as holographiclike references. It is shown that the reference structure can be many other things besides the rectangle used in [22] and that a complex-valued image can be reconstructed provided that a suitable linear differential operator can be found. This approach provides the flexibility to choose a reference shape that can be optimally manufactured to assure high quality in the final image. For example, for some manufacturing processes it may be more practical to fabricate a sharp tip, thin wire or slit, or even a parallelogram than a point or a rectangle. Application of HERALDO (and Podorov’s approach) requires a priori knowledge of the orientation and structure of the extended reference. Figure 1(a) shows an experimental setup for lensless imaging. For image reconstruction by phase retrieval, no reference wave is required, as indicated in Fig. 1(b), so the resolution is only limited by the wavelength and the largest scattered angle collected. It is also required that the transverse coherence length of the illuminating beam is at least as wide as the object. Figure 1(c) shows the object plane, containing the object and a point source, prepared for FT holographic reconstruction. In this case the required transverse coherence length is twice as large as that for phase retrieval, since we need coherent interference between the reference wave (from the point source) and the object while maintaining the holographic separation condition for the reference wave (to avoid overlap of the reconstruction and the object autocorrelation). Additionally the sampling (pixel size) in Fourier domain must be twice as fine as it is for the phase retrieval case. Figures 1(d) and (e) show Podorov’s approach. Although in [22] the authors were mainly

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Fig. 1. (a) Experimental setup for lensless imaging. (b) Image reconstruction by phase retrieval does not require a reference wave. (c) Sample prepared with properly separated reference point source for conventional FT holography. Sample prepared for HERALDO with object (d) contained within a rectangular reference and (e) object sitting outside of the rectangular reference (both described by Podorov et al.). (f) Sample prepared for HERALDO with an obscuring tip. Notice that the farthest corner of the rectangles from the object in (d) and (e) and the tip apex in (f) satisfy the conventional holographic separation condition.

concerned with the case of the object contained within the rectangle reference [Fig. 1(d)], they also mention the applicability of their method when the object and the reference rectangle do not overlap, as shown in Fig. 1(e). Figure 1(f) shows an example implementation of HERALDO with an obscuring tip. In Figs. 1(d), (e) and (f), the separation condition to avoid overlap with the object autocorrelation is that the reference feature (farthest corner of the rectangle or tip apex) must satisfy the conventional holographic condition. If our reference feature in these examples is the corner that is farthest from the object, then the sampling and coherence length requirements become the same as for FT holography. In Section 2 of this paper, the basic assumptions of HERALDO are outlined and the general formulation is derived. Section 3 provides the HERALDO separation conditions required to avoid overlap of the reconstructed object field with the multiple field autocorrelation terms. In Section 4 we address differential operators and reconstruction examples for different extended structures that can be used to produce a reference wave. Sections 5 and 6 analyze the effects on image quality due to the presence of apertures in Fourier domain (due to finite detector or beam stop), manufacturing limitations of the extended reference and noise.

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2. General formulation Let us assume that the field in object space can be represented by the sum of an object additive modulation o(x, y) and an extended reference r(x, y), i.e. f (x, y) = o(x, y) + r(x, y), where (x, y) are the Cartesian transverse coordinates. Notice that the additive modulation o(x, y), will only be equal to the object amplitude transmissivity, t(x, y), if the object and the reference do not overlap. In Podorov’s approach, on the other hand, where the object lies entirely within a rectangular aperture, f (x, y) = t(x, y)r(x, y) = o(x, y) + r(x, y), making o(x, y) = [t(x, y) − 1]r(x, y). In general we can say that for an extended uniform reference o(x, y) ∝ t(x, y) − 1 when the object is inside r(x, y) and o(x, y) ∝ t(x, y) when the object is outside r(x, y). In the paraxial approximation, the Fraunhofer field (far field) [2] can be described as the product of a quadratic phase factor and F(u, v) = O(u, v) + R(u, v), where (u, v) are the Cartesian transverse coordinates in Fourier space, and the FT is defined as F(u, v) = F { f (x, y)} =

f (x, y) exp [−i2π (ux + vy)] dx dy .

(1)

Throughout this paper all integrals are to be taken in the range (−∞, ∞) and upper-case letters will refer to the FTs of their lower-case counterpart. In a typical lensless imaging approach, an intensity detector array is used to measure the Fourier intensity |F(u, v)| 2 in the far field, which is related to the object-space field autocorrelation by means of an inverse FT F −1 |F(u, v)|2 = f ⊗ f = o ⊗ o + r ⊗ r + o ⊗ r + r ⊗ o, (2) where o⊗r ≡

o(x , y )r∗ (x − x, y − y)dx dy

(3)

is the cross-correlation of o(x, y) and r(x, y), and ( ∗ ) denotes complex conjugation. For brevity we omit the functional dependence on (x, y) of the cross-correlations whenever it does not lead to ambiguity. Notice that if either r ⊗ o or o ⊗ r in Eq. (2) can be separated from the other terms and the reference object is known, we might attempt recovery of the original object field by means of a deconvolution [18, 23, 24]. We emphasize that HERALDO is fundamentally different from this approach, as it involves the application of linear differential operators on the field autocorrelation. Furthermore for HERALDO the entire r(x, y) does not have to satisfy the holographic separation condition from o(x, y). Let us assume that we selected an extended reference and a linear differential operator, L (n) {·}, such that when we apply the latter onto r(x, y) we get the sum of a point Dirac delta function at (x0 , y0 ) and some other function, namely, L (n) {r(x, y)} = Aδ (x − x0 )δ (y − y0 ) + g(x, y),

