## Measurement of coherent x-ray focused beams by phase retrieval with transverse translation diversity

Optics Express, Vol. 17, Issue 4, pp. 2670-2685 (2009)

http://dx.doi.org/10.1364/OE.17.002670

Acrobat PDF (5043 KB)

### Abstract

We describe a method for characterizing focused x-ray beams using phase retrieval, with diversity achieved by transversely translating a phase-shifting or absorbing structure close to the beam focus. The required measurements can be taken with an experimental setup that is similar to that already used for fluorescent scan testing. The far-field intensity pattern is measured for each position of the translating structure, and the collected measurements are jointly used to estimate the beam profile by using a nonlinear optimization gradient search algorithm. The capability to reconstruct 1D and 2D beam foci is demonstrated through numerical simulations.

© 2009 Optical Society of America

## 1. Introduction

1. 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]

4. 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]

5. 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]

9. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett . **3**, 27–29 (1978). [CrossRef] [PubMed]

16. J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express **14**, 498–508 (2006). [CrossRef] [PubMed]

18. 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. Physics **4**, 394–398 (2008). [CrossRef]

24. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett . **93**, 023903 (2004). [CrossRef] [PubMed]

27. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**, 7264–7278 (2008). [CrossRef] [PubMed]

28. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science **321**, 379–382 (2008). [CrossRef] [PubMed]

29. M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum . **78**, 093707 (2007). [CrossRef] [PubMed]

31. H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett . **96**, 127401 (2006). [CrossRef] [PubMed]

32. A. Stein, K. Evans-Lutterodt, N. Bozovic, and A. Taylor, “Fabrication of silicon kinoform lenses for hard x-ray focusing by electron beam lithography and deep reactive ion etching,” J. Vac. Sci. Technol . B **26**, 122–127 (2008). [CrossRef]

*in situ*to directly assess the quality of the focused beam.

31. H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett . **96**, 127401 (2006). [CrossRef] [PubMed]

32. A. Stein, K. Evans-Lutterodt, N. Bozovic, and A. Taylor, “Fabrication of silicon kinoform lenses for hard x-ray focusing by electron beam lithography and deep reactive ion etching,” J. Vac. Sci. Technol . B **26**, 122–127 (2008). [CrossRef]

*et al*. reconstructed a 2D soft x-ray beam focus (obtained by a zone plate), using phase retrieval, from a measurement of its far-field diffraction pattern [33

33. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Physics **2**, 101–104 (2006). [CrossRef]

*et al*. measured the intensity profile of a hard x-ray 1D focus, and used phase retrieval to recover the beam phase profile [30]. Both approaches used a single intensity measurement and a support constraint at the plane of the focusing optic as constraints for the phase retrieval algorithm.

34. Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun . **30**, 304–308 (1979). [CrossRef]

*et al*. for the image reconstruction problem [24

24. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett . **93**, 023903 (2004). [CrossRef] [PubMed]

*e.g*. a moveable aperture, substantial phase shifting is enough to provide the desired diversity, and can be more easily obtained than large amplitude variations for hard x-rays with a relatively thin sample, which allows the transmitted field to be better approximated by a product of the incident beam and the structure transmissivity. For each position of the structure, the far-field intensity pattern of the beam is measured. These measurements are then used, along with the knowledge of the structure transmissivity and the translations, to reconstruct the incident x-ray field. The experimental arrangement is very similar to the fluorescent scanning approach and the reconstruction can be carried out using the same phase retrieval algorithms used for the analogous image reconstruction problem [24–28

24. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett . **93**, 023903 (2004). [CrossRef] [PubMed]

16. J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express **14**, 498–508 (2006). [CrossRef] [PubMed]

18. 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. Physics **4**, 394–398 (2008). [CrossRef]

*i.e*. both phase and amplitude) of the beam incident on the translating structure. Because the reconstructed field can be propagated numerically to any other plane, the translating structure need not be positioned within the beam depth-of-focus. This will greatly relax the longitudinal alignment requirements compared with the scanning approach. Furthermore, we will show that, after the beam is reconstructed, the distance from the structure to the beam focus can be easily estimated by maximizing a sharpness metric, so that this distance does not need to be known.

