## Micro-focusing of attosecond pulses by grazing-incidence toroidal mirrors |

Optics Express, Vol. 21, Issue 11, pp. 13040-13051 (2013)

http://dx.doi.org/10.1364/OE.21.013040

Acrobat PDF (1530 KB)

### Abstract

The design of optical systems for micro-focusing of extreme-ultraviolet (XUV) attosecond pulses through grazing-incidence toroidal mirrors is presented. Aim of the proposed configuration is to provide a micro-focused image through a high demagnification of the XUV source with the following characteristics: i) almost negligible aberrations; ii) long exit arm to easily accommodate at the output the experimental setups required for the applications of the focused attosecond pulses; iii) possibility to have an intermediate region where the XUV beam is collimated, in order to insert a plane split-mirror for the generation of two XUV pulse replicas to be used in a XUV-pump/XUV-probe setup. We present the analytical and numerical study of two optical configurations characterized by two sections based on the use of toroidal mirrors. The first section provides a demagnified image of the source in an intermediate focus that is free from defocusing but has a large coma aberration. The second section consists of a relay mirror that is mounted in Z-shaped geometry with respect to the previous one, in order to give a stigmatic image with a coma that is opposite to that provided by the first section. An example is provided to demonstrate the capability to achieve spot sizes in the 5-15 μm range with a demagnification higher than 10 in a compact envelope.

© 2013 OSA

## 1. Introduction

2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. **81**(1), 163–234 (2009). [CrossRef]

3. M. Nisoli and G. Sansone, “New frontiers in attosecond science,” Prog. Quantum Electron. **33**(1), 17–59 (2009). [CrossRef]

4. G. Sansone, L. Poletto, and M. Nisoli, “High-energy attosecond light sources,” Nat. Photonics **5**(11), 655–663 (2011). [CrossRef]

## 2. Analytical study of the configurations

14. T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, Y. Nabekawa, K. Yamanouchi, and K. Midorikawa, “Observation and analysis of an interferometric autocorrelation trace of an attosecond pulse train,” Phys. Rev. A **75**(3), 033817 (2007). [CrossRef]

15. A. M. Malvezzi, L. Garifo, and G. Tondello, “Grazing-incidence high-resolution stigmatic spectrograph with two optical elements,” Appl. Opt. **20**(14), 2560–2565 (1981). [CrossRef] [PubMed]

17. A. M. Malvezzi and G. Tondello, “Grazing incidence toroidal mirror pairs in imaging and spectroscopic applications,” Appl. Opt. **22**(16), 2444–2447 (1983). [CrossRef] [PubMed]

18. G. R. Bennett, “Advanced laser-backlit grazing-incidence x-ray imaging systems for inertial confinement fusion research. I. Design,” Appl. Opt. **40**(25), 4570–4587 (2001). [CrossRef] [PubMed]

19. G. R. Bennett and J. A. Folta, “Advanced laser-backlit grazing-incidence x-ray imaging systems for inertial confinement fusion research. I. Tolerance analysis,” Appl. Opt. **40**(25), 4588–4607 (2001). [CrossRef] [PubMed]

20. H. Haber, “The torus grating,” J. Opt. Soc. Am. **40**(3), 153–165 (1950). [CrossRef]

21. L. Poletto, P. Nicolosi, and G. Tondello, “Optical design of a stigmatic extreme-ultraviolet spectroscopic system for emission and absorption studies of laser-produced plasmas,” Appl. Opt. **41**(1), 172–181 (2002). [CrossRef] [PubMed]

*F*, of a ray emitted from the point source A, which passes through the focal point B after reflection at the point P(

*x*,

*y*,

*z*) on the mirror surface is defined as

*F*= <AP> + <PB>. Taking into account the equation of the toroidal surface, the distances <AP> and <PB> can be expressed as functions of the variables α,

*p*,

*q*,

*y*and

*z*, where α is the angle of incidence,

*p*and

*q*are the lengths of the entrance and exit arms respectively (i.e. the distances between A and the mirror center, O, and between O and B respectively),

