## The theoretical analysis of the hard X-ray block-structure supermirror |

Optics Express, Vol. 21, Issue 7, pp. 8638-8651 (2013)

http://dx.doi.org/10.1364/OE.21.008638

Acrobat PDF (2609 KB)

### Abstract

We present an analytical study to provide guide lines to design a block structure hard X-ray supermirror. The block structure supermirror is a kind of layered structure consisting of several “blocks” of multilayer of different d-spacing to obtain broad energy bandwidth response. This structure has been widely applied in X-ray telescopes because it is easy to fabricate. To examine the propagation of X-rays in a supermirror structure, further simplified approximation of Kozhevnikov’s theory has been developed. The supermirror structure is described by a structure function. The spectral function of the structure, which is the Laplace transformation of the structure function, turns out to be proportional to the reflectivity profile against X-ray energy. By analyzing the expression of the spectral function, we found the reflectivity of the supermirror could be smooth due to the box-car shaped spectral function if the d-spacing and layer number of each block is arranged with appropriate constraints.

© 2013 OSA

## 1. Introduction

2. M. C. Weisskopf, H. D. Tananbaum, L. P. Van Speybroeck, and S. L. O'Dell, “Chandra x-ray observatory (CXO): overview,” proc. SPIE **4012**, 2–16 (2000). [CrossRef]

3. F. Jansen, D. Lumb, B. Altieri, J. Clavel, M. Ehle, C. Erd, C. Gabriel, M. Guainazzi, P. Gondoin, R. Much, R. Munoz, M. Santos, N. Schartel, D. Texier, and G. Vacanti, “XMM-Newton observatory,” Astron. Astrophys. **365**(1), L1–L6 (2001). [CrossRef]

^{44}is a critical clue for understanding the evolution of SNR [4]. A mirror coated with a single layer no longer meets the requirement in hard X-ray bands above 10 keV, because the critical angle of the hard X-ray becomes much smaller than for soft X-rays. This situation inevitably reduces the effective area of the telescope for hard X-rays. In order to overcome the disadvantage of the performance of single-layer coating, several telescopes with supermirror coating have been developed. For example, the InFOCμS balloon telescope [5

5. S. M. Owens, T. Okajima, Y. Ogasaka, F. Berendse, and P. J. Serlemitsos, “Multilayer coated thin foil mirrors for InFOCuS,” Proc. SPIE **4012**, 619–625 (2000). [CrossRef]

7. H. Kunieda, H. Awaki, A. Furuzawa, Y. Haba, R. Iizuka, K. Ishibashi, T. Miyaza, H. Mori, Y. Namba, Y. Ogasaka, K. Ogi, T. Okajima, Y. Suzuki, K. Tamura, Y. Tawara, K. Uesugi, K. Yamashita, and S. Yamauchi, “Hard X-ray Telescope to be onboard ASTRO-H,” Proc. SPIE **7732**, 773214, 773214-12 (2010). [CrossRef]

8. K. D. Joensen, P. Voutov, A. Szentgyorgyi, J. Roll, P. Gorenstein, P. Høghøj, and F. E. Christensen, “Design of grazing-incidence multilayer supermirrors for hard-x-ray reflectors,” Appl. Opt. **34**(34), 7935–7944 (1995). [CrossRef] [PubMed]

9. I. V. Kozhevnikov, I. N. Bukreeva, and E. Ziegler, “Design of X-ray supermirrors,” Nucl. Instrum. Methods Phys. Res. A **460**(2-3), 424–443 (2001). [CrossRef]

10. X. Cheng, Z. Wang, Z. Zhang, F. Wang, and L. Chen, “Design of X-ray super-mirrors using simulated annealing algorithm,” Opt. Commun. **265**(1), 197–206 (2006). [CrossRef]

11. C. Morawe, E. Ziegler, J. C. Peffen, and I. V. Kozhevnikov, “Design and fabrication of depth-graded x-ray multilayers,” Nucl. Instrum. Methods Phys. Res. A **493**(3), 189–198 (2002). [CrossRef]

