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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 11 — Jun. 3, 2013
  • pp: 13040–13051
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Micro-focusing of attosecond pulses by grazing-incidence toroidal mirrors

L. Poletto, F. Frassetto, F. Calegari, S. Anumula, A. Trabattoni, and M. Nisoli  »View Author Affiliations


Optics Express, Vol. 21, Issue 11, pp. 13040-13051 (2013)
http://dx.doi.org/10.1364/OE.21.013040


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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

High-order harmonic generation (HHG) through the interaction of an intense femtosecond laser pulses with a gas, is widely used to produce coherent, brilliant, ultrashort table-top sources in the extreme ultraviolet (XUV) and soft x-ray spectral regions [1

1. P. Jaeglè, Coherent Sources of XUV Radiation (Springer, 2006).

]. In the last decade, the radiation produced by HHG using few-optical-cycles driving pulses has become a very important tool for the investigation of physical and chemical processes in atoms, molecules and solids with sub-femtosecond resolution [2

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

, 3

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

]. The access to this unexplored temporal domain opens new frontiers in atomic, molecular and solid-state science.

Although many applications of attosecond pulses have been demonstrated, the widespread adoption of attosecond sources is still limited, not only for the complexity of the laser tools required for a reliable generation, but also for the low photon flux characteristics of typical attosecond sources, particularly in the case of isolated pulses. Instead many experiments demand for high photon flux and tight focusing, such as attosecond pump–probe spectroscopy, which requires high intensities to produce observable two-XUV-photon processes [4

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

]. Therefore, tight focusing of the XUV radiation onto the experimental target is a general requirement for the success of a large class of experiments.

The geometry of the optical system demanded to focus attosecond pulses can be very different depending on the requirements on the XUV intensity on the target. Both multilayer-coated spherical mirrors [5

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 328(5986), 1658–1662 (2010). [CrossRef] [PubMed]

] and grazing-incidence toroidal mirrors [6

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 302(5650), 1540–1543 (2003). [CrossRef] [PubMed]

] have been used. Multilayer-coated mirrors are operated at near normal-incidence to manage high-aperture beams with low aberrations. In the case of a spherical mirror with incidence ae of a few degrees, the main residual aberration is the astigmatism [7

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 7360, 736003, 736003-12 (2009). [CrossRef]

]. Spot sizes of the order of a few tens of micrometres can be easily achieved. The aberrations can be fully eliminated by using a Cartesian surface. Spot sizes smaller than 1 μm have been obtained by using a SiC-Mg off-axis parabola with a short (6 cm) focal length, thus producing an intensity of 1014 W/cm2 [8

8. 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]

]. On the other hand, the focusing in a broad and tunable band requires the use of grazing-incidence mirrors. The simplest way to focus the beam with a single element is the use of a toroidal mirror with unity magnification (Rowland configuration), which cancels the astigmatism and the coma: this is the operative condition of many beamlines [9

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. 75(11), 4413 (2004). [CrossRef]

]. When a strong demagnification is required, coma aberration is unavoidable. Spot sizes of 7 μm in diameter and corresponding XUV intensity of 1012 W/cm2 have been reported, but at the expense of a strong reduction of the XUV beam aperture [10

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. 28(12), 1049–1051 (2003). [CrossRef] [PubMed]

]. The aberrations can be almost eliminated by using an ellipsoidal surface at grazing incidence. Spot sizes of 2.4 μm and intensities of 6 × 1013 W/cm2 in the 25–40 nm region have been reported using a platinum-coated ellipsoidal mirror at moderate grazing incidence (60°) [11

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. 45(3), 573–577 (2006). [CrossRef] [PubMed]

]. It is worth to point out that the residual geometric aberrations can significantly stretch the attosecond pulses [12

12. C. Bourassin-Bouchet, S. de Rossi, F. Delmotte, and P. Chavel, “Spatiotemporal distortions of attosecond pulses,” J. Opt. Soc. Am. A 27(6), 1395–1403 (2010). [CrossRef] [PubMed]

, 13

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 19(18), 17357–17371 (2011). [CrossRef] [PubMed]

].

