## Rytov approximation for x-ray phase imaging |

Optics Express, Vol. 21, Issue 3, pp. 2674-2682 (2013)

http://dx.doi.org/10.1364/OE.21.002674

Acrobat PDF (1462 KB)

### Abstract

In this study, we check the accuracy of the first-order Rytov approximation with a homogeneous sphere as a candidate for application in x-ray phase imaging of large objects e.g., luggage at the airport, or a human patient. Specifically, we propose a validity condition for the Rytov approximation in terms of a parameter *V* that depends on the complex refractive index of the sphere and the Fresnel number, for Fresnel numbers larger than 1000. In comparison with the exact Mie solution, we provide the accuracy of the Rytov approximation in predicting the intensity and phase profiles after the sphere. For large objects, where the Mie solution becomes numerically impractical, we use the principle of similarity to predict the accuracy of the Rytov approximation without explicit calculation of the Mie solution. Finally, we provide the maximum radius of the sphere for which the first order Rytov approximation remains valid within 1% accuracy.

© 2013 OSA

## 1. Introduction

2. P. C. Diemoz, A. Bravin, and P. Coan, “Theoretical comparison of three X-ray phase-contrast imaging techniques: propagation-based imaging, analyzer-based imaging and grating interferometry,” Opt. Express **20**(3), 2789–2805 (2012). [CrossRef] [PubMed]

3. E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Kristall und Technik **15**(8), 937–945 (1980). [CrossRef]

5. M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys. **41**(Part 2, No. 9A/B), L1016–L1018 (2002). [CrossRef]

6. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. **45**(6A), 5254–5262 (2006). [CrossRef]

8. T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. **68**(3Suppl), S13–S17 (2008). [CrossRef] [PubMed]

9. S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature **384**(6607), 335–338 (1996). [CrossRef]

11. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x‐ray phase contrast microimaging by coherent high‐energy synchrotron radiation,” Rev. Sci. Instrum. **66**(12), 5486–5492 (1995). [CrossRef]

12. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. **1**(4), 153–156 (1969). [CrossRef]

13. A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. **6**(8), 374–376 (1981). [CrossRef] [PubMed]

15. F. Lin and M. Fiddy, “The Born-Rytov controversy: I. Comparing analytical and approximate expressions for the one-dimensional deterministic case,” J. Opt. Soc. Am. A **9**(7), 1102–1110 (1992). [CrossRef]

16. E. Kirkinis, “Renormalization group interpretation of the Born and Rytov approximations,” J. Opt. Soc. Am. A **25**(10), 2499–2508 (2008). [CrossRef] [PubMed]

*V*defined in terms of the complex refractive index of the sphere and the Fresnel number. The parameter

*V*relieves us of the need to calculate the exact Mie solution for very large spheres, when the number of terms required in the Mie series expansion becomes prohibitive. We show that the error can be calculated from the value of

*V*, without having to explicitly calculate the Mie series.

## 2. Scattered field calculation under the first Rytov approximation

13. A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. **6**(8), 374–376 (1981). [CrossRef] [PubMed]

19. M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Tech. **32**(8), 860–874 (1984). [CrossRef]

## 3. Explicit validity condition of the first Rytov approximation for a homogeneous sphere

*R*and complex refractive index

*n*,

*α*is the order of the Bessel function. From this, we can explicitly calculate the neglected term from the Rytov approximation aswhere

*A(F)*is defined as follows:We obtain

*A(F)*by numerically integrating Eq. (13) for different values of

*β*and searching for the maximum. From the projection approximation and the known profile of the sphere, we should expect

*β*; 1. Figure 2 shows

*A(F)*calculated for seven different values of

*F*, indicating that for large enough values of

*F*,

*A(F)*is well approximated by a linear fit (

*R*= 0.9993), asWe found that the square-root approximation works very well for

^{2}*F*≳ 10

^{7}(95% confidence interval, [0.5 0.5001]). For smaller values of F, the accuracy of (14) as an approximation to (13) worsens.

*V*in Eq. (12) is a function of known parameters, and thus it can be used to predict the validity of the approximation without the need for a full calculation of the scattered field. More importantly, two spheres of a different size may provide a same

*V*value when the other parameters such as the wavelength

*λ*and the distance

*z*are properly chosen; the two cases can be simulated with the same accuracy using the Rytov approximation. This is analogous to the principle of similarity in fluid mechanics, where the Reynolds number similarly serves as a scaling parameter. As expected, the parameter

*V*depends on the size of the object being imaged, although in the case of a sphere the size also determines the radius of curvature “imparted” on the scattered wavefront.