*y*and

*z*span on the mirror surface. The light-path function is expressed as a power series of

*y*and

*z*:where the series has been truncated to the third-order terms. For a toroidal surface with tangential radius R and sagittal radius ρ, the F

_{ij}terms are:

*y*= δF/δ

*z*= 0, which must be satisfied simultaneously by any pair of

*y*and

*z*values. This is possible only if all F

_{ij}terms are equal to zero. The F

_{20}and F

_{02}terms control the tangential and sagittal defocusing respectively, which are the main optical aberrations to be cancelled. Therefore, in order to have stigmatic imaging, two conditions must be fulfilled: F

_{20}= 0 and F

_{02}= 0, which give:from which it is possible to calculate the mirror radii.

*F*give rise to the aberration terms. Indeed, since the partial derivatives have the geometrical significance of angles, the maximum tangential (

*y*) and sagittal (

*z*) displacements of the reflected rays from the true focus B can be calculated aswhere (2

*L*

_{tan}) × (2

*L*

_{sag}) is the illuminated area on the mirror surface. For the partial derivatives of order

*n*that do not vanish, these displacements correspond to aberrations of order

*n*in the focal plane. Since the second-order terms, namely the astigmatism, have been canceled by Eq. (6), the main residual aberrations are the third-order terms, namely the tangential and sagittal coma, which are controlled respectively by F

_{30}and F

_{12}. Let us indicate as M =

*p*/

*q*the ratio between the entrance and exit arms. For M > 1, the mirror is used to demagnify the source. The sizes of the illuminated portion on the mirror are

*L*

_{tan}= Dp/cosα,

*L*

_{sag}= Dp, where D is the half-divergence of the source. After some elaborations, the tangential ΔC

_{tan}and sagittal ΔC

_{sag}coma from a toroidal mirror in a stigmatic configuration are calculated from Eq. (7) as For a toroidal mirror with unity magnification in the so-called Rowland mounting, i.e.

*p*=

*q*= R cosα and Μ = 1, the coma aberration is fully canceled since both F

_{30}and F

_{12}are null. On the contrary, the higher the demagnification, the higher the coma, as stated by Eqs. (8) and (9).

_{1}and M

_{1}are respectively the entrance arm and the demagnification of MIR1.

_{1}is the entrance arm of MIR1, and M

_{1}is the demagnification, which is the ratio between the entrance arm of MIR1 and the exit arm of MIR2. Therefore, the two configurations have the same analytical formula for the aberrations at the output of the section 1.

_{2}and M

_{2}are respectively the output arm and the demagnification of section 2. The incidence angle α has been assumed to be the same for both sections.

_{tan,1}= ΔC

_{tan,2}, that iswhere the approximation holds for M

_{1}>>1, i.e. large demagnification.

*p*

_{1}, which is the distance between the attosecond source and the center of the first mirror, 2) the length of the output arm

*q*

_{2}, which is the distance between the center of the last mirror and the focal point in the center of the experimental chamber, and 3,4) the demagnifications of the two sections M

_{1}and M

_{2}, being M = M

_{1}⋅M

_{2}the global demagnification. Three of these parameters have to be fixed from the experimental requirements, typically p

_{1}, q

_{2}and M

_{1}, the forth parameter M

_{2}is calculated by Eq. (12).

## 3. Design of demagnifying configurations

*p*

_{1}≤ 1500 mm, to limit the overall size of the XUV beamline; (ii)

*q*

_{2}= 600 mm, to easily accommodate at the output various experimental setups (e.g., an electron time-of-flight, a velocity-map imaging spectrometer, etc.); (iii) M > 10, in order to achieve the required source demagnification. A device consisting of two plane mirrors mounted side by side is required to split spatially the XUV beam in two replicas, whose mutual delay can be changed by the linear translation of one of the two mirrors that is mounted on a piezo-actuator. For this purpose, configuration C2 is adopted. The optical layout of the beamline is shown in Fig. 3. The design parameters are resumed in Table 1. The incidence angle on the mirrors is chosen to be 80 deg to have high reflectivity in the 20-80 nm region.