**93**, 023903 (2004). [CrossRef] [PubMed]

## 2. 2D focused beam diagnostics example

32. A. Stein, K. Evans-Lutterodt, N. Bozovic, and A. Taylor, “Fabrication of silicon kinoform lenses for hard x-ray focusing by electron beam lithography and deep reactive ion etching,” J. Vac. Sci. Technol . B **26**, 122–127 (2008). [CrossRef]

*μ*m. The first lens (

*L*

_{1}) has a focal length of

*f*= 10 cm and is oriented to focus in the

*x*-direction, and the second lens (

*L*

_{2}) is placed

*z*

_{2}= 1 cm downstream and has a focal length of

*f*-

*z*

_{2}= 9 cm and focuses in the

*y*-direction. The different focal lengths were chosen as to provide the best focus position at 10 cm downstream of

*L*

_{1}for both the

*x*and

*y*directions.

*L*

_{1}and

*L*

_{2}. The simulated aberrations (deviation from cylinder) of the two lenses are shown in Figs. 2(b) and 2(c). Aberrations for the kinoforms were introduced by dividing the lens into 40 sections along the focusing direction, each section having the same number of waves (peak to valley) of the quadratic phase that is responsible for the focusing. Each section was assigned a zero-mean random smooth phase error (independent from one section to the next) with a root-mean-squared (RMS) value of 0.05 waves (0.314 radians) and a random piston error with 0.1 waves (0.628 radians) RMS.

*π*phase steps upon transition from one kinoform section to the next.

*L*

_{1},

*u*

_{1}(

*x*

_{1},

*y*

_{1}), is given by

*t*

_{1}(

*x*

_{1},

*y*

_{1}) are shown in Figs. 2(a) and 2(b), respectively.

*L*

_{2}was computed, assuming scalar paraxial diffraction theory, by numerically propagating

*u*

_{1}(

*x*

_{1},

*y*

_{1}) by a distance

*z*

_{2}. Sampling requirements were minimized by first propagating

*u*

_{1}(

*x*

_{1},

*y*

_{1}) to the lens nominal focus and then propagating back to the plane of

*L*

_{2}using, for both propagations, the numerical propagation approach for cylindrical wavefronts described in Appendix A [Eq. (A5)].

*u*

_{2}(

*x*

_{2},

*y*

_{2}), the field immediately after

*L*

_{2}, by multiplying by the transmissivity of the second lens. The amplitude and phase (deviation from sphere) of the field after

*L*

_{2}are shown in Figs. 3(a) and 3(b), respectively. Notice that the phase aberrations in

*L*

_{1}have caused amplitude variations on the field after

*L*

_{2}.

*u*

_{2}(

*x*

_{2},

*y*

_{2}) is a spherically converging beam, the beam at focus,

*u*

_{3}(

*x*

_{3},

*y*

_{3}), was computed efficiently through a single 2D FFT Fresnel transform as given by Eq. (A1). The beam at focus, shown in Fig. 3(c), has a width of about 100 nm (peak to first null). Sampling at focus was ∆

*x*

_{3}= ∆

*y*

_{3}≃ 19.6 nm.

*π*radians phase shift to the beam inside a 392 nm (20 pixels) radius. The structure was placed in the path of the beam at ∆

*z*= 1 mm downstream from the nominal focus position. The beam at this longitudinal position was significantly larger than the beam at focus. This allows for using a relatively large structure, which is easier to make lithographically. Additionally, the distance from focus does not need to be known for the algorithm to work, which relaxes the alignment requirements for the measurement approach. The beam amplitude at 1 mm downstream of focus, calculated from the beam at focus using angular spectrum [3], is shown in Fig. 4(a).

