## Holography with extended reference by autocorrelation linear differential operation

Optics Express, Vol. 15, Issue 26, pp. 17592-17612 (2007)

http://dx.doi.org/10.1364/OE.15.017592

Acrobat PDF (3504 KB)

### 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

## 1. Introduction

1. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. **52**, 1123–1130 (1962). [CrossRef]

3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

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 Fe_{65}Al_{35},” Phys. Rev. B **76**, 014204 (2007). [CrossRef]

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). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

*a priori*or estimated from the object’s autocorrelation function [15

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

17. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A **3**, 1897–1907 (1986). [CrossRef]

1. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. **52**, 1123–1130 (1962). [CrossRef]

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). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

21. W. F. Schlotter*et al*., “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. **89**, 163112 (2006). [CrossRef]

*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

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). [CrossRef] [PubMed]

*a priori*knowledge of the orientation and structure of the extended reference.

## 2. General formulation

*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).

*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

*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.

*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

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

*r*(

*x,y*) does not have to satisfy the holographic separation condition from

*o*(

*x,y*).

*ℒ*

^{(n)}{·}, such that when we apply the latter onto

*r*(

*x,y*) we get the sum of a point Dirac delta function at (

*x*

_{0},

*y*

_{0}) and some other function, namely,

*A*is an arbitrary complex-valued constant,

*n*-th order linear differential operator and

*a*are constant coefficients. The term

_{k}*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.

*o*(

*x*+

*x*+

_{0},y*y*

_{0}), of the object and a twin image,

*o**(

*x*), which is complex-conjugated and inverted through the origin.

_{0}-x,y_{0}-y*r*(

*x,y*)=

*Aδ*(

*x-x*

_{0})δ(

*y-y*

_{0}),

*g*(

*x,y*)=0, and

*ℒ*

^{(0)}{·}=I is the identity operator.

## 3. HERALDO separation conditions

*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*+

*x*+

_{0},y*y*

_{0}), are as follows.

*ℒ*

^{(n)}{

*o*⊗

*o*}: Because the autocorrelation of

*o*(

*x,y*) will be contained by a circle of radius 2

*ρ*centered at the origin, any

_{o}*ℒ*

^{(n)}{

*o*⊗

*o*} is also confined to a circle of radius 2

*ρ*

_{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*

^{2}

_{0}+

*y*

^{2}

_{0})

^{1/2}>3

*ρ*

_{0}. 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 2

*ρ*

_{0}, which is the same separation condition as for a conventional holographic reconstruction.

^{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 centrosym-metrical 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 2

*ρ*

_{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.

*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).

*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*.

*r*(

*x,y*) in the direction of the slit,

*α*̂. 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.

*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

*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*

*H*(

*x*) is the Heaviside function and rect(

*x*) is the rectangle function, defined as

*x*

^{′}, we obtain

*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

*α*̂=

*x*̂cosα+

*y*̂sin

*α*and

*ρ*

_{0}. HERALDO separation Condition 2, on the other hand, requires the length of the slit to be no less than 2

*ρ*

_{0}. 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).

*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

*et al*. for the particular case of a rectangle [22

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). [CrossRef] [PubMed]

*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

*g*(

*x,y*)=0.

*x*), is shown in Fig. 4(b). Arrows indicate the direction of the derivatives. An object is flood illuminated and an 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.

_{0},y_{0}*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

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). [CrossRef]

*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) 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.

*ρ*

_{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 2

*ρ*

_{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

*ρ*

_{0}. A similar stitching approach could be used with any reference that produces more than one reconstruction to relax HERALDO separation Condition 2.

*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.

*ρ*

_{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

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). [CrossRef] [PubMed]

*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.

*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

*H*(

*u,v*) is given by

*F*(

_{H}*u,v*)=

*F*(

*u,v*)

*H*(

*u,v*).

*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,

*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.

*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

*O*(

*u,v*)

*H*(

*u,v*), omitting a linear phase.

^{12}, 10

^{10}and 10

^{9}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.

^{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.

*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

*r*(

_{d}*x,y*) as the imperfect reference, we introduce the modified Eq. (4),

*d*(

*x,y*) is the imperfect point delta function resulting from the operator.

### 6.1. Imperfect from convolution

^{7}photons on the brightest pixel. The magnitude of the result after multiplication by

*i*2

*π 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

**15**, 9954–9962 (2007). [CrossRef] [PubMed]

**15**, 9954–9962 (2007). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

*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.

21. W. F. Schlotter*et al*., “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. **89**, 163112 (2006). [CrossRef]

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). [CrossRef]

## Appendix A. Linear differential operators and the cross-correlation

*ℒ*

^{(n)}{·}, as defined in Eq. (5), thus obtaining Eq. (7).

## Appendix B. Application of linear differential operator to a corner

*θ*̂, 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

*θ*, arising from the angle of the edge to the directional derivative unit vector.

*β*>

*α*. Taking the directional derivatives along its edges,

*β*is parallel to

*β*̂, thus giving an obliquity factor of sin(0)=0 for the [

*β*̂·∇]

*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

*β*̂ makes an angle of

*β-α*with the edge at angle

*α*and is pointing from dark to bright, using Eq. (B1) we obtain

*α*is constant in the direction of

*α*̂. By using Eq. (B1) again we can apply [

*α*̂·∇], noting that the unit vector

*α*̂ makes an angle

*π*+α-

*β*with the edge at angle

*β*and is pointing from dark to bright. We then have

*π+α-β*)=sin(

*β-α*).

## References and links

1. | E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. |

2. | J. W. Goodman, |

3. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

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 |

5. | P. S. Idell, J. R. Fienup, and R. S. Goodman, “Image synthesis from nonimaged laser-speckle patterns,” Opt. Lett. |

6. | J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrieval from experimental far-field speckle data,” Opt. Lett. |

7. | J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express |

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 |

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 |

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 |

11. | D. Shapiro |

12. | H. N. Chapman |

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 Fe |

14. | H. N. Chapman |

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. |

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 |

17. | J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A |

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. |

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. |

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 |

21. | W. F. Schlotter |

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 |

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. |

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 |

25. | R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. |

26. | R. N. Bracewell, |

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 |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(100.5070) Image processing : Phase retrieval

(110.7440) Imaging systems : X-ray imaging

(090.1995) Holography : Digital holography

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Holography

**History**

Original Manuscript: October 5, 2007

Revised Manuscript: November 29, 2007

Manuscript Accepted: December 1, 2007

Published: December 11, 2007

**Citation**

Manuel Guizar-Sicairos and James R. Fienup, "Holography with extended reference by autocorrelation linear differential
operation," Opt. Express **15**, 17592-17612 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-26-17592

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### References

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- J. R. Fienup and C. C. Wackerman, "Phase-retrieval stagnation problems and solutions," J. Opt. Soc. Am. A 3, 1897-1907 (1986). [CrossRef]
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