## Fresnel diffraction in the case of an inclined image plane

Optics Express, Vol. 16, Issue 7, pp. 5141-5149 (2008)

http://dx.doi.org/10.1364/OE.16.005141

Acrobat PDF (492 KB)

### Abstract

An extension of the theoretical formalism of Fresnel diffraction to the case of an inclined image plane is proposed. The resulting numerical algorithm speeds up computation times by typically three orders of magnitude, thus opening the possibility of utilizing previously inapplicable image analysis algorithms for this special type of a non shift-invariant imaging system. This is exemplified by adapting an iterative phase retrieval algorithm developed for electron microscopy to the case of hard x-ray imaging with asymmetric Bragg reflection (the so-called “Bragg Magnifier”). Numerical simulations demonstrate the convergence and feasibility of the iterative phase retrieval algorithm for the case of x-ray imaging with the Bragg Magnifier.

© 2008 Optical Society of America

## 1. Introduction

2. C.E. Metz and K. Doi, “Transfer function analysis of radiographic imaging systems,” Phys. Med. Biol. **24**, 1079–1106 (1979). [CrossRef] [PubMed]

3. P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays,” Appl. Phys. Lett. **75**, 2912–2914 (1999). [CrossRef]

4. V. Yu. Ivanov, V. P. Sivokon, and M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A **9**,1515–1524 (1992). [CrossRef]

5. W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. **69**, 3743–3746 (1992). [CrossRef] [PubMed]

6. I. A. Cunningham and R. Shaw, “Signal-to-noise optimization of medical imaging systems,” J. Opt. Soc. Am. A **16**621–632 (1999). [CrossRef]

7. A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, “Deterministic retrieval of surface waviness by means of topography with coherent X-rays,” J. Synchrotron Rad. **9**, 223–228 (2002). [CrossRef]

10. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified phase imaging using asymmetric Bragg reflection: experiment and theory,” Phys. Rev. B **74**054107-1–054107-10 (2006). [CrossRef]

## 2. Experimental example

13. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. **42**2015–2025 (1997). [CrossRef] [PubMed]

10. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified phase imaging using asymmetric Bragg reflection: experiment and theory,” Phys. Rev. B **74**054107-1–054107-10 (2006). [CrossRef]

*µ*m thick copper mesh is normally used for calibration and has a periodicity of 63.5

*µ*m in both directions. A photon energy of 8.048 keV corresponding to a wavelength of 1.54 Åwas used, yielding 40.4-fold magnification in vertical and 41.6-fold magnification in horizontal direction. A highly efficient CCD camera (Bruker AXS Smart Apex 2) with a pixel size of 15

*µ*m at 4096

^{2}pixels was used for image acquisition. The measurement was performed at the beamline ID19 at the European Synchrotron Radiation Facility in Grenoble (France). Figure 1(b) shows the intensity distribution of the first and last period along the horizontal line in Fig. 1(a). The intensity was averaged over seven horizontal lines to increase the signal-to-noise ratio.

10. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified phase imaging using asymmetric Bragg reflection: experiment and theory,” Phys. Rev. B **74**054107-1–054107-10 (2006). [CrossRef]

## 3. Theory

*D̂*(

_{in}*q*) of the input wave field

*D*(

_{in}*x*) to the observable output wave field

*D*(

_{out}*x*) after free-space propagation over the distance z+z

_{0}. The Fourier transform variable q corresponds to the spatial variable

*x*. For a one-dimensional object (the extension to two object dimensions will be discussed below) it can be written as [1]

*K*=2

*π*/λ is given by the wavelength λ and z

_{0}is an arbitrary distance between object and image plane (the mean propagation distance would be a convenient choice; see Fig. 2). As long as the image plane is perpendicular to the main beam direction (i.e.

*z*is constant) Eq. (1) constitutes a simple Fourier transform.

*α*≠

*π*/2), the propagation distance becomes dependent on the position in the image (i.e.

*z*=

*z*(

*x*)). As stated in the introduction,

*D*can be calculated by performing Fourier transforms for many planes of constant propagation distances (typically several hundred slices) and interpolating the results to the coordinates of the inclined image plane s. However, this is obviously a time-consuming approach.

_{out}*x*=

*s*sin

*α*and

*z*=

*s*cos

*α*(see Fig. 2) have been used. The idea of the following substitution is to reduce the last factor to exp(

*is f*), thus converting the integral (1) to a simple Fourier transform. Obviously, this can be achieved by choosing

*q*=

*K*tan

*α*divides the graph of Eq. (4) into two branches) is resolved as follows. The Fourier component

*q*corresponds to a plane wave propagating with an angular deviation of ≈

*q*/

*K*with respect to the main beam direction. But the validity of the Fresnel diffraction integral is limited to small angular deviations from the main beam direction [1]. This implies the condition

*q*≪

*K*which can only be fulfilled by choosing the negative sign in Eq. (4).

