## Framework for computing the spatial coherence effects of polycapillary x-ray optics |

Optics Express, Vol. 20, Issue 4, pp. 3975-3982 (2012)

http://dx.doi.org/10.1364/OE.20.003975

Acrobat PDF (985 KB)

### Abstract

Despite the extensive use of polycapillary x-ray optics for focusing and collimating applications, there remains a significant need for characterization of the coherence properties of the output wavefield. In this work, we present the first quantitative computational method for calculation of the spatial coherence effects of polycapillary x-ray optical devices. This method employs the coherent mode decomposition of an extended x-ray source, geometric optical propagation of individual wavefield modes through a polycapillary device, output wavefield calculation by ray data resampling onto a uniform grid, and the calculation of spatial coherence properties by way of the spectral degree of coherence.

© 2012 OSA

## 1. Introduction

1. C. Schroer and B. Lengeler, “X-ray optics,” in *Springer Handbook of Lasers and Optics*, F. Träger, ed., (Springer-Verlag, 2007). [CrossRef]

2. C. A. MacDonald and W. M. Gibson, “Applications and advances in polycapillary optics,” X-ray Spectrom . **32**, 258–268 (2003). [CrossRef]

3. Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, and M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res., Sect. B **134**, 174–180 (1998). [CrossRef]

6. S. B. Dabagov, “Wave theory of x-ray scattering in capillary structures,” X-Ray Spectrom. **32**, 223–228 (2003). [CrossRef]

7. L. Vincze, K. Janssens, F. Adams, and A. Rindby, “Detained ray-tracing code for capillary optics,” X-Ray Spectrom. **24**, 27–37 (1995). [CrossRef]

9. S. V. Kukhlevsky, “Interference and diffraction in capillary x-ray optics,” X-Ray Spectrom. **32**, 223–228 (2003). [CrossRef]

10. Q. F. Xiao and S. V. Poturaev, “Polycapillary-based X-ray optics,” Nucl. Instrum. Methods Phys. Res., Sect. A **347**, 376–383 (1994). [CrossRef]

11. A. Liu, “The X-ray distribution after a focussing polycapillary a shadow simulation,” Nucl. Instrum. Methods Phys. Res., Sect. B **243**, 223–226 (2006). [CrossRef]

12. C. Welnak, G. J. Chen, and F. Cerrina, “Shadow: a synchrotron radiation and X-ray optics simulation tool,” Nucl. Instrum. Methods Phys. Res., Sect. A **347**, 344–347 (1994). [CrossRef]

13. D. Hampai, S. B. Dabagov, G. Cappuccio, and G. Cibin, “X-ray propagation through hollow channel: PolyCAD - a ray tracing code,” Nucl. Instrum. Methods Phys. Res., Sect. B **244**, 481–488 (2006). [CrossRef]

14. D. Hampai, S. B. Dabagov, G. Cappuccio, and G. Cibin, “X-ray propagation through polycapillary optics studied through a ray tracing approach,” Spectrochim. Acta, Part B **62**, 608–614 (2007). [CrossRef]

15. S. B. Dabagov, M. A. Kumakhov, S. V. Nikitina, V. A. Murashova, R. V. Fedorchuk, and M. N. Yakimenko, “Observation of interference effects at the focus of an x-ray lens,” J. Synchrotron Radiat. **2**, 132–135 (1995). [CrossRef] [PubMed]

18. A. Bjeoumikhov, S. Bjeoumikhova, H. Riesemeier, M. Radtke, and R. Wedell, “Propagation of synchrotron radiation through nanocapillary structures,” Phys. Lett. A **366**, 283–288 (2007). [CrossRef]

19. S. B. Dabagov, R. V. Fedorchuk, V. A. Murashova, S. V. Nikitina, and M. N. Yakimenko, “Interference phenomenon under focusing of synchrotron radiation by a Kumakhov lens,” Nucl. Instrum. Methods Phys. Res., Sect. B **108**, 213–218 (1996). [CrossRef]

20. L. Vincze, K. Janssens, and S. V. Kukhlevsky, “Simulation of polycapillary lenses for coherent and partially coherent x-rays,” Proc. SPIE **5536**, 81–85 (2004). [CrossRef]

## 2. Methods

### 2.1. Coherence theory

*Ũ*(

_{s}**r**,

*t*), are characterized by the autocorrelation function,

*W*(

_{s}**r**

_{1},

**r**

_{2},

*ω*) = ∫

*dτ*Γ

*(*

_{s}**r**

_{1},

**r**

_{2},

*τ*)