(4)

where A is an arbitrary complex-valued constant, L (n) {·} ≡

n

∂n

∑ ak ∂ xn−k ∂ yk

(5)

k=0

is an n-th order linear differential operator and a k are constant coefficients. The term g(x, y) could be another point, or line Dirac delta, or any arbitrary extended function. Notice that Eq. (4) imposes a very special relationship between the differential operator and the extended reference. #88310 - $15.00 USD

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Upon application of the linear operator to the field autocorrelation, we obtain L (n) { f ⊗ f } = L (n) {o ⊗ o} + L (n) {r ⊗ r} + (−1)n o ⊗ L (n) {r} + L (n) {r} ⊗ o , (6) where we have used the identity (see Appendix A), L (n) {h ⊗ g} = (−1)n h ⊗ L (n) {g} = L (n) {h} ⊗ g .

(7)

Using Eq. (4) and the sifting property of the Dirac delta [2], L (n) { f ⊗ f } = L (n) {r ⊗ r} + L (n) {o ⊗ o} + (−1)no ⊗ g + g ⊗ o + (−1)n A∗ o(x + x0, y + y0 ) + Ao∗(x0 − x, y0 − y), (8) or alternatively 1 1 L (n) { f ⊗ f } = (−1)n [A∗ r(x + x0 , y + y0) + r ⊗ g] + [Ar∗ (x0 − x, y0 − y) + g ⊗ r] 2 2 (n) n n ∗ + L {o ⊗ o} + (−1) o ⊗ g + g ⊗ o + (−1) A o(x + x0, y + y0 ) + Ao∗(x0 − x, y0 − y). (9) The last two terms of Eqs. (8) and (9) are the main result of this paper as they enable direct recovery of the original object field, provided that the reconstruction does not overlap with other cross-correlation terms. As for conventional FT holography, we reconstruct both a translated image, o(x + x0 , y + y0 ), of the object and a twin image, o ∗ (x0 − x, y0 − y), which is complexconjugated and inverted through the origin. Conventional holography is a special case of this formulation where r(x, y) = Aδ (x − x0 )δ (y − y0 ), g(x, y) = 0, and L (0) {·} = I is the identity operator. 3. HERALDO separation conditions When Eq. (4) holds, Eqs. (8) and (9) show that we could use this approach to directly retrieve the object field, provided we can separate it from other cross-correlation terms. It is then of interest to analyze the conditions under which these terms would not overlap with the reconstruction. Let us assume that the object is spatially finite and can be contained in a circle of radius ρ 0 , as shown in Fig. 2(a). Without loss of generality we can assume that the object is centered at (0, 0). The conditions to separate undesired terms from the desired reconstruction term, o(x + x0 , y + y0), are as follows. 1. L (n) {o ⊗ o}: Because the autocorrelation of o(x, y) will be contained by a circle of radius 2ρo centered at the origin, any L (n) {o ⊗ o} is also confined to a circle of radius 2ρ 0 . We can then avoid overlap with the reconstruction by constraining the separation, from the center of the object, of the Dirac delta in Eq. (4) to (x 20 + y20 )1/2 > 3ρ0 . This is equivalent to separating the reference feature [the feature of r(x, y) that gives rise to the desired Dirac delta] from the edge of the object by a distance 2ρ 0 , which is the same separation condition as for a conventional holographic reconstruction. 2. (−1)n o ⊗ g + g ⊗ o: Notice that, much like the delta function, any feature in g(x, y) will replicate the object, or its complex-conjugated inversion, at any position where g(x, y) or its centrosymmetrical inversion about the object are non-zero. Since g(x, y) may be extended, it will create continuously overlapped images that are not suited for direct reconstruction. If we want no overlap between this cross-correlations and the reconstruction, g(x, y) and g(−x, −y) must be zero in a radius of 2ρ 0 around the Dirac delta. Notice that if L (n) {r} results in more than one point Dirac delta, resulting in additional reconstructions, this separation condition prevents overlap between the different reconstructions. #88310 - $15.00 USD

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Fig. 2. (a) Field f (x, y) with a slit reference. (b) Field autocorrelation f ⊗ f (inverse FT of far-field intensity). (c) HERALDO separation conditions; the reference feature must satisfy separation conditions from o(x, y), g(x, y), g∗ (−x, −y) and r ⊗ r. (d) Directional derivative of the autocorrelation in the direction of the slit, αˆ .

3. L (n) {r ⊗ r}: Overlap of this term with the reconstruction can be avoided by having r ⊗ r be zero at a radius ρ 0 around the Dirac delta. However, if overlap of this term with the reconstruction cannot be avoided, it may be possible to subtract it if it is known from previous knowledge of r(x, y) or if the setup enables recording of the Fourier intensity without the object, thus enabling measurement of |R(u, v)| 2 . Furthermore, it will be shown that under special circumstances the overlap of this term and the reconstruction is actually desirable (for example when the object sits within a parallelogram that is used as a reference). Figure 2 shows a particular example that illustrates the three HERALDO separation conditions. The field in object space f (x, y) is shown in Fig. 2(a) which includes o(x, y) confined to a circle of radius ρ 0 and a thin bright slit that is used as an extended reference. Figure 2(b) shows the field autocorrelation f ⊗ f . Notice that in this case r(x, y) does not satisfy the conventional holographic condition, so that o ⊗ r overlaps with o ⊗ o and cannot be separated directly. This makes it impossible to recover the object field by deconvolution of o ⊗ r. It will be shown in Section 4 that two point Dirac deltas can be obtained at the ends of the slit by taking the directional derivative of r(x, y) in the direction of the slit, αˆ . If we attempt reconstruction by the rightmost end of the slit (our reference feature), then the leftmost end becomes g(x, y). Figure 2(c) shows that the three separation conditions are satisfied for the reference feature, thus enabling direct reconstruction upon taking the directional derivative of f ⊗ f . Conditions 2 and 3 are illustrated in Fig. 2(c) as circles around the terms that require separation from the Dirac delta, which is equivalent to specifying a circle around the Dirac delta from which those terms must be separated. #88310 - $15.00 USD

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Fig. 3. (a) Finite line Dirac delta along the direction of αˆ . (b) After applying the the directional derivative along αˆ , we obtain two point Dirac delta functions at the line ends.