_{1}= 10, to fulfill the requirement on total demagnification M > 10. Since

*p*

_{1}and α have been already chosen, the parameters of the mirrors of section 1, i.e. the tangential and sagittal radii, are univocally determined. M

_{2}is then calculated from Eq. (12), therefore also the parameters of the output mirror are determined since

*q*

_{2}is already given. Finally, a ray-tracing optimization of the output aberrations is performed by varying M

_{2}around the analytical solution. The simulations have been performed by a ray-tracing program written in the laboratory for applications to synchrotron radiation beamlines and modified by LP to calculate the length of the various ray trajectories in case of the use with ultrashort pulses. The results of the optimization procedure are shown in Fig. 4. The analytical solution gives M

_{2}= 1.12, the ray-tracing optimization around the condition calculated by the analytical formula gives M

_{2}= 1.10. The slightly different results are due to the fact that, while the analytical solution cancel only the coma, the ray-tracing procedure minimizes the total aberrations. Therefore, the analytical solution is assumed to be the starting point for the successive fine optimization. From the practical point of view, the difference in the calculated aberrations between the analytical and numerical solutions are almost unnoticeable. The optical parameters of the beamline as determined by the optimization procedure are listed in Table 2. Note that the three-mirrors-system here described is able to provide a demagnification of a factor 10, an intermediate section with a collimated beam and a 0.6-m exit arm in an envelope shorter than 3 m, indeed is suitable for table-top applications.

## 4. Temporal pulse distortion

## 5. Effects of the slope errors

23. L. Eybert, M. Wulff, W. Reichenbach, A. Plech, F. Schotte, E. Gagliardini, L. Zhang, O. Hignette, A. Rommeveaux, and A. Freund, “The toroidal mirror for single-pulse experiments on ID09B,” SPIE Proc. **4782**, 246–257 (2002). [CrossRef]

24. F. Siewert, J. Buchheim, S. Boutet, G. J. Williams, P. A. Montanez, J. Krzywinski, and R. Signorato, “Ultra-precise characterization of LCLS hard X-ray focusing mirrors by high resolution slope measuring deflectometry,” Opt. Express **20**(4), 4525–4536 (2012). [CrossRef] [PubMed]

## 6. Conclusions

## Acknowledgments

## References and links

1. | P. Jaeglè, |

2. | F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. |

3. | M. Nisoli and G. Sansone, “New frontiers in attosecond science,” Prog. Quantum Electron. |

4. | G. Sansone, L. Poletto, and M. Nisoli, “High-energy attosecond light sources,” Nat. Photonics |

5. | M. Schultze, M. Fiess, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, Th. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdörfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, and V. S. Yakovlev, “Delay in Photoemission,” Science |

6. | Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science |

7. | B. Mills, E. T. F. Rogers, J. Grant-Jacob, S. L. Stebbings, M. Praeger, A. M. de Paula, C. A. Froud, R. T. Chapman, T. J. Butcher, W. S. Brocklesby, and J. G. Frey, “EUV off-axis focussing using a high harmonic source,” Proc. SPIE |

8. | H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 10 |

9. | L. Poletto, S. Bonora, M. Pascolini, and P. Villoresi, “Instrumentation for analysis and utilization of extreme-ultraviolet and soft x-ray high-order harmonics,” Rev. Sci. Instrum. |

10. | C. Valentin, D. Douillet, S. Kazamias, Th. Lefrou, G. Grillon, F. Augé, G. Mullot, Ph. Balcou, P. Mercère, and Ph. Zeitoun, “Imaging and quality assessment of high-harmonic focal spots,” Opt. Lett. |

11. | H. Mashiko, A. Suda, and K. Midorikawa, “Focusing multiple high-order harmonics in the extreme-ultraviolet and soft-x-ray regions by a platinum-coated ellipsoidal mirror,” Appl. Opt. |

12. | C. Bourassin-Bouchet, S. de Rossi, F. Delmotte, and P. Chavel, “Spatiotemporal distortions of attosecond pulses,” J. Opt. Soc. Am. A |