12. H. Jiang, A. Michette, S. Pfauntsch, Z. Wang, J. Zhu, and D. Li, “Determination of the evolution of layer thickness errors and interfacial imperfections in ultrathin sputtered Cr/C multilayers using high-resolution transmission electron microscopy,” Opt. Express **19**(12), 11815–11824 (2011). [CrossRef] [PubMed]

13. K. Yamashita, H. Kunieda, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, Y. Hidaka, S. Ichimaru, S. Takahashi, A. Gotou, H. Kitou, T. Okajima, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard x-ray optics,” Proc. SPIE **3766**, 327–335 (1999). [CrossRef]

13. K. Yamashita, H. Kunieda, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, Y. Hidaka, S. Ichimaru, S. Takahashi, A. Gotou, H. Kitou, T. Okajima, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard x-ray optics,” Proc. SPIE **3766**, 327–335 (1999). [CrossRef]

16. I. V. Kozhevnikov and C. Montcalm, “Design of x-ray multilayer mirrors with maximal integral efficiency,” Nucl. Instrum. Methods Phys. Res. A **624**(1), 192–202 (2010). [CrossRef]

9. I. V. Kozhevnikov, I. N. Bukreeva, and E. Ziegler, “Design of X-ray supermirrors,” Nucl. Instrum. Methods Phys. Res. A **460**(2-3), 424–443 (2001). [CrossRef]

## 2. Mathematical description of the supermirror

9. I. V. Kozhevnikov, I. N. Bukreeva, and E. Ziegler, “Design of X-ray supermirrors,” Nucl. Instrum. Methods Phys. Res. A **460**(2-3), 424–443 (2001). [CrossRef]

**460**(2-3), 424–443 (2001). [CrossRef]

**460**(2-3), 424–443 (2001). [CrossRef]

**460**(2-3), 424–443 (2001). [CrossRef]

_{+}is the amplitude of the E-M wave propagating from the top to the bottom; and U

_{-}is the amplitude of the E-M wave propagating from the bottom to the top.

*k*is the wave number, which is equal to

_{n}and y(z) comes from Eq. (4), and L is the total thickness of the multilayer. S

_{+}(z) and S

_{-}(z) are the terms which may oscillate rapidly as was described in [9

**460**(2-3), 424–443 (2001). [CrossRef]

_{-}from the top to position z. U

_{-}is the amplitude of the E-M wave propagating upward. Equation (7) can be understood in the same way. Obviously, the reflectivity is equal to the amplitude U

_{-}(0) of the E-M wave on the top of the structure.

18. E. Spiller, “Characterization of multilayer coating by x-ray reflection,” Rev. Phys. Appl. (Paris) **23**(10), 1687–1700 (1988). [CrossRef]

_{+}(z)≈1. Though this approximation leads to overestimation of the reflectivity, the reflectivity profile can be reproduced qualitatively,

- 1. Amplitude reflectivity of each boundary is low for high energy X-rays at grazing angles larger than critical angles. Therefore the amplitude of the X-ray propagating in the structure is still comparable with unity.
- 2. The vector model is very simple and very suitable for qualitative analysis of the shape of the reflectivity (oscillation).
- 3. Reflectivity curves calculated by the exact method and the vector model will be compared with each other below to demonstrate the validity of this approximation.

_{-}(z) will be discussed later. Considering the fact that

## 3. Analysis of the block-structure supermirror

_{-}(0). The structure function of each block with periodic multilayer can be expressed as follows:

_{0}is the thickness of the block.

_{i}” stands for the structure function of i-th block of the supermirror.

_{0}’s structure function should be:

_{-}(0) of one block will be discussed as follows.

**460**(2-3), 424–443 (2001). [CrossRef]

_{-}(z) is derived as follows,While we apply the “vector model”, i.e. U

_{+}(z) = 1 and z = 0 to obtain the reflectivity, thenThe amplitude reflectivity is proportional to the Laplace transform of a cosine function which describes the varying of the dielectric constant. However, the textbook told us that the spectrum of the cosine function consists of a positive frequency component and a negative frequency component which are symmetrical with Y-axis.

_{0}. Considering that the tail of the negative component drops so much at the position of the peak of positive component, it is concluded that S

_{-}(0) can be neglected in our reflectivity expression.