*e*, where

^{iωτ}*ω*is the temporal frequency. The cross-spectral density of stochastic, extended sources is Hermitian and thus has a Mercer expansion, or coherent mode decomposition (CMD) [22

22. E. Wolf, “New spectral representation of random sources and of the partially coherent field that they generate,” Opt. Commun. **38**, 3–6 (1981). [CrossRef]

^{*}indicates complex conjugation. Here, the cross-spectral density is equal to the incoherent sum of statistically independent modes,

*ϕ*(

_{n}**r**,

*ω*), each weighted by a quantity

*β*(

_{n}*ω*).

*Ũ*(

**r**,

*ω*), which is the propagated wavefield

*Ũ*(

_{s}**r**,

*ω*), is given by where

*ψ*(

_{n}**r**,

*ω*) are found by propagating

*ϕ*(

_{n}**r**,

*ω*). That is, the cross-spectral density of a propagated wavefield is the incoherent sum of the propagated modes of the wavefield at the source, multiplied by the original weights

*β*(

_{n}*ω*). The propagated modes can be found via the method of Green functions [23

23. H. Liu, G. Mu, and L. Lin, “Propagation theories of partially coherent electromagnetic fields based on coherent or separated-coordinate mode decomposition,” J. Opt. Soc. Am. A **23**, 2208–2218 (2006). [CrossRef]

24. A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. **95**, 043904 (2005). [CrossRef] [PubMed]

25. R. W. Schoonover, A. M. Zysk, and P. S. Carney, “Geometrical optics limit of stochastic electromagnetic fields,” Phys. Rev. A **77**, 043831 (2008). [CrossRef]

*μ*(

**r**

_{1},

**r**

_{2},

*ω*)| = 1, the wavefield is fully coherent between the two points. When |

*μ*(

**r**

_{1},

**r**

_{2},

*ω*)| = 0, the wavefield is completely incoherent between the two points.

*ϕ*(

_{n}**r**,

*ω*) in Eq. (1). (2) The wavefield from each point source, or mode, is propagated through the polycapillary device using geometrical ray tracing. (3) The wavefield on the output plane is constructed by combining and resampling the ray data. (4) The wavefield from the output plane is optionally propagated away from the polycapillary device. (5) The spectral degree of coherence is computed by way of the CMD in Eq. (2).

### 2.2. Modeling an extended source and polycapillary device

*ϕ*(

_{n}**r**,

*ω*) =

*δ*(

**r**–

**r**

*) and*

_{n}*S*(

*ω*) is the spectrum of the anode,

*ν*is the radius of the anode spot, and

**r**

*are the sampled points on the anode. Each source point, or mode, generates a spherical wave centered at*

_{n}**r**

*. An anode consisting of 49 source points laid out on a rectangular grid with radius*

_{n}*ν*= 2.1

*μ*m (corresponding to a full-width half maximum value of 5

*μ*m) and maximum radial anode position 6.4

*μ*m is simulated.

*U*(

**r**,

*ω*) =

*A*(

**r**)exp[

*iωS*(

**r**)/

*c*], where

*A*(

**r**), the amplitude, and

*S*(

**r**), the eikonal or optical path length, are independent of frequency, and

*c*is the speed of light.

*x*,

_{l}*y*) denote the center of the

_{l}*l*th capillary at the input,

*r*

_{0}is the radius of the capillary at the input, and

*f*(

_{l}*z*) and

*g*(

_{l}*z*) are second-order polynomials in

*z*chosen such that the capillary input face is normal to the device input focus location and the capillary exit plane is normal to the optical axis [26

26. A. Liu, “Simulation of x-ray propagation in a straight capillary,” Math. Comput. Simulat. **65**, 251–256 (2004). [CrossRef]

27. A. Liu, “Simulation of x-ray beam collimation by polycapillaries,” Nucl. Instrum. Methods Phys. Res., Sect. B **234**, 555–562 (2005). [CrossRef]

*ɛ*= 1 – 9.115 × 10

^{−6}+

*i*1.145 × 10

^{−7}at an energy of 8 keV [28

28. Q. Xiao, I. Ponomarev, A. Kolomitsev, and J. Kimball, “Numerical simulations for capillary-based x-ray optics,” Proc. SPIE **1736**, 227–238 (1992). [CrossRef]

*μ*m, an input focus location 3 cm from the input face, and a polycapillary device length of 23 cm. For each point source, or mode,

*N*= 1 × 10

^{7}ray trajectories are uniformly randomly angularly distributed to model an outgoing spherical wavefield centered at the point source location. Each ray, labeled

*m*, has initial amplitude

*A*= 1.