**93**, 023903 (2004). [CrossRef] [PubMed]

26. O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy **108**, 481–487 (2008). [CrossRef]

^{10}photons, which corresponds to an incident flux of 10

^{6}photons/μm

^{2}/s on the lens with 1 second exposures.

27. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**, 7264–7278 (2008). [CrossRef] [PubMed]

**93**, 023903 (2004). [CrossRef] [PubMed]

28. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science **321**, 379–382 (2008). [CrossRef] [PubMed]

*μ*m lens with 10 cm focal length. For obtaining the initial estimate we assumed that the translating structure was located at the beam focus, rather than at ∆

*z*= 1 mm downstream from the focal plane, to prove that the longitudinal displacement of the translating structure from the focal plane need not be known. The reconstruction after 150 iterations is shown in Fig. 6(a), where we define an iteration as every time the gradient of the error metric is computed. The result matches the true beam, shown in Fig. 4(a), very well.

27. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**, 7264–7278 (2008). [CrossRef] [PubMed]

*α*, an arbitrary complex-valued constant, is considered successful, we assess the quality of the reconstruction through

*E*, a normalized translation-invariant RMS error between the true field and its estimate,

*û*(

*x,y*), given by [35

35. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt . **36**, 8352–8357 (1997). [CrossRef]

35. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt . **36**, 8352–8357 (1997). [CrossRef]

*E*= 0.2484, computed by registering the beam and the reconstruction to within 1/100 of a pixel using an efficient subpixel registration algorithm [36

36. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett . **33**, 156–158 (2008). [CrossRef] [PubMed]

*y*= 3

*μ*m. Within the composite support of the translating structure shown in Fig. 4(b),

*E*= 0.0423, which indicates that the field within this support was accurately recovered. Most of the error in the reconstruction comes from the inaccuracy of the low-amplitude sidelobes over a large area (20×20

*μ*m

^{2}).

*μ*m computational window and we have used the same number of diversity images, with same (relative) overlap of the translating structure in an equivalent position grid.

*E*= 0.0874, computed over the entire computational window. The error is significantly reduced on account of a better agreement for the outer sidelobes of the reconstruction. Figures 6(c) and 6(d) show horizontal cuts through the amplitudes of the initial estimate (dashed curves), the true x-ray field (solid curves), the reconstructions using the 392 nm radius structure (circles) and the 981 nm structure (points). Notice that both reconstructions have an excellent agreement with the true x-ray beam for the cut through the origin, shown in Fig. 6(c). However, there is a significantly better agreement of the reconstruction using the larger translating structure (points) for the cut through

*y*= 3

*μ*m, shown in Fig. 6(d). This improvement can also be obtained with the smaller structure by increasing the number of positions of the translating structure so as to cover an increased area of the beam.

*μ*m and 1.358

*μ*m upstream of the nominal focus for the reconstructions using the small and large translating structures, respectively. Recovering a distance different from the nominal focus is not due to an error in the reconstruction. For this beam, the aberrations introduced by the lenses have slightly shifted the best focus position. We confirmed this by also finding the best focus (maximum sharpness) of the true x-ray beam. This was found at 1.318

*μ*m upstream of the nominal focus, in very good agreement with the reconstruction from the 981 nm translating structure.

## 3. 1D beam focus diagnostics example

*μ*m cylindrical lens with a

*f*= 10 cm focal length, as shown in Fig. 1(b). The transmissivity and aberrations of the lens (deviation from cylinder) are shown in Figs. 2(a) and 2(b), respectively. The field after the lens was numerically propagated to the beam focus using Eqs. (A5) and (A6) with a

*N*×

*N*= 1024 × 1024 computational window. The sampling rate at the plane of the lens was ∆

*x*

_{1}= ∆

*y*

_{1}≃ 543 nm. The sampling at the focal plane was (∆

*x*

_{3}, ∆

*y*

_{3}) ≃ (543,19.6) nm.