The result of applying the directional derivative on f ⊗ f is shown in Fig. 2(d). All inverted images (twin images) are also complex-conjugated. Since the rightmost end of the slit satisfies all three separation conditions, we are able to directly retrieve the object field (and its twin image). Notice that since the leftmost end of the slit does not satisfy HERALDO separation Condition 1, it cannot be used for reconstruction as the reconstruction formed by it partially overlaps with the object autocorrelation. 4. Examples of extended references Throughout this section we will discuss examples of particular extended references and differential operators that can be used for reconstruction. Conventional holography is a special case of this formulation where the reference is a point source r(x, y) = Aδ (x − x 0 )δ (y − y0 ), g(x, y) = 0 and L (0) {·} = I is the identity operator. 4.1. Wire or slit reference holography The natural extension of a point reference is the slit reference as illustrated in Fig. 2. Assume that we can manufacture a linear source or a slit that is thinner than the desired resolution. First consider the case where it is parallel to a x -axis, in a rotated coordinate frame as shown in Fig. 3(a). In that case, that reference can be described as a line Dirac delta of length L r(x , y ) = Aδ (y )rect(x /L) = Aδ (y ) H(x + L/2) − H(x − L/2) , (10) where H(x) is the Heaviside function and rect(x) is the rectangle function, defined as ⎧ ⎧ ⎪ ⎪ if |x| < 1/2 if x < 0 ⎨1, ⎨0, H(x) = 1, if |x| > 1/2 . if x > 0 and rect(x) = 0, ⎪ ⎪ ⎩ ⎩ 1/2, if |x| = 1/2 1/2, if x = 0

(11)

After taking the partial derivative with respect to x , we obtain

where we have used

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∂ r(x , y ) = Aδ (y ) δ (x + L/2) − δ (x − L/2) ∂x

(12)

∂ H(x) = δ (x). ∂x

(13)



Notice that we have obtained two point Dirac deltas at the ends of the wire. The different signs of the deltas are due to the positive and negative transitions of r(x , y ) along the x -axis. To generalize this result to a line delta making an angle α with respect to the x-axis, we define the non-rotated coordinates as x = y =

x cos α − y sin α , y cos α + x sin α .

(14a) (14b)

In this case αˆ = xˆ cos α + ysin ˆ α and L (1) {·} =

∂ ∂ ∂ + sin α = αˆ · ∇, = cos α ∂ x ∂x ∂y

(15)

and we find that the operation required to obtain point deltas at the ends of the wire is reduced to the directional derivative along the slit, as illustrated in Fig. 3. For a reconstruction with a slit reference, HERALDO separation Condition 1 restricts the separation of one of the ends of the slit from the center of the object to be greater than 3ρ 0 . HERALDO separation Condition 2, on the other hand, requires the length of the slit to be no less than 2ρ0 . This case was illustrated in Fig. 2. Different signs on the reconstruction and twin image in Fig. 2(d) arise from the (−1) n term in Eq. (8). Throughout the derivation we have assumed that a bright slit is used as a reference, however this result can be straightforwardly extended to the case of a thin wire that obscures the beam. This could be a very practical thing to implement experimentally, for example with a carbon nanotube, and achieve good reconstructions provided that the wire is straight for a distance ρ 0 from its end and is thin compared to the object features we wish to resolve. The result for a semi-infinite line Dirac delta is achieved by letting one of the line ends go to infinity which would result in a single reconstructed image with its twin. 4.2. Corner reference holography As was demonstrated by Podorov et al. for the particular case of a rectangle [22], a sharp corner can act a reference provided that a second order differential operator is applied on the field autocorrelation. Here the result is generalized to an arbitrary corner and later applied to reconstruction using a parallelogram as a reference. A bright corner with edges making angles α and β with respect to the x-axis (β > α ), as shown in Fig. 4(a), can be described by the product of two Heaviside functions r(x, y) = A H(y cos α − x sin α )H(x sin β − y cos β ).

(16)

A point Dirac delta is obtained from a corner upon taking two directional derivatives along its edges. For the case illustrated in Fig. 4(a) the appropriate operator is L (2) {·} =

1 [αˆ · ∇] βˆ · ∇ |sin (β − α )|

(17)

so that 1 [αˆ · ∇] βˆ · ∇ H(y cos α − x sin α )H(x sin β − y cos β ) = δ (x)δ (y). |sin (β − α )|

(18)

The reader is referred to Appendix B for a derivation of this equation. In this case g(x, y) = 0. A sample implementation of HERALDO with a corner, located at (x 0 , y0 ), is shown in Fig. 4(b). Arrows indicate the direction of the derivatives. An object is flood illuminated and an #88310 - $15.00 USD