_{m}*R*. After each reflection, the ray amplitude is updated

*A*→

_{m}*A*. As each ray propagates through the capillary, the ray’s eikonal is tabulated. This process continues until the ray has traveled to the exit plane of the capillary. At the end of the computation, the exit plane location, eikonal, and amplitude of each ray is recorded. In this simulation, rays that do not enter a capillary are assumed to be lost from the system and rays are not allowed to move between capillaries.

_{m}R### 2.3. Calculation of the output wavefield

29. M. Popov, “A new method of computation of wave fields using gaussian beams,” Wave Motion **4**, 85–97 (1982). [CrossRef]

31. G. Forbes and M. Alonso, “Using rays better. I. theory for smoothly varying media,” J. Opt. Soc. Am. A **18**, 1132–1145 (2001). [CrossRef]

32. T. Heilpern, E. Heyman, and V. Timchenko, “A beam summation algorithm for wave radiation and guidance in stratified media,” J. Acoust. Soc. Am. **121**, 1856–1864 (2007). [CrossRef] [PubMed]

33. E. Svensson, “Gaussian beam summation in shallow waveguides,” Wave Motion **45**, 445–456 (2008). [CrossRef]

34. J. Jackson, C. Meyer, D. Nishimura, and A. Macovski, “Selection of a convolution function for fourier inversion using gridding [computerised tomography application],” IEEE Trans. Med. Imag. **10**, 473–478 (1991). [CrossRef]

34. J. Jackson, C. Meyer, D. Nishimura, and A. Macovski, “Selection of a convolution function for fourier inversion using gridding [computerised tomography application],” IEEE Trans. Med. Imag. **10**, 473–478 (1991). [CrossRef]

36. B. Hargreaves and P. Beatty, “Gridding functions,” http://mrsrl.stanford.edu/brian/gridding/.

*et al.*[34

34. J. Jackson, C. Meyer, D. Nishimura, and A. Macovski, “Selection of a convolution function for fourier inversion using gridding [computerised tomography application],” IEEE Trans. Med. Imag. **10**, 473–478 (1991). [CrossRef]

*ψ*(

_{n}**r**,

*ω*), of the propagated wavefield from the extended source which are combined in step 5 via the CMD in Eq. (2) to compute the spectral degree of coherence in Eq. (3).

### 2.4. Validation of wavefield generation method

*et al.*[8

8. S. V. Kukhlevsky, F. Flora, A. Marinai, G. Nyitray, A. Ritucci, L. Palladino, A. Reale, and G. Tomassetti, “Diffraction of X-ray beams in capillary waveguides,” Nucl. Instrum. Methods Phys. Res., Sect. B **168**, 276–282 (2000). [CrossRef]

_{2}device (length = 21 cm, plane separation = 300

*μ*m) illuminated by a quasi-monochromatic source (

*λ*= 20 nm, diameter = 50

*μ*m) [35

35. S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Ya. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzinni, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. E. Turcu, K. Vigli-Papadaki, and G. Schina, “Soft X-ray plasma source for atmospheric-pressure microscopy, radiobiology and other applications,” Il Nuovo Cimento **20**, 1685–1701 (1998).

*m*“field modes,” where each mode corresponds to the initial field having been reflected

*m*times.

8. S. V. Kukhlevsky, F. Flora, A. Marinai, G. Nyitray, A. Ritucci, L. Palladino, A. Reale, and G. Tomassetti, “Diffraction of X-ray beams in capillary waveguides,” Nucl. Instrum. Methods Phys. Res., Sect. B **168**, 276–282 (2000). [CrossRef]

*et al.*, the simulated field is reported to be consistent with one in which

*m*= 1. The field produced with the method in this paper also consists of

_{max}*m*= 1 reflections. It is also noteworthy that both simulations neglected the polychromaticity of the source [35

_{max}35. S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Ya. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzinni, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. E. Turcu, K. Vigli-Papadaki, and G. Schina, “Soft X-ray plasma source for atmospheric-pressure microscopy, radiobiology and other applications,” Il Nuovo Cimento **20**, 1685–1701 (1998).