13. | C. Bourassin-Bouchet, M. Stephens, S. de Rossi, F. Delmotte, and P. Chavel, “Duration of ultrashort pulses in the presence of spatio-temporal coupling,” Opt. Express |

14. | T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, Y. Nabekawa, K. Yamanouchi, and K. Midorikawa, “Observation and analysis of an interferometric autocorrelation trace of an attosecond pulse train,” Phys. Rev. A |

15. | A. M. Malvezzi, L. Garifo, and G. Tondello, “Grazing-incidence high-resolution stigmatic spectrograph with two optical elements,” Appl. Opt. |

16. | D. E. Aspnes, “Imaging performance of mirror pairs for grazing-incidence applications: a comparison,” Appl. Opt. |

17. | A. M. Malvezzi and G. Tondello, “Grazing incidence toroidal mirror pairs in imaging and spectroscopic applications,” Appl. Opt. |

18. | G. R. Bennett, “Advanced laser-backlit grazing-incidence x-ray imaging systems for inertial confinement fusion research. I. Design,” Appl. Opt. |

19. | G. R. Bennett and J. A. Folta, “Advanced laser-backlit grazing-incidence x-ray imaging systems for inertial confinement fusion research. I. Tolerance analysis,” Appl. Opt. |

20. | H. Haber, “The torus grating,” J. Opt. Soc. Am. |

21. | L. Poletto, P. Nicolosi, and G. Tondello, “Optical design of a stigmatic extreme-ultraviolet spectroscopic system for emission and absorption studies of laser-produced plasmas,” Appl. Opt. |

22. | W. J. Smith, |

23. | L. Eybert, M. Wulff, W. Reichenbach, A. Plech, F. Schotte, E. Gagliardini, L. Zhang, O. Hignette, A. Rommeveaux, and A. Freund, “The toroidal mirror for single-pulse experiments on ID09B,” SPIE Proc. |

24. | F. Siewert, J. Buchheim, S. Boutet, G. J. Williams, P. A. Montanez, J. Krzywinski, and R. Signorato, “Ultra-precise characterization of LCLS hard X-ray focusing mirrors by high resolution slope measuring deflectometry,” Opt. Express |

**OCIS Codes**

(080.1010) Geometric optics : Aberrations (global)

(320.7160) Ultrafast optics : Ultrafast technology

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

(080.4035) Geometric optics : Mirror system design

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: March 25, 2013

Revised Manuscript: April 26, 2013

Manuscript Accepted: May 6, 2013

Published: May 20, 2013

**Citation**

L. Poletto, F. Frassetto, F. Calegari, S. Anumula, A. Trabattoni, and M. Nisoli, "Micro-focusing of attosecond pulses by grazing-incidence toroidal mirrors," Opt. Express **21**, 13040-13051 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13040