*x*and

*y*. The focused beam has a width of approximately 100 nm (peak to first null). Notice that there is substantial variation of the beam along the

*y*direction and that the beam extent in that direction is still close to the original 150

*μ*m.

34. Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun . **30**, 304–308 (1979). [CrossRef]

*μ*m pixel pitch located at 7.37 m downstream the translating structure.

*x*and

*y*directions and that the beam vertical extent is still close to the original 150

*μ*m. The simulated measurement, shown in Fig. 9(b), was obtained after integrating the beam intensity over 40

*μ*m square detector pixels. The field intensity along the vertical direction was severely undersampled and only covered about 5 pixels. Poisson-distributed noise was added to the intensity patterns after normalizing them to a total of 2.25×10

^{10}photons. This corresponds to an incident flux on the cylindrical lens of 10

^{6}photons/

*μ*m

^{2}/s with 1 second exposure per intensity pattern.

*y*direction will make difficult the retrieval of the 2D field. Instead we can integrate the measured intensity over the

*y*direction and attempt retrieval of a projection of the field, which is a function of

*x*. The integrated intensity along the

*y*direction for the pattern shown in Fig. 9(b) is shown in Fig. 9(c).

_{x,y}{․} is the 2D Fresnel diffraction propagation integral, given in Eq. (A1), and 𝓟

_{x}{․} is its 1D counterpart. Equation (3) shows that the projection of an arbitrary beam along a Cartesian coordinate follows the paraxial propagation rule of a 1D beam.

*i.e*. |∬

*u*(

*x,y*)dy|

^{2}, but the quantity we measure is the projection of the intensity,

*i.e*. ∬ |

*u*(

*x,y*)|

^{2}dy. However, if the fluctuations of the field along the

*y*direction are not large compared to the mean, the projection of the intensity gives a reasonably good estimate of the intensity of the field projection. This mismatch makes the use of diverse measurements even more critical for the 1D focus measurement. The implicit assumption of a 1D field also restricts the translating structure to be a 1D function, hence the choice of a phase-shifting structure that is constant along the

*y*direction.

*μ*m aperture and 10 cm focal length, shown in Fig. 10(a) (dashed curve). No defocusing was added to the initial estimate to account for the fact that the distance from the focal plane to the plane of the translating structure may not be known.

**16**, 7264–7278 (2008). [CrossRef] [PubMed]

*E*= 0.117.

*μ*m downstream the nominal focus. Figure 10(b) shows the amplitude of the projection of the true beam (solid line) and the reconstructed beam (points) at this position. The recovered projected beam at focus is in very good agreement with the true beam projection.

## 4. Conclusions

**16**, 7264–7278 (2008). [CrossRef] [PubMed]

*e.g*. a pinhole, as suggested in [28

28. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science **321**, 379–382 (2008). [CrossRef] [PubMed]

37. G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express **17**, 624–639 (2009). [CrossRef] [PubMed]

37. G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express **17**, 624–639 (2009). [CrossRef] [PubMed]

**321**, 379–382 (2008). [CrossRef] [PubMed]

**16**, 7264–7278 (2008). [CrossRef] [PubMed]

**16**, 7264–7278 (2008). [CrossRef] [PubMed]

37. G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express **17**, 624–639 (2009). [CrossRef] [PubMed]

38. S. T. Thurman, R. T. DeRosa, and J. R. Fienup, “Amplitude metrics for field retrieval with hard-edge and uniformly-illuminated apertures,” J. Opt. Soc. Am . A , doc. ID 101701 (posted 26 January 2009, in press). [CrossRef]

*e.g*., extreme ultraviolet or optical.

## Appendix A: Propagation of cylindrical wavefronts

*z*can be computed by the Fresnel diffraction integral [3],

*ikz*) factor,

*λ*is the illumination wavelength,

*k*is the wavenum-ber, and uin and uout are the fields before and after propagation, respectively. Notice that Eq. (A1) is in the form of a 2D FT. To efficiently compute Eq. (A1) one must seek a representation that does not require sampling of a large quadratic phase, because this can greatly increase the number of points required to avoid aliasing. It is also convenient to carry the quadratic phase outside of the integral analytically.