## 3. Results and discussion

27. A. Liu, “Simulation of x-ray beam collimation by polycapillaries,” Nucl. Instrum. Methods Phys. Res., Sect. B **234**, 555–562 (2005). [CrossRef]

37. D. Hampai, S. B. Dabagov, G. Cappuccio, G. Cibin, and V. Sessa, “X-ray micro-imaging by capillary optics,” Spectrochim. Acta, Part B **64**, 1180–1184 (2009). [CrossRef]

**r**

_{1}in the output plane. The simulated optic has a highly structured coherence response that, in the output plane, is characterized by several “bright spots” corresponding to a high spatial degree of correlation with

**r**

_{1}. As the wavefields propagate away from the device, the structured spatial response begins to disperse, causing a more uniform effective spectral degree of coherence of approximately

*μ*

_{eff}= 0.3, especially when referenced to

**r**

_{1}= (0,0).

17. A. Bjeoumikhov, “Observation of peculiarities in angular distributions of X-ray radiation after propagation through polycapillary structures,” Phys. Lett. A **360**, 405–410 (2007). [CrossRef]

18. A. Bjeoumikhov, S. Bjeoumikhova, H. Riesemeier, M. Radtke, and R. Wedell, “Propagation of synchrotron radiation through nanocapillary structures,” Phys. Lett. A **366**, 283–288 (2007). [CrossRef]

8. S. V. Kukhlevsky, F. Flora, A. Marinai, G. Nyitray, A. Ritucci, L. Palladino, A. Reale, and G. Tomassetti, “Diffraction of X-ray beams in capillary waveguides,” Nucl. Instrum. Methods Phys. Res., Sect. B **168**, 276–282 (2000). [CrossRef]

16. S. B. Dabagov, M. A. Kumakhov, and S. V. Nikitina, “On the interference of X-rays in multiple reflection optics,” Phys. Lett. A **203**, 279–282 (1995). [CrossRef]

## Acknowledgments

## References and links

1. | C. Schroer and B. Lengeler, “X-ray optics,” in |

2. | C. A. MacDonald and W. M. Gibson, “Applications and advances in polycapillary optics,” X-ray Spectrom . |

3. | Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, and M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res., Sect. B |

4. | S. B. Dabagov and A. Marcelli, “Single-reflection regime of x rays that travel into a monocapillary,” Appl. Opt. |

5. | S. V. Kukhlevsky, F. Flora, A. Marinai, G. Nyitray, Zs. Kozma, A. Ritucci, L. Palladino, A. Reale, and G. Tomassetti, “Wave-optics treatment of x-rays passing through tapered capillary guides,” X-Ray Spectrom . |

6. | S. B. Dabagov, “Wave theory of x-ray scattering in capillary structures,” X-Ray Spectrom. |

7. | L. Vincze, K. Janssens, F. Adams, and A. Rindby, “Detained ray-tracing code for capillary optics,” X-Ray Spectrom. |

8. | S. V. Kukhlevsky, F. Flora, A. Marinai, G. Nyitray, A. Ritucci, L. Palladino, A. Reale, and G. Tomassetti, “Diffraction of X-ray beams in capillary waveguides,” Nucl. Instrum. Methods Phys. Res., Sect. B |

9. | S. V. Kukhlevsky, “Interference and diffraction in capillary x-ray optics,” X-Ray Spectrom. |

10. | Q. F. Xiao and S. V. Poturaev, “Polycapillary-based X-ray optics,” Nucl. Instrum. Methods Phys. Res., Sect. A |

11. | A. Liu, “The X-ray distribution after a focussing polycapillary a shadow simulation,” Nucl. Instrum. Methods Phys. Res., Sect. B |

12. | C. Welnak, G. J. Chen, and F. Cerrina, “Shadow: a synchrotron radiation and X-ray optics simulation tool,” Nucl. Instrum. Methods Phys. Res., Sect. A |

13. | D. Hampai, S. B. Dabagov, G. Cappuccio, and G. Cibin, “X-ray propagation through hollow channel: PolyCAD - a ray tracing code,” Nucl. Instrum. Methods Phys. Res., Sect. B |

14. | D. Hampai, S. B. Dabagov, G. Cappuccio, and G. Cibin, “X-ray propagation through polycapillary optics studied through a ray tracing approach,” Spectrochim. Acta, Part B |