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Fig. 4. (a) Bright corner with angle (β − α ), edges run along the unit vectors αˆ and βˆ . (b) Example holographic setup with an obscuring tip. (c) Result of the double directional derivative on the autocorrelation (along αˆ and βˆ ) of the field shown in (b). The object is well separated from the tip and its transmissivity t(x, y) can be directly computed.

opaque tip placed near the object to create the corner reference. This approach could be very useful when manufacturing a sharp tip is easier than implementing a suitable point reference. Note that in Fig. 4(b) the tip blocks the illuminating beam so the reference is given by r(x, y) = A {1 − H [(y − y0) cos α − (x − x0 ) sin α ] H [(x − x0) sin β − (y − y0) cos β ]} ,

(19)

and upon application of the differential operator we obtain a point Dirac delta with a negative value 1 [αˆ · ∇] βˆ · ∇ r(x, y) = −Aδ (x − x0)δ (y − y0 ). (20) |sin (β − α )| Figure 4(c) shows the result of applying this operator on the autocorrelation of the field shown in Fig. 4(b). All terms of Eq. (8) are shown. The object reconstruction is clearly separable from other terms in this example. Since HERALDO separation Condition 3 is not satisfied by the corner in this case, there is an additional bias term arising from L (n) {r ⊗ r}. This bias has a similar effect as that arising from a reconstruction where the object is contained in a parallelogram reference and will be discussed later. 4.3. Parallelogram reference holography Among contiguous references, the parallelogram is of special interest as it is the only shape that can be transformed into four point Dirac deltas at the corners upon application of the operator in Eq. (17) with no additional g(x, y) terms. As demonstrated in Appendix B, in order to produce a point Dirac delta we must apply the double directional derivative operator along the edges of a corner. The parallelogram is the only four-sided contiguous shape that accommodates four corners that share the direction of their edges. Because the rectangle is a special case of a parallelogram, the analysis provided in this section also applies to the reconstruction procedure considered in [22]. For the particular case of a parallelogram, FT theorems for rotation and shear [25] can be used to prove that the operator in Eq. (17) produces point Dirac deltas at its corners. Let us examine the L (2) {r ⊗ r} term for the parallelogram. Since L (2) {r(x, y)} results in four point Dirac deltas at the corners of the parallelogram, as shown in Figs. 5(a) and (b), it proves more instructive to see the overlap effect of this term on the HERALDO equation, Eq. (9). Direct inspection of this term readily shows that L (2) {r ⊗ r} results in four centrosymmetrical inversions of the original parallelogram that are displaced to the corners. Figure 5(c) #88310 - $15.00 USD

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Fig. 5. (a) Parallelogram r(x, y) with sides making angles α and β with respect to the x-axis. (b) Directional derivatives of r(x, y) taken along αˆ and βˆ produce four point Dirac deltas. (c) The term L (2) {r ⊗ r} term produces a uniform amplitude parallelogram of twice the size of the original one.

shows the L (2) {r ⊗ r} term. Notice that two parallelograms have sign inversion because two of the point deltas are themselves negative. Thus in this case the amplitude of L (2) {r ⊗ r} is uniform, except along the boundaries of the four parallelograms. If the object lies outside the parallelogram, as shown in Fig. 6(a), HERALDO separation Condition 1 indicates that the center of the object must be separated from the farthest corner of the parallelogram by at least 3ρ 0 to ensure that at least one reconstructed image does not overlap with the object autocorrelation. The resulting terms from L (2) { f ⊗ f } are shown in Fig. 6(d). In Fig. 6(a) HERALDO separation Condition 1 is satisfied for all the corners, so none of the reconstructions shown in Fig. 6(d) overlap with L (2) {o ⊗ o}. Since four point deltas are obtained from the operator, we should expect four object reconstructions and four twin images. HERALDO separation Condition 2 puts a lower bound on the parallelogram sides so that the different reconstructions do not overlap with one another. Direct reconstruction can only be achieved if the parallelogram is of at least the same size as the object (sides of length 2ρ 0 ), thus having non-overlapping reconstructions as shown in Fig. 6(d). If we are, however, willing to selectively stitch different reconstructions together to arrive at a full reconstruction, and the four Dirac deltas satisfy HERALDO Conditions 1 and 3, we can tolerate partial overlap of the object with itself. In this case, illustrated in Figs. 6(b) and (e), the length of the parallelogram sides can be ρ0 . A similar stitching approach could be used with any reference that produces more than one reconstruction to relax HERALDO separation Condition 2. Notice that in Fig. 6(d) one of the reconstructions does not satisfy HERALDO separation Condition 3 and partially overlaps with L (2) {r ⊗ r}. When the object is outside of the parallelogram, the object transmittance t(x, y) ∝ o(x, y), so this partial overlap would not allow the use of that object term for direct reconstruction (although we still have other three reconstructions that do satisfy HERALDO separation Condition 3). So to achieve successful reconstructions at least one corner of the parallelogram must satisfy HERALDO separation Condition 3. For the case of the object contained within the parallelogram, as shown in Fig. 6(c), HERALDO separation Condition 1 places a lower limit on the size of the parallelogram and conditions the position of the object within. To avoid overlap with the autocorrelation of the object at least one corner of the parallelogram must be separated from the center of the object by 3ρ0 . If the object is contained in the parallelogram, this implies that the latter must be at least twice the size of the object. If the side length of the parallelogram is exactly 4ρ 0 then the object must be at one of the corners. This is the condition imposed on the reconstruction example in Podorov’s work [22]. Figure 6(c) shows the extended reference plus the object term. #88310 - $15.00 USD

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Fig. 6. Example holographic setups with object outside of the parallelogram reference are shown in (a) and (b). (c) Example holographic setup with object inside the parallelogram reference. Directional derivatives are taken along αˆ and βˆ . Reconstruction result, L (2) { f ⊗ f }, of fields in (a), (b) and (c) are shown in (d), (e) and (f), respectively. The support of L (2) {o ⊗ o} is shown as a circle in (d), (e) and (f). Since HERALDO separation Condition 2 is not satisfied by the corners of the parallelogram in (b) the reconstructions, shown in (e), overlap with one another. However, since the parallelogram sides are greater than ρ0 , and HERALDO separation Conditions 1 and 3 are satisfied, we can arrive at a full reconstruction by selectively stitching reconstruction pixels that do not overlap. For example, the object upper-left quadrant (open eye) can be obtained from the upper-left reconstruction.