*L*

_{i}is the total thickness of

*i*-th block. L

_{-1}is defined as 0, which means the surface of the first block is at 0 point. N is the number of blocks.

18. E. Spiller, “Characterization of multilayer coating by x-ray reflection,” Rev. Phys. Appl. (Paris) **23**(10), 1687–1700 (1988). [CrossRef]

_{0}, and the inverse lattice is in the arithmetic series with common difference of

_{0}and the inverse lattice between the adjacent block is

_{0}= 50nm). The layer number of each block is set incrementally from 17 to 24. The grazing angle is set at 0.28deg. Then, Eq. (23) has been calculated and the profile of the spectral function is shown as follows.

## 4. An example of the block structure supermirror in high energy band

- 1. Total number of the layer pairs is limited for ease of fabrication (N~150-200).
- 2. Roughness is set at 0.4nm, which corresponds to our fabrication ability and materials (Pt/C).
- 3. All the blocks are far from saturated (Details will be published in a future work).

- 1. The d-spacing of the first block is set at 2.94nm, corresponding to 45keV.
- 2. The number of the first block is arbitrarily set at 17. So, the thickness of each block is 50nm
- 3. Following our design rules introduced in Fig. 7, the d-spacing and layer pairs of all other blocks can be decided sequentially.
- 4. Add the blocks in the bottom of the structure until all the target band is covered.
- 5. If the total layer pairs N does not achieve 150-200, return to step 1 and add 1 layer pair to the first block and do step 2-4 and let the number of layer pairs meet the requirement. (In this example, the number of layer pairs is 168.)

8. K. D. Joensen, P. Voutov, A. Szentgyorgyi, J. Roll, P. Gorenstein, P. Høghøj, and F. E. Christensen, “Design of grazing-incidence multilayer supermirrors for hard-x-ray reflectors,” Appl. Opt. **34**(34), 7935–7944 (1995). [CrossRef] [PubMed]

_{i}is the discrete energy point defined within [45keV, 60keV]. R

_{0}is the target reflectivity which is set at 30% (without the roughness).

## 5. Conclusion

## References and links

1. | E. Spiller, |

2. | M. C. Weisskopf, H. D. Tananbaum, L. P. Van Speybroeck, and S. L. O'Dell, “Chandra x-ray observatory (CXO): overview,” proc. SPIE |

3. | F. Jansen, D. Lumb, B. Altieri, J. Clavel, M. Ehle, C. Erd, C. Gabriel, M. Guainazzi, P. Gondoin, R. Much, R. Munoz, M. Santos, N. Schartel, D. Texier, and G. Vacanti, “XMM-Newton observatory,” Astron. Astrophys. |

4. | A. F. Iyudin, R. Diehl, H. Bloemen, W. Hermsen, G. G. Lichti, D. Morris, J. Ryan, V. Schonfelder, H. Steinle, M. Varendorff, C. Vries, and C. Winkler, “COMPTEL observations of |

5. | S. M. Owens, T. Okajima, Y. Ogasaka, F. Berendse, and P. J. Serlemitsos, “Multilayer coated thin foil mirrors for InFOCuS,” Proc. SPIE |

6. | F. A. Harrison, F. E. Christensen, W. Craig, C. Hailey, W. Baumgartner, C. M. H. Chen, J. Chonko, W. R. Cook, J. Koglin, K. K. Madsen, M. Pivavoroff, S. Boggs, and D. Smith, “Development of the HEFT and NuSTAR focusing telescopes,” ExA |

7. | H. Kunieda, H. Awaki, A. Furuzawa, Y. Haba, R. Iizuka, K. Ishibashi, T. Miyaza, H. Mori, Y. Namba, Y. Ogasaka, K. Ogi, T. Okajima, Y. Suzuki, K. Tamura, Y. Tawara, K. Uesugi, K. Yamashita, and S. Yamauchi, “Hard X-ray Telescope to be onboard ASTRO-H,” Proc. SPIE |

8. | K. D. Joensen, P. Voutov, A. Szentgyorgyi, J. Roll, P. Gorenstein, P. Høghøj, and F. E. Christensen, “Design of grazing-incidence multilayer supermirrors for hard-x-ray reflectors,” Appl. Opt. |