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### References

- P. Jaeglè, Coherent Sources of XUV Radiation (Springer, 2006).
- F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys.81(1), 163–234 (2009). [CrossRef]
- M. Nisoli and G. Sansone, “New frontiers in attosecond science,” Prog. Quantum Electron.33(1), 17–59 (2009). [CrossRef]
- G. Sansone, L. Poletto, and M. Nisoli, “High-energy attosecond light sources,” Nat. Photonics5(11), 655–663 (2011). [CrossRef]
- M. Schultze, M. Fiess, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, Th. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdörfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, and V. S. Yakovlev, “Delay in Photoemission,” Science328(5986), 1658–1662 (2010). [CrossRef] [PubMed]
- Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science302(5650), 1540–1543 (2003). [CrossRef] [PubMed]
- B. Mills, E. T. F. Rogers, J. Grant-Jacob, S. L. Stebbings, M. Praeger, A. M. de Paula, C. A. Froud, R. T. Chapman, T. J. Butcher, W. S. Brocklesby, and J. G. Frey, “EUV off-axis focussing using a high harmonic source,” Proc. SPIE7360, 736003, 736003-12 (2009). [CrossRef]
- H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 1014 W/cm2.,” Opt. Lett.29(16), 1927–1929 (2004). [CrossRef] [PubMed]
- L. Poletto, S. Bonora, M. Pascolini, and P. Villoresi, “Instrumentation for analysis and utilization of extreme-ultraviolet and soft x-ray high-order harmonics,” Rev. Sci. Instrum.75(11), 4413 (2004). [CrossRef]
- C. Valentin, D. Douillet, S. Kazamias, Th. Lefrou, G. Grillon, F. Augé, G. Mullot, Ph. Balcou, P. Mercère, and Ph. Zeitoun, “Imaging and quality assessment of high-harmonic focal spots,” Opt. Lett.28(12), 1049–1051 (2003). [CrossRef] [PubMed]
- H. Mashiko, A. Suda, and K. Midorikawa, “Focusing multiple high-order harmonics in the extreme-ultraviolet and soft-x-ray regions by a platinum-coated ellipsoidal mirror,” Appl. Opt.45(3), 573–577 (2006). [CrossRef] [PubMed]
- C. Bourassin-Bouchet, S. de Rossi, F. Delmotte, and P. Chavel, “Spatiotemporal distortions of attosecond pulses,” J. Opt. Soc. Am. A27(6), 1395–1403 (2010). [CrossRef] [PubMed]
- C. Bourassin-Bouchet, M. Stephens, S. de Rossi, F. Delmotte, and P. Chavel, “Duration of ultrashort pulses in the presence of spatio-temporal coupling,” Opt. Express19(18), 17357–17371 (2011). [CrossRef] [PubMed]
- T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, Y. Nabekawa, K. Yamanouchi, and K. Midorikawa, “Observation and analysis of an interferometric autocorrelation trace of an attosecond pulse train,” Phys. Rev. A75(3), 033817 (2007). [CrossRef]
- A. M. Malvezzi, L. Garifo, and G. Tondello, “Grazing-incidence high-resolution stigmatic spectrograph with two optical elements,” Appl. Opt.20(14), 2560–2565 (1981). [CrossRef] [PubMed]
- D. E. Aspnes, “Imaging performance of mirror pairs for grazing-incidence applications: a comparison,” Appl. Opt.21(14), 2642–2646 (1982). [CrossRef] [PubMed]
- A. M. Malvezzi and G. Tondello, “Grazing incidence toroidal mirror pairs in imaging and spectroscopic applications,” Appl. Opt.22(16), 2444–2447 (1983). [CrossRef] [PubMed]
- G. R. Bennett, “Advanced laser-backlit grazing-incidence x-ray imaging systems for inertial confinement fusion research. I. Design,” Appl. Opt.40(25), 4570–4587 (2001). [CrossRef] [PubMed]
- G. R. Bennett and J. A. Folta, “Advanced laser-backlit grazing-incidence x-ray imaging systems for inertial confinement fusion research. I. Tolerance analysis,” Appl. Opt.40(25), 4588–4607 (2001). [CrossRef] [PubMed]
- H. Haber, “The torus grating,” J. Opt. Soc. Am.40(3), 153–165 (1950). [CrossRef]
- L. Poletto, P. Nicolosi, and G. Tondello, “Optical design of a stigmatic extreme-ultraviolet spectroscopic system for emission and absorption studies of laser-produced plasmas,” Appl. Opt.41(1), 172–181 (2002). [CrossRef] [PubMed]
- W. J. Smith, Modern Lens Design (McGraw Hill, 2005).
- L. Eybert, M. Wulff, W. Reichenbach, A. Plech, F. Schotte, E. Gagliardini, L. Zhang, O. Hignette, A. Rommeveaux, and A. Freund, “The toroidal mirror for single-pulse experiments on ID09B,” SPIE Proc. 4782, 246–257 (2002). [CrossRef]
- F. Siewert, J. Buchheim, S. Boutet, G. J. Williams, P. A. Montanez, J. Krzywinski, and R. Signorato, “Ultra-precise characterization of LCLS hard X-ray focusing mirrors by high resolution slope measuring deflectometry,” Opt. Express20(4), 4525–4536 (2012). [CrossRef] [PubMed]

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