*u*can be sampled by a reasonable number of points (is not a strongly converging or diverging field) and we propagate a large distance

_{in}*z*, then the propagation can be computed by a single 2D FFT as expressed in Eq. (A1). When using the FFT, the output sampling will be

*x*=

*λz*/(

*N*∆

*x*′), where

*N*×

*N*is the size of the computational array, and ∆

*x*′ and ∆

*x*are the sample spacings at the input and output planes, respectively. However, if the propagation distance is small, sampling the quadratic phase kernel in Eq. (A1) may become an issue. In that case computing the propagation in Fourier domain by a two-step transfer function approach [3] (paraxial angular spectrum) is preferable,

*u*is a focusing field, we can avoid sampling of the large spherical wavefront component by using a single FT computation if

_{in}*u*needs to be computed close to the beam nominal focus. The quadratic factor on the field will then nearly cancel the quadratic kernel of the Fresnel integral, and the output sampling will be given by ∆

_{out}*x*=

*λ*

*z*|(

*N*∆

*x*′). For cases where the focusing field needs to be propagated far from the nominal focus (either close to the original field or on the opposite side of focus) a two step approach is more convenient, propagating the field to the nominal focus and then to the plane of interest. For this case the relation of output and input sampling is ∆

*x*= |

*zf*-

*z*|∆

*x*′|

*zf*, where

*zf*is the distance from

*u*to the nominal focus and

_{in}*z*is the distance from

*u*to

_{in}*u*.

_{out}*u*in this case can be sought by noting that paraxial propagation is separable in Cartesian coordinates. So that the integrals can be computed independently for the

_{out}*x*and

*y*directions following the guidelines given above.

*u*is the field transmitted by a cylindrical lens of focal length

_{in}*f*(focusing only along the

*x*direction), then

*t*(

*x,y*) is a complex-valued function. The amplitude of

*t*(

*x,y*) describes the field amplitude right after the lens and its phase describes the wavefront deviation from a cylinder.

_{y→fy}{․} is the 1D FT.

*y*= ∆

*y*′ and ∆

*x*=

*λf*/(

*N*∆

*x*′). This different sampling along the horizontal and vertical directions is appropriate for this problem because focusing occurs only along x and the beam at focus has a widely different extent along the

*x*and

*y*directions.

## References and links

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32. | A. Stein, K. Evans-Lutterodt, N. Bozovic, and A. Taylor, “Fabrication of silicon kinoform lenses for hard x-ray focusing by electron beam lithography and deep reactive ion etching,” J. Vac. Sci. Technol . B |

33. | H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Physics |

34. | Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun . |

35. | J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt . |

36. | M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett . |

37. | G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express |

38. | S. T. Thurman, R. T. DeRosa, and J. R. Fienup, “Amplitude metrics for field retrieval with hard-edge and uniformly-illuminated apertures,” J. Opt. Soc. Am . A , doc. ID 101701 (posted 26 January 2009, in press). [CrossRef] |

39. | M. Guizar-Sicairos and J. R. Fienup, “Focused x-ray beam characterization by phase retrieval with a move-able phase-shifting structure,” in |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(110.7440) Imaging systems : X-ray imaging

(340.7480) X-ray optics : X-rays, soft x-rays, extreme ultraviolet (EUV)

(140.3295) Lasers and laser optics : Laser beam characterization

(110.3200) Imaging systems : Inverse scattering

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 7, 2008

Revised Manuscript: December 10, 2008

Manuscript Accepted: January 16, 2009

Published: February 10, 2009

**Virtual Issues**

Vol. 4, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Manuel Guizar-Sicairos and James R. Fienup, "Measurement of coherent x-ray focused
beams by phase retrieval with transverse
translation diversity," Opt. Express **17**, 2670-2685 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2670

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