The resulting terms from L (2) { f ⊗ f } are shown in 6(f). Different signs on the reconstructions shown in Figs. 6(d) and (f) arise from signs on the point Dirac deltas. For the case of the object being completely inside the parallelogram, HERALDO separation Condition 2 is automatically satisfied. Notice, however, that by restricting the object to be inside the parallelogram, HERALDO separation Condition 3 is necessarily not satisfied. As shown in 6(f), the reconstructed images overlap with the term L (2) {r ⊗ r}, and direct recovery of o(x, y) cannot be achieved. However, for the parallelogram L (2) {r ⊗ r} consists of four shifted replicas of the original parallelogram r(x, y), and the overlap of this term with the reconstruction will then give the original field at that position f (x, y) = o(x, y) + r(x, y) ∝ t(x, y) which is proportional to the object transmissivity (which is typically the physical quantity of interest). So HERALDO separation Condition 3 does not need to be satisfied when the object is contained within the parallelogram as was shown by Podorov et al. for the specific case of a rectangle. 5. Apertures and noise in the Fourier domain Due to the finite diameter of the detector array, only a portion of the signal can be detected in the Fourier plane. Furthermore, for the particular application to x-ray imaging, it is usually #88310 - $15.00 USD

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necessary to block a central portion of the beam (using a beam stop) to avoid damaging the detector. It is then of great importance to understand the effect on the quality of the reconstruction of apertures in the detected Fourier intensity. We will keep this derivation general, so the results apply to any reference function proposed in Section 4. The detected intensity through a Fourier amplitude transfer function, an aperture transmittance, a stop, or a post-detection weighting function, H(u, v) is given by |FH (u, v)|2 = |H(u, v)|2 |F(u, v)|2 ,

(21)

where FH (u, v) = F(u, v)H(u, v). Following the derivation from Eq. (2) to Eq. (6), we find L (n) { fh ⊗ fh } = L (n) {oh ⊗ oh} + L (n) {rh ⊗ rh } + (−1)n oh ⊗ L (n) {rh } + L (n) {rh } ⊗ oh , (22) where the subscript h indicates convolution with h(x, y), the impulse response due to H(u, v), given by the inverse FT of H(u, v). For example, oh (x, y) = o ∗ h(x, y) =

o(x , y )h(x − x, y − y )dx dy .

(23)

Using Eq. (4) and the sifting property of the Dirac delta, we obtain L (n) {rh (x, y)} = Ah(x − x0, y − y0) + gh(x, y).

(24)

Inserting this result into Eq. (22) and rearranging terms, we find L (n) { fh ⊗ fh } = L (n) {oh ⊗ oh} + L (n) {rh ⊗ rh } + (−1)noh ⊗ gh + gh ⊗ oh + (−1)nA∗ oh⊗h (x + x0, y + y0 ) + Ao∗h⊗h(x0 − x, y0 − y). (25) Notice that, if the intensity is captured through a Fourier aperture, we can directly retrieve oh⊗h (x + x0 , y + y0) = [o ∗ (h ⊗ h)] (x + x0 , y + y0),

(26)

i.e., the reconstruction will be convolved with the autocorrelation of the aperture impulse response. The FT of the reconstructed image is O(u, v)|H(u, v)| 2 , where we have omitted a linear phase. For the specific case of a binary Fourier aperture, we have |H(u, v)| 2 = H(u, v), consistent with a finite detector or a binary beam stop. For that case, h ⊗ h(x, y) = h(x, y) and oh⊗h (x + x0, y + y0 ) = (o ∗ h)(x + x0, y + y0 ),

(27)

which has a FT O(u, v)H(u, v), omitting a linear phase. Thus for the particular case of a binary Fourier aperture, we can retrieve the object as would be acquired through an optical system with an entrance pupil of the same size and shape as the Fourier aperture. In this context the finite diameter of the detector would restrict the highest spatial frequencies of the reconstruction, while a beam stop will block the lowest spatial frequencies. Figure 7 shows a numerical simulation for a reconstruction example using HERALDO. The 128 × 128 object, shown in Fig. 7(a), was embedded into a 2048 × 2048 array and illuminated #88310 - $15.00 USD

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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 7. (a) Original 128 × 128 object. (b) Object space field amplitude f (x, y). The original object is illuminated by a Gaussian beam and a 40◦ tip is used to partially obstruct the beam. (c) Simulated intensity measurement, including the effect of finite detector diameter. (d) Downsampled, 64 × 64, version of original object shown in (a); resolution was halved in Fourier domain. (e) Differential operator, Eq. (17), is applied as a polynomial product on the measured intensity. (f) Magnitude of the inverse FT of (e). Due to their large dynamic ranges, the sixth and fourth roots of the magnitude are shown in (c) and (e) respectively.