## 4. Error estimation of the Rytov approximation: comparison with Mie solution

*R⁄λ*is on the order of 10

^{9}and, hence, so is the

*F*parameter. Mie theory provides an exact formula for the complex vector field scattered from a homogeneous sphere [18, 22]; therefore, it can serve as a gold standard with which the accuracy of the Rytov approximation can be checked. Since the Mie solution is an infinite series of vector spherical harmonics, in practice higher order terms are truncated to provide a solution within a given accuracy [23

23. A. A. Neves and D. Pisignano, “Effect of finite terms on the truncation error of Mie series,” Opt. Lett. **37**(12), 2418–2420 (2012). [CrossRef] [PubMed]

*R⁄λ*, where

*R*is the radius of the sphere and

*λ*is the wavelength of the incident beam. However, in case the ratio

*R⁄λ*is extremely large as in our case, alternative approaches need be sought, e.g., adopting an asymptotic formula for the Mie series, transforming the series into an equivalent but a more rapidly converging sum, etc [24]. Here, we suggest an alternative method, scaling down the sphere with

*V*in Eq. (12) as a scaling parameter and predict the accuracy of the Rytov approximation based on the model system.

*n*= 1-

*δ*+

*iβ*, where

*δ*= 6.844 × 10

^{−8}and

*β*= 3.379 × 10

^{−11}[25

25. B. Henke, E. Gullikson, and J. C. X. Davis, “X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables **54**(2), 181–342 (1993). [CrossRef]

*V*in this case is 0.0027. Note that the full Mie series cannot be directly evaluated because of the large

*R⁄λ*ratio (2.42 × 10

^{9}). Instead, we consider a model system with the same

*V*value, but with

*R⁄λ*= 10

^{4}. We put the measurement plane at

*z*= 4

*πR*to guarantee the scalar field assumption in Eq. (1) holds. For these choices of parameters, the value of

*F*is 10

^{4}. Now, we find the wavelength of the incident beam to guarantee the same

*V*value. Since most materials are highly dispersive in x-rays, we need to solve

*λ*= 0.256 nm, and the other parameters are accordingly determined as

*R*= 2.56

*μ*m,

*z*= 32.2

*μ*m. In a nutshell, the accuracy of the Rytov approximation for the model system (

*λ*= 0.256 nm,

*R*= 2.56

*μ*m, and

*z*= 32.2

*μ*m) is the same as that for the original system (

*λ*= 0.0207 nm,

*R*= 5 cm, and

*z*= 1 m). Since the radius of sphere in the model system is small enough, we can check the validity of the Rytov approximation using the Mie solution.

26. F. Slimani, G. Grehan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt. **23**(22), 4140–4148 (1984). [CrossRef] [PubMed]

*t*indicates the total (sum of the incident and scattered) field, and the superscript * represents the complex conjugate. The variables

*θ*and

*ϕ*represent the polar and azimuthal coordinates and as subscripts they represent the corresponding components of a vector field, respectively. On the other hand, the phase profile after the sphere can be obtained from the argument of the polar-angle component of the electric field when the profile is calculated at a large distance that the radial component can be ignored. The error of the Rytov approximation in predicting the intensity and phase profiles after a homogeneous sphere may be defined as where

*I*and

*φ*are the intensity and phase profiles after the sphere, respectively.

*Err*and

_{I}*Err*for different values of

_{φ}*V*. For less than 1% accuracy in intensity profile, one needs

*V*< 0.0049, while for the same accuracy in phase profile, one needs

*V*< 0.030. The validity condition is less stringent in case of estimating the phase profile than the intensity profile. However, the error quickly increases for a large value of

*V*.

*V*= 0.0027, and this corresponds to intensity error of 0.38% and phase error of 0.18%. Figure 4 compares the results obtained with the Rytov approximation and the Mie series for the model system (

*λ*= 0.256 nm,

*R*= 2.56

*μ*m, and

*z*= 32.2

*μ*m). Figures 4(c) and 4(d) are the cross-sections of Figs. 4(a) and 4(b), respectively. Figures 4(e,f) plot the difference in the intensity and profiles, respectively, provided by the Rytov approximation and the Mie series. Note that the center part of the profile in Fig. 4(c) is smaller than one due to the material absorption of the sphere, while the oscillation near the edges is the phase signature. Figure 4(e) shows that the Rytov approximation provides a more accurate profile in the center part than the edge regions.

*μ*m. However, in the inner part, the profile given by the Rytov approximation is shifted to the inside compared the Mie solution, which is responsible for the large error near the edge regions in Fig. 4(a). The reason of this shift is not clear.

*R*

_{max}, which renders the Rytov approximation to be valid within 1% accuracy. For the water sphere, we obtain

*R*

_{max}= 9.16 cm for the intensity measurement and

*R*

_{max}= 58.0 cm for the phase measurement. These values are reasonable for medical applications or luggage inspection.

## 5. Conclusion

*V*, which is defined in terms of the complex refractive index of sphere and the Fresnel number. In comparison with the exact Mie solution, we calculated the accuracy of the Rytov approximation for different values of

*V*. Using the principle of similarity, we estimated the maximum size of water sphere that can be accurately simulated with the first-order Rytov approximation.