*P̂*(

*q*) to relate input and output wave field in Fourier space. While in the usual case of shift-invariant imaging techniques the relation is given by

*D̂*(

_{out}*q*)=

*P̂*(

*q*)

*D̂*(

_{in}*q*), it will turn out that in the present case the relation needs to be modified. Therefore, we use the propagator in a generalized sense.

*q*’s in equations (5) and (6) are functions of

*f*according to Eq. (4) with negative sign.

*x*and

*y*direction) is simple if the image plane is only rotated around the

*y*-axis and the orientations of the

*x*,

*y*coordinate axes in the object plane are chosen in a convenient way. By setting the

*y*-direction to be perpendicular to the normal of the image plane, the propagation distance in

*y*-direction is constant in the image plane. Hence, the

*y*-direction can be treated as usual, yielding

*p*corresponds to the spatial variable

*y*.

## 4. Iterative phase retrieval algorithm for the Bragg Magnifier

*et al.*[13

13. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. **42**2015–2025 (1997). [CrossRef] [PubMed]

14. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. **90**, 193501-1–193501-3 (2007). [CrossRef]

*et al.*for electron microscopy [5

5. W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. **69**, 3743–3746 (1992). [CrossRef] [PubMed]

**74**054107-1–054107-10 (2006). [CrossRef]

*θ*defines the angular position of the main beam direction on the analyzer reflection curve (working point). The observable intensity of a known input wave field is then given by

3. P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays,” Appl. Phys. Lett. **75**, 2912–2914 (1999). [CrossRef]

5. W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. **69**, 3743–3746 (1992). [CrossRef] [PubMed]

*θ*is the natural choice of the free experimental parameter realizing the independence of the corresponding experimental images.

*I*be the

^{exp}_{m}*m*th experimentally obtained intensity of a total of

*M*pictures(e.g.

*M*=2 for two pictures recorded on opposite slope positions of the reflection curve). Further, let

*I*(

_{m}*s*) be the calculated intensity according to Eq. (9). Then the quadratic difference is given by

*D̂*

_{0}(

*f*). The

*j*+1th iteration is performed by calculating (a direct adaptation of Eq. (6) in [5

**69**, 3743–3746 (1992). [CrossRef] [PubMed]

*γ*a convergence parameter,

*F*{…} (

*F*

^{-1}{…}) indicating the (inverse) Fourier transform of its argument and

*I*(

_{mj}*s*) the intensity (calculated according to Eq. 9) corresponding to the current wave field

*D̂*.

_{j}*I*(

^{exp}_{m}*s*)=

*I*(

_{mj}*s*)) while the function of the iteration can be understood by reading Eq. (11) from inside to outside. First, the current wave field

*D̂*is propagated and transformed to real space (

_{j}*F*

^{-1}{

*D̂*(

_{j}*f*)

*P̂*(

_{m}*f*)}), yielding the current input wave field in real space. This wave field is now weighted with the difference between experiment and calculation (

*I*(

^{exp}_{m}*s*)-

*I*(

_{mj}*s*)) and the result is then back-transformed to Fourier space. The multiplication with the conjugated propagator

*P̂**(

_{m}*f*) corresponds to a back-propagation of the modified wave field. Finally, the new wave field

*D̂*

_{j+1}is obtained by the weighted sum over all modified wave fields.

*f*. So, before obtaining the wave field in real space it has to be rewritten in terms of the original Fourier argument

*q*. In this first attempt, this is done by simply calculating

*q*=

*f*/sin

*α*, which is a linearized version of Eq. (3) avoiding the need of interpolation.

### A demonstration of the feasibility

*D*(

_{in}*x*). In order to give a smooth appearance and to minimize the discontinuity at the edges, the input wave field was convoluted with a small rectangular filter of 2.5

*µ*m width.

*σ*reflection at 8.048 keV and 40-fold magnification corresponding to the first analyzer crystal of the Bragg Magnifier.

## 5. Conclusion

**74**054107-1–054107-10 (2006). [CrossRef]

16. Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Appl. Phys. D: Appl. Phys. **38**4259–4271 (2005). [CrossRef]

*full*description of the image formation process and thus open the possibility of quantitative tomography on a sub-micrometer scale with the Bragg Magnifier.