by a Gaussian beam with a 1/e radius of 410 pixels, as shown in Fig. 7(b). The beam is partially obscured by a 40 ◦ tip placed in the vicinity of the object. Figure 7(c) shows the simulated measurement of the far-field intensity pattern. This was obtained by computing the fast FT (FFT) of the field shown in 7(b) and taking its squared modulus. Only the central 1024 × 1024 portion of the FT is used (and displayed), thus accounting for a finite detector diameter. Since we have only used the central 1024 × 1024 portion of the 2048 × 2048 intensity array we expect a final reconstructed image of half the resolution as implied by Eq. (27). For comparison purposes we downsampled the 128 × 128 object shown in Fig. 7(a) by computing its FFT and inverse Fourier transforming only the central 64 × 64 portion of the FT. Thus arriving at an ideal image of reduced resolution, shown in Fig. 7(d), that can be compared to the reconstructions. Although the analysis and intuition for HERALDO are best presented in object space, directional derivatives require interpolating and are not trivial to compute in a Cartesian sampled grid. Fortunately, a directional derivative can be efficiently applied in object space by a suitable polynomial multiplication in Fourier space [26]. For the numerical simulations shown throughout this manuscript we compute the directional derivatives of the autocorrelation by application of a polynomial product in Fourier space. The second-order differential operator, Eq. (17), was applied in Fourier space as a polynomial

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12

10

8

Photons

10

4

10

0

10

0

100

(b)

200

300

400 r [pixels] (a)

(c)

500

600

700

(d)

Fig. 8. (a) Cut through (from DC to upper right corner) simulated noisy measurements for 1012 , 1010 and 109 photons on the brightest pixel. Magnitude of the 64 × 64 reconstructions directly obtained from noisy patterns with (b) 1012 , (c) 1010 and (d) 109 photons on the brightest pixel.

multiplication [26], and the magnitude of the result is shown in Fig. 7(e). Upon taking the inverse FFT we obtain the result shown in Fig. 7(f) where an object reconstruction (and twin image) are clearly seen just outside the saturated area near the center of the picture. Much like conventional holography, the image reconstruction is obtained by directly taking the object pixel values from the result shown in Fig. 7(f). Sample reconstructions with different levels of noise are shown in Fig. 8. The simulated measurement, as shown in Fig. 7(c), was renormalized to a finite number of photons on the brightest pixel and Poisson distributed noise was applied to each pixel. Three reconstructions were made with noisy far-field intensity patterns that had 10 12 , 1010 and 109 photons on the brightest pixel respectively. Cuts through the noisy intensity patterns (from DC to the upper right corner) vs. radial distance in pixels are shown in Fig. 8(a). Notice that for 10 9 photons on the brightest pixel (black line), and frequencies beyond 350 pixels from DC, the average number of photons per pixel is low enough that many single photon (and zero photon) detections occur. Consequently at that noise level we would not expect the higher spatial frequencies to be recovered. Following the procedure outlined in Fig. 7 and directly separating the image pixels, we arrive at the 64 × 64 images shown in Fig. 8(b), (c) and (d). The reconstructions were found to be

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(a)

(b)

(c)

Fig. 9. (a) A square beam stop (42 pixel diagonal) was added to the simulated intensity shown in Fig. 7(c). (b) After polynomial multiplication in Fourier domain. (c) 64 × 64 reconstruction directly obtained from the inverse FT of (b).

affected by an overall positive bias term that results from the non-uniform illumination. The magnitude of the recovered field is displayed by assigning black to its lowest value and white to its highest value. The measurement with 10 12 photons on the central pixel has a high SNR at all frequencies, thus having no noticeable effect due to noise on the reconstruction, shown in Fig. 8(b). For this case we were able to reconstruct a faithful image of the original object, as can be seen from direct comparison of the reconstruction and the downsampled object, shown in Figs. 8(b) and 7(d), respectively. The image quality degrades gracefully as the amount of noise increases, consistent with Podorov’s results. Figure 9 shows the effect of a beam stop on the reconstruction. The 1024 × 1024 simulated intensity measured [shown in Fig. 7(c)] was multiplied by a binary square of 42 pixels diagonal, rotated by 45 ◦ and centered at the origin, as shown in Fig. 9(a). The differential operator, Eq. (17), was applied as a polynomial product on the measured intensity, shown in Fig. 9(b). Upon taking the inverse FFT of the result shown in Fig. 9(b) we can directly separate the 64×64 image shown in Fig. 9(c). Due to the removal of the DC term by the beam stop, L (n) { f ⊗ f } is zero mean, and the image shown in Fig. 9(c) has negative values. The real part of the image was displayed so that the minimum value is represented by black and the highest by white. Notice that the principal effect of the beam stop in this reconstruction is that it reduces the background uniformity. When adding noise to the detected intensity using a beam stop we found similar results as those shown in Fig. 8. 6. Effect of an imperfect reference Because in any a real implementation there will be some departure from the ideal reference, it is important to study the effect of not achieving a perfect point Dirac delta after applying the differential operator. Reconstruction by any of the sample extended references provided in Section 4 will be affected by edges that are tapered or not perfectly straight, or by a rounded corner. Returning to the original derivation at Eq. (6), but now denoting r d (x, y) as the imperfect reference, we introduce the modified Eq. (4), L (n) {rd (x, y)} = Ad(x − x0, y − y0) + gd (x, y),

(28)

where d(x, y) is the imperfect point delta function resulting from the operator. #88310 - $15.00 USD

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Using o(x, y) ⊗ Ad(x − x0, y − y0) = Ad(x − x0, y − y0) ⊗ o(x, y) =