In this work we present the design of optical systems for micro-focusing of XUV attosecond pulses, based on the use of grazing-incidence toroidal mirrors. The proposed configurations present the following characteristics: i) generation of a XUV spot-size in the range 5-15 µm, with almost negligible aberrations and ii) long exit arm suitable to accommodate various experimental setups in the beam focus (e.g., a time-of-flight spectrometer, a velocity map imaging detector, etc.), which can be used for the application of attosecond pulses. Moreover, the use of grazing-incidence metallic-coated mirrors gives high flexibility in the selection of the spectral region of operation, with almost constant reflectivity in the whole bandwidth. Toroidal mirrors are a cheaper alternative to the use of more expensive Cartesian surfaces (i.e. elliptical and parabolic), with the drawback of high aberrations when used to provide high demagnification, since in this case they have to be operated highly off-Rowland. Furthermore, the drawback of any configuration with high demagnification based on the use of a single mirror is given by the short length of the exit arm: indeed, once the entrance arm has been fixed, the higher the demagnification the shorter the exit arm.

2. Analytical study of the configurations

Figure 1
Fig. 1 Configurations for micro-focusing with toroidal mirrors: a) two mirrors, intermediate focus between MIR1 and MIR2, section 1 consists of MIR1 and section 2 of MIR2; b) three mirrors, parallel beam between Mir1 and MIR2, intermediate focus between MIR2 and MIR3, section 1 consists of MIR1 and MIR2 and section 2 of MIR3.
shows the optical configurations discussed in this work. The first configuration (C1, see Fig. 1(a)) consists of two grazing-incidence toroidal mirrors: MIR1 provides the high demagnification, MIR2 is the relay section required to increase the length of the exit arm and to compensate for the high-order aberrations introduced by MIR1. C1 can be used to provide a micro-focusing image of the XUV attosecond source leaving enough space at the output for the accommodation of various measurement setups. The second configuration (C2, see Fig. 1(b)) consists of three grazing-incidence toroidal mirrors. This scheme is particularly suitable for the realization of the experimental setup for XUV-pump/XUV-probe measurements, where two XUV pulse replicas are generated by using a plane split-mirror [14

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]

], which must be placed where the XUV beam is collimated. Indeed, only in this case the two pulse replicas can be focused in the same position irrespective of their relative temporal delay. On the contrary, if the split-mirror is inserted in a divergent/convergent beam, the focal spot of the replica that is delayed moves with the delay, i.e. with the displacement of the plane mirror that is translated. This requires to replace the first single-mirror section of C1 with a double-mirror section, as shown in Fig. 1(b): MIR1 is the collimating mirror, MIR2 is the condenser mirror to focus and demagnify the two replicas after the splitting, MIR3 is the relay section. The split-mirror, not shown in Fig. 1(b), is accommodated between MIR1 and MIR2.

The main characteristic of both configurations is the use of a pair of toroidal mirrors to compensate for the high-order aberrations that would be unacceptable for a single-mirror configuration. Aim of the optical design is to identify a configuration in which the aberrations of the mirrors compensate reciprocally. The distinct possible configurations with toroidal mirrors have been investigated by calculating their aberration patterns and the best performance for imaging applications have been obtained for a Z-shaped geometry where the two mirrors deflect in opposite directions across a common interior sagittal and tangential focus [15

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

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]

]. However, the application of such a configuration for high demagnification has not been studied. The combined use of grazing-incidence spherical mirrors to realize Kirkpatrick-Baez compound microscopes has been later analytically and numerically studied [18

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

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]

], although the system has not been extended to the use of toroidal surfaces. We provide here an analytical treatment of both configurations, which is based on the geometric approach that has been introduced for a toroidal grating [20

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

] and successively used to calculate the tangential coma given by a spherical mirror in grazing incidence [21

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]

].