9. | I. V. Kozhevnikov, I. N. Bukreeva, and E. Ziegler, “Design of X-ray supermirrors,” Nucl. Instrum. Methods Phys. Res. A |

10. | X. Cheng, Z. Wang, Z. Zhang, F. Wang, and L. Chen, “Design of X-ray super-mirrors using simulated annealing algorithm,” Opt. Commun. |

11. | C. Morawe, E. Ziegler, J. C. Peffen, and I. V. Kozhevnikov, “Design and fabrication of depth-graded x-ray multilayers,” Nucl. Instrum. Methods Phys. Res. A |

12. | H. Jiang, A. Michette, S. Pfauntsch, Z. Wang, J. Zhu, and D. Li, “Determination of the evolution of layer thickness errors and interfacial imperfections in ultrathin sputtered Cr/C multilayers using high-resolution transmission electron microscopy,” Opt. Express |

13. | K. Yamashita, H. Kunieda, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, Y. Hidaka, S. Ichimaru, S. Takahashi, A. Gotou, H. Kitou, T. Okajima, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard x-ray optics,” Proc. SPIE |

14. | K. Yamashita, “Development of Pt/C multilayer supermirrors for hard x-ray optics,” Nucl. Instrum. Methods Phys. Res. A |

15. | Y. Tawara, K. Yamashita, H. Kunieda, K. Tamura, A. Furuzawa, K. Haga, N. Nakajo, T. Okajima, H. Takata, P. J. Serlemitsos, J. Tueller, R. Petre, S. Yang, K. W. Chan, G. S. Lodha, Y. Namba, and J. Yu, “Development of a multilayer supermirror for hard x-ray telescopes,” Proc. SPIE |

16. | I. V. Kozhevnikov and C. Montcalm, “Design of x-ray multilayer mirrors with maximal integral efficiency,” Nucl. Instrum. Methods Phys. Res. A |

17. | E. Spiller, |

18. | E. Spiller, “Characterization of multilayer coating by x-ray reflection,” Rev. Phys. Appl. (Paris) |

19. | A. R. S. Bahai, B. R. Saltzber, and M. Ergen, |

20. | D. L. Windt, “IMD—software for modeling the optical properties of multilayer films,” Comput. Phys. |

**OCIS Codes**

(340.6720) X-ray optics : Synchrotron radiation

(340.7470) X-ray optics : X-ray mirrors

(350.1260) Other areas of optics : Astronomical optics

(310.4165) Thin films : Multilayer design

(310.6805) Thin films : Theory and design

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: February 5, 2013

Revised Manuscript: March 13, 2013

Manuscript Accepted: March 16, 2013

Published: April 2, 2013

**Citation**

Youwei Yao, Hideyo Kunieda, and Zhanshan Wang, "The theoretical analysis of the hard X-ray block-structure supermirror," Opt. Express **21**, 8638-8651 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8638