A∗ o(x + x0, y + y0 ) ∗ d ∗(−x, −y), ∗

Ao (x0 − x, y0 − y) ∗ d(x, y),

(29a) (29b)

and Eq. (6), we obtain L (n) { f ⊗ f } = L (n) {o ⊗ o} + L (n) {rd ⊗ rd } + (−1)no ⊗ gd + gd ⊗ o + (−1)nA∗ o(x + x0, y + y0) ∗ d ∗ (−x, −y) + Ao∗(x0 − x, y0 − y) ∗ d(x, y), (30) Notice that the effect on the image is that the reconstruction is cross-correlated with the imperfect delta. Thus, similar to conventional FT holography, the reference [e.g. the sharpness of the tip shown in Fig. 7(b)] only needs to be as good as the smallest feature we try to resolve on the object. Furthermore, knowledge of d(x, y) would enable image improvement through deconvolution. 6.1. Imperfect from convolution A particular case of the above formulation is when we have an imperfect reference that can be modeled as a perfect reference convolved with some kernel. This model is well suited to include the effect of the width of a wire reference or a gradual intensity tapering across an edge. For this special case, (31) rd (x, y) = (r ∗ d) (x, y), and

L (n) {rd (x, y)} = L (n) {r} ∗ d (x, y) = Ad(x − x0 , y − y0) + (g ∗ d)(x, y)

(32)

where we have used the identity in Eq. (7) and Eq. (4) for the perfect reference. Upon comparing this with Eq. (28), we see that the convolution kernel of our reference became the imperfect Dirac delta itself and that g d (x, y) = (g ∗ d)(x, y). So if our kernel was a smooth-edged corner that can be modeled as a perfect corner convolved with a Gaussian kernel, then the reconstruction will be cross-correlated with that same Gaussian kernel. This confirms the conclusion that the reference feature only needs to be as sharp as the finest detail we want to resolve on the object. A numerical simulation was implemented to show the effect of an imperfect reference. A 220 × 220 complex-valued object (amplitude and phase shown in Fig. 10(a) and (b) respectively) was embedded in a 1024 × 1024 array and placed close to a bright slit reference of finite width, as shown in Fig. 10(c). The reference slit, one end of which is shown in Fig. 10(i), was created by convolving a single pixel line with a Gaussian with a 1/e radius of 1.5 pixels. Figure 10(d) shows the detected far-field intensity pattern, obtained by computing the field FFT and taking the square modulus. Poisson noise was included after renormalizing the data to have 10 7 photons on the brightest pixel. The magnitude of the result after multiplication by i2π u is shown in Fig. 10(e). Upon taking the inverse FFT we arrive at the field autocorrelation derivative with respect to x, shown in Fig. 10(f), which contains two reconstructed images and their twin images that are clearly separated from other terms. Figures 10(g) and (h) show the recovered magnitude and phase, respectively, of the bottom-left image reconstruction. Notice that the reference width resulted in blurring of the image as expected. Some noise artifacts are also visible on the recovered object phase in Fig. 10(h). 7. Conclusions We have generalized the FT holographic technique to the case of extended references by means of application of a linear differential operator on the field autocorrelation. We call this holography with extended reference by autocorrelation linear differential operation (HERALDO). #88310 - $15.00 USD

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 10. Original 220 × 220 object (a) amplitude (b) and phase. (c) Object space field amplitude f (x, y). (d) Far-field noisy intensity pattern. (e) Magnitude after multiplication by i2π u. (f) Inverse FFT of (e). (g) Amplitude (h) and phase of the 220 × 220 bottom left reconstructed image. (i) 40 × 40 pixel inset of (c). Fifth root of magnitude is shown in (d) and (e). Phase in (b) and (h) has a peak-to-valley range of 3π /2 and is shown in the range [−π , π ].

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Conventional holography and Podorov’s non-iterative method [22] are special cases of this formulation. In this scheme, a feature on the extended reference is chosen such that when we apply a linear differential operator on it, we obtain a point Dirac delta that is adequately separated from other resulting terms. The reconstruction resolution was found to be limited by the quality of this feature of the reference. This generalized holographic technique allows the formation of a reference from the boundary wave of an extended object. We have shown that many other shapes besides the rectangle proposed by Podorov [22] can serve as a reference. This provides the flexibility of selecting a reference that can be precisely manufactured and we believe will prove to be a substantial advance for x-ray lensless imaging. The developed technique is fundamentally different from previously reported holographic methods that use an extended reference and recover the object field by means of a deconvolution [18, 23, 24]. In our approach a suitable differential operator and reference pair are found such that when the former is applied onto the latter, a point reference is obtained. New holographic-like separation conditions were obtained under which we can avoid overlap of the reconstruction with the rest of the terms in the field autocorrelation. In particular when considering the object to be centered at the origin: 1) the object has to be well separated from the reference feature (similar to the FT holographic separation condition). 2) The feature has to be isolated from other terms resulting from applying the differential operator on the reference, g(x, y), [and g(−x, −y)] by a length equal to the spatial extent of the object. For example, if we attempt reconstruction with a tip (corner), the tip should be of good quality (straight, abrupt edges) for a length equal to the width of the object. Notice this is a requirement on the quality of the tip neighborhood and not of the tip itself (sharpness) in order to avoid the extra g(x, y) terms. 3) The feature has to be well separated from the autocorrelation of the reference (r ⊗ r). We found that when using an obscuring tip or a parallelogram reference that contains the object, successful reconstruction is possible even when this condition cannot be satisfied. The effect of apertures in Fourier domain on the reconstruction were analyzed. These apertures arise because of the finite diameter of the detector array and a possible beam stop that might be introduced to avoid damage to the detector. Additionally, the impact on image quality of an imperfect feature on the extended reference was also considered, such as the effect of the tip sharpness. In particular it was shown that if the reference feature is imperfect because of a convolution with a kernel d(x, y), the reconstruction will be cross-correlated with d(x, y). Examples of reconstructions with different extended references were outlined and simulated numerically. HERALDO has requirements on the transverse coherence and uniformity of the illuminating beam that are more stringent than for phase retrieval and are comparable to FT holography. However, it advances conventional FT holography by providing flexibility in tailoring the reference to something that can be accurately manufactured. Numerical simulations showed that HERALDO is robust in the presence of noise. Furthermore, for references where more than one reconstruction is obtained, different reconstructions may be combined to improve the final image SNR in a way analogous to spatially multiplexed FT holography [21, 27], thus enabling reconstructions with decreased intensity in the illumination beam. Because of manufacturing limitations for holographic optical elements and suitable point reference sources at x-ray wavelengths, we believe that this technique will have its greatest impact in x-ray lensless imaging. However our approach is directly applicable to other wavelengths, e.g. visible and ultra-violet, as well.