## References and links

1. | M. Born and E. Wolf, |

2. | C.E. Metz and K. Doi, “Transfer function analysis of radiographic imaging systems,” Phys. Med. Biol. |

3. | P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays,” Appl. Phys. Lett. |

4. | V. Yu. Ivanov, V. P. Sivokon, and M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A |

5. | W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. |

6. | I. A. Cunningham and R. Shaw, “Signal-to-noise optimization of medical imaging systems,” J. Opt. Soc. Am. A |

7. | A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, “Deterministic retrieval of surface waviness by means of topography with coherent X-rays,” J. Synchrotron Rad. |

8. | T. Panzner, G. Dleber, T. Sant, W. Leitenberger, and U. Pietsch, “Coherence experiments at the white-beam beamline of BESSY II,” Thin Sol. Films |

9. | K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. |

10. | P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified phase imaging using asymmetric Bragg reflection: experiment and theory,” Phys. Rev. B |

11. | N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A |

12. | N. Delen and B. Hooker, “Verification and comparison of a fast Fourier transform-based full diffraction method for tilted and offset planes,” Appl. Opt. |

13. | D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. |

14. | P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. |

15. | A.C. Kak and M. Slaney “Principles of Computerized Tomographic Imaging,” IEEE Press. (1988). |

16. | Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, “Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging,” J. Appl. Phys. D: Appl. Phys. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(110.2990) Imaging systems : Image formation theory

(110.7440) Imaging systems : X-ray imaging

(340.7460) X-ray optics : X-ray microscopy

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: December 14, 2007

Revised Manuscript: January 25, 2008

Manuscript Accepted: January 27, 2008

Published: March 28, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Peter Modregger, Daniel Lübbert, Peter Schäfer, Rolf Köhler, Timm Weitkamp, Michael Hanke, and Tilo Baumbach, "Fresnel diffraction in the case of an inclined image plane," Opt. Express **16**, 5141-5149 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-5141

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### References

- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge 7th ed. 1999)
- C. E. Metz and K. Doi, "Transfer function analysis of radiographic imaging systems," Phys. Med. Biol. 24, 1079-1106 (1979). [CrossRef] [PubMed]
- P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron raditaion x-rays," Appl. Phys. Lett. 75, 2912-2914 (1999). [CrossRef]
- V. Yu. Ivanov, V. P. Sivokon, and M. A. Vorontsov, "Phase retrieval from a set of intensity measurements: theory and experiment," J. Opt. Soc. Am. A 9, 1515-1524 (1992). [CrossRef]
- W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, "Phase retrieval through focus variation for ultraresolution in field-emission transmission electron microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992). [CrossRef] [PubMed]
- I. A. Cunningham and R. Shaw, "Signal-to-noise optimization of medical imaging systems," J. Opt. Soc. Am. A 16621-632 (1999). [CrossRef]
- A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent X-rays," J. Synchrotron Rad. 9, 223-228 (2002). [CrossRef]
- T. Panzner, G. Dleber, T. Sant, W. Leitenberger, and U. Pietsch, "Coherence experiments at the white-beam beamline of BESSY II," Thin Sol. Films 5155563-5567 (2007). [CrossRef]
- K. D. Mielenz, "Algorithms for Fresnel diffraction at rectangular and circular apertures," J. Res. Natl. Inst. Stand. Technol. 103497-509 (1998). [CrossRef]
- P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Magnified phase imaging using asymmetric Bragg reflection: experiment and theory," Phys. Rev. B 74 054107-1-054107-10 (2006). [CrossRef]
- N. Delen and B. Hooker, "Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach," J. Opt. Soc. Am. A 15, 857-867 (1998). [CrossRef]
- N. Delen and B. Hooker, "Verification and comparison of a fast Fourier transform-based full diffraction method for tilted and offset planes," Appl. Opt. 40, 3525-3531 (2001). [CrossRef]
- D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced x-ray imaging," Phys. Med. Biol. 422015-2025 (1997). [CrossRef] [PubMed]
- P. Modregger, D. Lubbert, P. Schafer, and R. Kohler, "Two dimensional diffraction enhanced imaging algorithm," Appl. Phys. Lett. 90, 193501-1-193501-3 (2007). [CrossRef]
- A. C. Kak and M. Slaney "Principles of Computerized Tomographic Imaging," IEEE Press. (1988).
- Ya. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "Polychromaticity in the combined propagationbased/ analyser-based phase-contrast imaging," J. Appl. Phys. D: Appl. Phys. 384259-4271 (2005). [CrossRef]

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