15. | S. B. Dabagov, M. A. Kumakhov, S. V. Nikitina, V. A. Murashova, R. V. Fedorchuk, and M. N. Yakimenko, “Observation of interference effects at the focus of an x-ray lens,” J. Synchrotron Radiat. |

16. | S. B. Dabagov, M. A. Kumakhov, and S. V. Nikitina, “On the interference of X-rays in multiple reflection optics,” Phys. Lett. A |

17. | A. Bjeoumikhov, “Observation of peculiarities in angular distributions of X-ray radiation after propagation through polycapillary structures,” Phys. Lett. A |

18. | A. Bjeoumikhov, S. Bjeoumikhova, H. Riesemeier, M. Radtke, and R. Wedell, “Propagation of synchrotron radiation through nanocapillary structures,” Phys. Lett. A |

19. | S. B. Dabagov, R. V. Fedorchuk, V. A. Murashova, S. V. Nikitina, and M. N. Yakimenko, “Interference phenomenon under focusing of synchrotron radiation by a Kumakhov lens,” Nucl. Instrum. Methods Phys. Res., Sect. B |

20. | L. Vincze, K. Janssens, and S. V. Kukhlevsky, “Simulation of polycapillary lenses for coherent and partially coherent x-rays,” Proc. SPIE |

21. | L. Mandel and E. Wolf, |

22. | E. Wolf, “New spectral representation of random sources and of the partially coherent field that they generate,” Opt. Commun. |

23. | H. Liu, G. Mu, and L. Lin, “Propagation theories of partially coherent electromagnetic fields based on coherent or separated-coordinate mode decomposition,” J. Opt. Soc. Am. A |

24. | A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. |

25. | R. W. Schoonover, A. M. Zysk, and P. S. Carney, “Geometrical optics limit of stochastic electromagnetic fields,” Phys. Rev. A |

26. | A. Liu, “Simulation of x-ray propagation in a straight capillary,” Math. Comput. Simulat. |

27. | A. Liu, “Simulation of x-ray beam collimation by polycapillaries,” Nucl. Instrum. Methods Phys. Res., Sect. B |

28. | Q. Xiao, I. Ponomarev, A. Kolomitsev, and J. Kimball, “Numerical simulations for capillary-based x-ray optics,” Proc. SPIE |

29. | M. Popov, “A new method of computation of wave fields using gaussian beams,” Wave Motion |

30. | A. Norris, “Complex point-source representation of real point sources and the gaussian beam summation method,” J. Opt. Soc. Am. A |

31. | G. Forbes and M. Alonso, “Using rays better. I. theory for smoothly varying media,” J. Opt. Soc. Am. A |

32. | T. Heilpern, E. Heyman, and V. Timchenko, “A beam summation algorithm for wave radiation and guidance in stratified media,” J. Acoust. Soc. Am. |

33. | E. Svensson, “Gaussian beam summation in shallow waveguides,” Wave Motion |

34. | J. Jackson, C. Meyer, D. Nishimura, and A. Macovski, “Selection of a convolution function for fourier inversion using gridding [computerised tomography application],” IEEE Trans. Med. Imag. |

35. | S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Ya. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzinni, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. E. Turcu, K. Vigli-Papadaki, and G. Schina, “Soft X-ray plasma source for atmospheric-pressure microscopy, radiobiology and other applications,” Il Nuovo Cimento |

36. | B. Hargreaves and P. Beatty, “Gridding functions,” http://mrsrl.stanford.edu/brian/gridding/. |

37. | D. Hampai, S. B. Dabagov, G. Cappuccio, G. Cibin, and V. Sessa, “X-ray micro-imaging by capillary optics,” Spectrochim. Acta, Part B |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(340.0340) X-ray optics : X-ray optics

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: October 17, 2011

Revised Manuscript: January 13, 2012

Manuscript Accepted: January 19, 2012

Published: February 2, 2012

**Citation**

Adam M. Zysk, Robert W. Schoonover, Qiaofeng Xu, and Mark A. Anastasio, "Framework for computing the spatial coherence effects of polycapillary x-ray optics," Opt. Express **20**, 3975-3982 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3975