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Fig. 11. Heaviside function and its directional derivative along θˆ .

Appendix A.

Linear differential operators and the cross-correlation

Similar to the derivative theorem for the convolution of two functions [26],

∂ [ f (x) ∗ g(x)] = f x (x) ∗ g(x) = f (x) ∗ gx (x), ∂x where fx (x) =

∂ f (x) , ∂x

(A1)

(A2)

we have the identity

∂ [h ⊗ g(x, y)] = −h ⊗ gx(x, y) = hx ⊗ g(x, y). ∂x

(A3)

This is easily seen from the derivative theorem for the FT [26]: the Fourier transform of Eq. (A3) is i2π u[H(u, v)G∗(u, v)] = −H(u, v)[i2π uG(u, v)] ∗ = [i2π uH(u, v)]G∗(u, v)

(A4)

and it is evident that the equality holds. Additionally, this identity applies for higher dimensional cross-correlations and FTs. Equation (A3) can be straightforwardly extended by induction to a n-th order linear differential operator L (n) {·}, as defined in Eq. (5), thus obtaining Eq. (7). Appendix B.

Application of linear differential operator to a corner

As an intermediate result we will need the directional derivative of an edge (Heaviside function). Let us assume that the directional derivative unit vector, θˆ , makes an angle θ with respect the edge, as shown in Fig. 11. For specificity we will assume that the derivative is taken along the dark-to-bright direction (going from bright to dark will differ by a minus sign). Then ∂ ∂ + sin θ θˆ · ∇ H(y) = cos θ (B1) H(y) = sin θ δ (y). ∂x ∂y The derivative is a line Dirac delta (along the edge) with an obliquity factor, sin θ , arising from the angle of the edge to the directional derivative unit vector. A corner with edges at angles α and β , as shown in Fig. 4(a), can be described as r(x, y) = H(y cos α − x sin α )H(x sin β − y cos β ), #88310 - $15.00 USD

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(B2)



where we make the assumption that β > α . Taking the directional derivatives along its edges, [αˆ · ∇] βˆ · ∇ r(x, y) = [αˆ · ∇] βˆ · ∇ [H(y cos α − x sin α )H(x sin β − y cos β )] = [αˆ · ∇] H(x sin β − y cos β ) βˆ · ∇ H(y cos α − x sin α ) ,(B3) where we used the derivative product rule and the fact that the edge at angle β is parallel to βˆ , thus giving an obliquity factor of sin(0) = 0 for the [ βˆ · ∇]H(x sin β − y cos β ) term. Notice that although the corner aperture in Fig. 4(a) ends at the origin, the function H(x sin β − y cos β ) stretches from −∞ to ∞ along the edge. Noting that the unit vector βˆ makes an angle of β − α with the edge at angle α and is pointing from dark to bright, using Eq. (B1) we obtain [αˆ · ∇] βˆ · ∇ r(x, y) = [αˆ · ∇] [H(x sin β − y cos β ) sin (β − α ) δ (y cos α − x sin α )] = sin (β − α ) δ (y cos α − x sin α ) [αˆ · ∇] H(x sin β − y cos β ), (B4) where we have used the fact that the line delta function at angle α is constant in the direction of αˆ . By using Eq. (B1) again we can apply [ αˆ · ∇], noting that the unit vector αˆ makes an angle π + α − β with the edge at angle β and is pointing from dark to bright. We then have [αˆ · ∇] βˆ · ∇ r(x, y) = sin2 (β − α ) δ (y cos α − x sin α )δ (x sin β − y cos β ), (B5) where we used sin(π + α − β ) = sin(β − α ). Now let us introduce a rotated set of Cartesian coordinates x y

= x cos α + y sin α , = y cos α − x sin α .

(B6a) (B6b)

Upon direct substitution and using the sifting and scaling property of the Dirac delta [2], we obtain [αˆ · ∇] βˆ · ∇ r(x, y) = sin2 (β − α ) δ (y )δ x sin(β − α ) − y cos(β − α ) = sin2 (β − α ) δ (y )δ x sin(β − α ) = |sin (β − α )| δ (y )δ (x ) (B7) = |sin (β − α )| δ (y)δ (x), where in the last step we have used the fact that a two-dimensional point Dirac delta remains a point Dirac delta upon rotation because it is an area-preserving operation. This proves that the selected operator, given by Eq. (17), achieves a point Dirac delta upon application to a corner.

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