## Acknowledgments

## References and links

1. | D. M. Paganin, |

2. | P. C. Diemoz, A. Bravin, and P. Coan, “Theoretical comparison of three X-ray phase-contrast imaging techniques: propagation-based imaging, analyzer-based imaging and grating interferometry,” Opt. Express |

3. | E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Kristall und Technik |

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

5. | M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys. |

6. | A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. |

7. | F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. |

8. | T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. |

9. | S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature |

10. | X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Methods Phys. Res. B |

11. | A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x‐ray phase contrast microimaging by coherent high‐energy synchrotron radiation,” Rev. Sci. Instrum. |

12. | E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. |

13. | A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. |

14. | J. B. Keller, “Accuracy and Validity of the Born and Rytov Approximations,” J. Opt. Soc. Am. |

15. | F. Lin and M. Fiddy, “The Born-Rytov controversy: I. Comparing analytical and approximate expressions for the one-dimensional deterministic case,” J. Opt. Soc. Am. A |

16. | E. Kirkinis, “Renormalization group interpretation of the Born and Rytov approximations,” J. Opt. Soc. Am. A |

17. | M. Slaney and A. Kak, |

18. | M. Born, E. Wolf, and A. B. Bhatia, |

19. | M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Tech. |

20. | Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express |

21. | E. M. Stein and G. L. Weiss, |

22. | C. F. Boliren and D. R. Huffman, |

23. | A. A. Neves and D. Pisignano, “Effect of finite terms on the truncation error of Mie series,” Opt. Lett. |

24. | W. G. Melbourne, |

25. | B. Henke, E. Gullikson, and J. C. X. Davis, “X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables |

26. | F. Slimani, G. Grehan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt. |

**OCIS Codes**

(340.7440) X-ray optics : X-ray imaging

(290.5825) Scattering : Scattering theory

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: November 30, 2012

Revised Manuscript: January 15, 2013

Manuscript Accepted: January 16, 2013

Published: January 28, 2013

**Virtual Issues**

Vol. 8, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Yongjin Sung and George Barbastathis, "Rytov approximation for x-ray phase imaging," Opt. Express **21**, 2674-2682 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-3-2674

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### References

- D. M. Paganin, Coherent X-ray Optics (Oxford University Press, 2006).
- P. C. Diemoz, A. Bravin, and P. Coan, “Theoretical comparison of three X-ray phase-contrast imaging techniques: propagation-based imaging, analyzer-based imaging and grating interferometry,” Opt. Express20(3), 2789–2805 (2012). [CrossRef] [PubMed]
- E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Kristall und Technik15(8), 937–945 (1980). [CrossRef]
- D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol.42(11), 2015–2025 (1997). [CrossRef] [PubMed]
- M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002). [CrossRef]
- A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006). [CrossRef]
- F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008). [CrossRef] [PubMed]
- T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol.68(3Suppl), S13–S17 (2008). [CrossRef] [PubMed]
- S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996). [CrossRef]
- X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Methods Phys. Res. B234(4), 563–572 (2005). [CrossRef]
- A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x‐ray phase contrast microimaging by coherent high‐energy synchrotron radiation,” Rev. Sci. Instrum.66(12), 5486–5492 (1995). [CrossRef]
- E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun.1(4), 153–156 (1969). [CrossRef]
- A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett.6(8), 374–376 (1981). [CrossRef] [PubMed]
- J. B. Keller, “Accuracy and Validity of the Born and Rytov Approximations,” J. Opt. Soc. Am.59, 1003–1004 (1969).
- F. Lin and M. Fiddy, “The Born-Rytov controversy: I. Comparing analytical and approximate expressions for the one-dimensional deterministic case,” J. Opt. Soc. Am. A9(7), 1102–1110 (1992). [CrossRef]
- E. Kirkinis, “Renormalization group interpretation of the Born and Rytov approximations,” J. Opt. Soc. Am. A25(10), 2499–2508 (2008). [CrossRef] [PubMed]
- M. Slaney and A. Kak, Principles of Computerized Tomographic imaging (SIAM, 1988).
- M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999).
- M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Tech.32(8), 860–874 (1984). [CrossRef]
- Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express17(1), 266–277 (2009). [CrossRef] [PubMed]
- E. M. Stein and G. L. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, 1971).
- C. F. Boliren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (J Wiley & Sons, 1983).
- A. A. Neves and D. Pisignano, “Effect of finite terms on the truncation error of Mie series,” Opt. Lett.37(12), 2418–2420 (2012). [CrossRef] [PubMed]
- W. G. Melbourne, Radio Occultations Using Earth Satellites: a Wave Theory Treatment (Wiley-Interscience, 2005).
- B. Henke, E. Gullikson, and J. C. X. Davis, “X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables54(2), 181–342 (1993). [CrossRef]
- F. Slimani, G. Grehan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt.23(22), 4140–4148 (1984). [CrossRef] [PubMed]

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