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### References

- E. Spiller, Soft X-ray Optics (SPIE, 1994), Chap. 4.
- M. C. Weisskopf, H. D. Tananbaum, L. P. Van Speybroeck, and S. L. O'Dell, “Chandra x-ray observatory (CXO): overview,” proc. SPIE4012, 2–16 (2000). [CrossRef]
- F. Jansen, D. Lumb, B. Altieri, J. Clavel, M. Ehle, C. Erd, C. Gabriel, M. Guainazzi, P. Gondoin, R. Much, R. Munoz, M. Santos, N. Schartel, D. Texier, and G. Vacanti, “XMM-Newton observatory,” Astron. Astrophys.365(1), L1–L6 (2001). [CrossRef]
- A. F. Iyudin, R. Diehl, H. Bloemen, W. Hermsen, G. G. Lichti, D. Morris, J. Ryan, V. Schonfelder, H. Steinle, M. Varendorff, C. Vries, and C. Winkler, “COMPTEL observations of 44Ti gamma-ray line emission from Cas A,” Astron. Astrophys.284, L1–L4 (1994).
- S. M. Owens, T. Okajima, Y. Ogasaka, F. Berendse, and P. J. Serlemitsos, “Multilayer coated thin foil mirrors for InFOCuS,” Proc. SPIE4012, 619–625 (2000). [CrossRef]
- F. A. Harrison, F. E. Christensen, W. Craig, C. Hailey, W. Baumgartner, C. M. H. Chen, J. Chonko, W. R. Cook, J. Koglin, K. K. Madsen, M. Pivavoroff, S. Boggs, and D. Smith, “Development of the HEFT and NuSTAR focusing telescopes,” ExA20, 131–137 (2005).
- H. Kunieda, H. Awaki, A. Furuzawa, Y. Haba, R. Iizuka, K. Ishibashi, T. Miyaza, H. Mori, Y. Namba, Y. Ogasaka, K. Ogi, T. Okajima, Y. Suzuki, K. Tamura, Y. Tawara, K. Uesugi, K. Yamashita, and S. Yamauchi, “Hard X-ray Telescope to be onboard ASTRO-H,” Proc. SPIE7732, 773214, 773214-12 (2010). [CrossRef]
- K. D. Joensen, P. Voutov, A. Szentgyorgyi, J. Roll, P. Gorenstein, P. Høghøj, and F. E. Christensen, “Design of grazing-incidence multilayer supermirrors for hard-x-ray reflectors,” Appl. Opt.34(34), 7935–7944 (1995). [CrossRef] [PubMed]
- I. V. Kozhevnikov, I. N. Bukreeva, and E. Ziegler, “Design of X-ray supermirrors,” Nucl. Instrum. Methods Phys. Res. A460(2-3), 424–443 (2001). [CrossRef]
- X. Cheng, Z. Wang, Z. Zhang, F. Wang, and L. Chen, “Design of X-ray super-mirrors using simulated annealing algorithm,” Opt. Commun.265(1), 197–206 (2006). [CrossRef]
- C. Morawe, E. Ziegler, J. C. Peffen, and I. V. Kozhevnikov, “Design and fabrication of depth-graded x-ray multilayers,” Nucl. Instrum. Methods Phys. Res. A493(3), 189–198 (2002). [CrossRef]
- H. Jiang, A. Michette, S. Pfauntsch, Z. Wang, J. Zhu, and D. Li, “Determination of the evolution of layer thickness errors and interfacial imperfections in ultrathin sputtered Cr/C multilayers using high-resolution transmission electron microscopy,” Opt. Express19(12), 11815–11824 (2011). [CrossRef] [PubMed]
- K. Yamashita, H. Kunieda, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, Y. Hidaka, S. Ichimaru, S. Takahashi, A. Gotou, H. Kitou, T. Okajima, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard x-ray optics,” Proc. SPIE3766, 327–335 (1999). [CrossRef]
- K. Yamashita, “Development of Pt/C multilayer supermirrors for hard x-ray optics,” Nucl. Instrum. Methods Phys. Res. A529(1-3), 59–62 (2004). [CrossRef]
- Y. Tawara, K. Yamashita, H. Kunieda, K. Tamura, A. Furuzawa, K. Haga, N. Nakajo, T. Okajima, H. Takata, P. J. Serlemitsos, J. Tueller, R. Petre, S. Yang, K. W. Chan, G. S. Lodha, Y. Namba, and J. Yu, “Development of a multilayer supermirror for hard x-ray telescopes,” Proc. SPIE3444, 569–575 (1998). [CrossRef]
- I. V. Kozhevnikov and C. Montcalm, “Design of x-ray multilayer mirrors with maximal integral efficiency,” Nucl. Instrum. Methods Phys. Res. A624(1), 192–202 (2010). [CrossRef]
- E. Spiller, Soft X-ray Optics (SPIE, 1994), Chap. 7.
- E. Spiller, “Characterization of multilayer coating by x-ray reflection,” Rev. Phys. Appl. (Paris)23(10), 1687–1700 (1988). [CrossRef]
- A. R. S. Bahai, B. R. Saltzber, and M. Ergen, Multi-Carrier Digital Communications Theory and Applications of OFDM (Sprigner, 2004), Chap. 1.
- D. L. Windt, “IMD—software for modeling the optical properties of multilayer films,” Comput. Phys.12(4), 360–370 (1998). [CrossRef]

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