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### References

- C. Schroer and B. Lengeler, “X-ray optics,” in Springer Handbook of Lasers and Optics, F. Träger, ed., (Springer-Verlag, 2007). [CrossRef]
- C. A. MacDonald and W. M. Gibson, “Applications and advances in polycapillary optics,” X-ray Spectrom. 32, 258–268 (2003). [CrossRef]
- Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, and M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res., Sect. B 134, 174–180 (1998). [CrossRef]
- S. B. Dabagov and A. Marcelli, “Single-reflection regime of x rays that travel into a monocapillary,” Appl. Opt. 38, 7494–7497 (1999). [CrossRef]
- S. V. Kukhlevsky, F. Flora, A. Marinai, G. Nyitray, Zs. Kozma, A. Ritucci, L. Palladino, A. Reale, and G. Tomassetti, “Wave-optics treatment of x-rays passing through tapered capillary guides,” X-Ray Spectrom. 29, 354–359 (2000). [CrossRef]
- S. B. Dabagov, “Wave theory of x-ray scattering in capillary structures,” X-Ray Spectrom. 32, 223–228 (2003). [CrossRef]
- L. Vincze, K. Janssens, F. Adams, and A. Rindby, “Detained ray-tracing code for capillary optics,” X-Ray Spectrom. 24, 27–37 (1995). [CrossRef]
- S. V. Kukhlevsky, F. Flora, A. Marinai, G. Nyitray, A. Ritucci, L. Palladino, A. Reale, and G. Tomassetti, “Diffraction of X-ray beams in capillary waveguides,” Nucl. Instrum. Methods Phys. Res., Sect. B 168, 276–282 (2000). [CrossRef]
- S. V. Kukhlevsky, “Interference and diffraction in capillary x-ray optics,” X-Ray Spectrom. 32, 223–228 (2003). [CrossRef]
- Q. F. Xiao and S. V. Poturaev, “Polycapillary-based X-ray optics,” Nucl. Instrum. Methods Phys. Res., Sect. A 347, 376–383 (1994). [CrossRef]
- A. Liu, “The X-ray distribution after a focussing polycapillary a shadow simulation,” Nucl. Instrum. Methods Phys. Res., Sect. B 243, 223–226 (2006). [CrossRef]
- C. Welnak, G. J. Chen, and F. Cerrina, “Shadow: a synchrotron radiation and X-ray optics simulation tool,” Nucl. Instrum. Methods Phys. Res., Sect. A 347, 344–347 (1994). [CrossRef]
- D. Hampai, S. B. Dabagov, G. Cappuccio, and G. Cibin, “X-ray propagation through hollow channel: PolyCAD - a ray tracing code,” Nucl. Instrum. Methods Phys. Res., Sect. B 244, 481–488 (2006). [CrossRef]
- D. Hampai, S. B. Dabagov, G. Cappuccio, and G. Cibin, “X-ray propagation through polycapillary optics studied through a ray tracing approach,” Spectrochim. Acta, Part B 62, 608–614 (2007). [CrossRef]
- S. B. Dabagov, M. A. Kumakhov, S. V. Nikitina, V. A. Murashova, R. V. Fedorchuk, and M. N. Yakimenko, “Observation of interference effects at the focus of an x-ray lens,” J. Synchrotron Radiat. 2, 132–135 (1995). [CrossRef] [PubMed]
- S. B. Dabagov, M. A. Kumakhov, and S. V. Nikitina, “On the interference of X-rays in multiple reflection optics,” Phys. Lett. A 203, 279–282 (1995). [CrossRef]
- A. Bjeoumikhov, “Observation of peculiarities in angular distributions of X-ray radiation after propagation through polycapillary structures,” Phys. Lett. A 360, 405–410 (2007). [CrossRef]
- A. Bjeoumikhov, S. Bjeoumikhova, H. Riesemeier, M. Radtke, and R. Wedell, “Propagation of synchrotron radiation through nanocapillary structures,” Phys. Lett. A 366, 283–288 (2007). [CrossRef]
- S. B. Dabagov, R. V. Fedorchuk, V. A. Murashova, S. V. Nikitina, and M. N. Yakimenko, “Interference phenomenon under focusing of synchrotron radiation by a Kumakhov lens,” Nucl. Instrum. Methods Phys. Res., Sect. B 108, 213–218 (1996). [CrossRef]
- L. Vincze, K. Janssens, and S. V. Kukhlevsky, “Simulation of polycapillary lenses for coherent and partially coherent x-rays,” Proc. SPIE 5536, 81–85 (2004). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
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