We have first calculated the aberrations introduced by a single toroidal mirror. The optical layout is shown in Fig. 2
Fig. 2 Optical layout of a toroidal mirror.
. The light-path function, 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:
F=p+q+F20y2+F02z2+F30y3+F12yz2+O(y4,z4).      
(1)
where the series has been truncated to the third-order terms. For a toroidal surface with tangential radius R and sagittal radius ρ, the Fij terms are:

F20=12cos2α(1p+1q2Rcosα).
(2)
F02=12(1p+1q2cosαρ).
(3)
F30=12sinαcosα[1p(cosαp1R)1q(cosαq1R)].
(4)
F12=12sinα[1p(1pcosαρ)1q(1qcosαρ)].
(5)

According to Fermat’s principle of least time, point B is located such that F will be an extreme for any point P. Since points A and B are fixed while point P can be any point on the surface of the mirror, aberration-free image focusing is obtained by the conditions δF/δy = δF/δz = 0, which must be satisfied simultaneously by any pair of y and z values. This is possible only if all Fij terms are equal to zero. The F20 and F02 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: F20 = 0 and F02 = 0, which give:
1p+1q=2Rcosα=2cosαρ.
(6)
from which it is possible to calculate the mirror radii.

The remaining parts of the derivatives of the optical path function 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 as
Δtan=qcosαFy|y=Ltan,z=Lsag,Δsag=qFy|y=Ltan,z=Lsag.
(7)
where (2Ltan) × (2Lsag) 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 F30 and F12. 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 Ltan = Dp/cosα, Lsag = Dp, where D is the half-divergence of the source. After some elaborations, the tangential ΔCtan and sagittal ΔCsag coma from a toroidal mirror in a stigmatic configuration are calculated from Eq. (7) as
ΔCtan=3qcosαF30Ltan2=34pD2M21Mtanα.
(8)
ΔCsag=2qF12LtanLsag=12pD2M21Mtanα=23ΔCtan.
(9)
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 F30 and F12 are null. On the contrary, the higher the demagnification, the higher the coma, as stated by Eqs. (8) and (9).

Let us first analyze configuration C1. The coma in the intermediate focus between MIR1 and MIR2 is expressed as
ΔCtan,1=34p1D2M121M1tanα,ΔCsag,1=23ΔCtan,1.
(10)
where p1 and M1 are respectively the entrance arm and the demagnification of MIR1.

In case of configuration C2, with a parallel beam between the two mirrors MIR1 and MIR2 of the section 1, the coma given by the two mirrors, that is calculated by Eqs. (4) and (5), can be combined, the first mirror having a parallel output, the second one a parallel input. The coma at the output of the section 1 is finally given again by Eq. (10), where p1 is the entrance arm of MIR1, and M1 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.

After some elaborations, the coma at the output of the section 2 is expressed for both configurations as
ΔCtan,2=34q2D2M12(M221)tanα,ΔCsag,2=23ΔCtan,2.
(11)
where q2 and M2 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.

The coma compensation is possible if the mirrors of the two sections, i.e. MIR1 and MIR2 in case of C1 and MIR2 and MIR3 in case of C2, are mounted in a Z-shaped geometry to deflect in opposite directions across a common interior focus. Since the comas of the two section must assume the same absolute value, although in two opposite directions, the condition to cancel the global coma is ΔCtan,1 = ΔCtan,2, that is
M2=1+p1q2M121M131+p1q2M1.
(12)
where the approximation holds for M1 >>1, i.e. large demagnification.

Four are the free parameters to define the configurations: 1) the length of the entrance arm p1, which is the distance between the attosecond source and the center of the first mirror, 2) the length of the output arm q2, 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 M1 and M2, being M = M1⋅M2 the global demagnification. Three of these parameters have to be fixed from the experimental requirements, typically p1, q2 and M1, the forth parameter M2 is calculated by Eq. (12).

3. Design of demagnifying configurations

In the following we will consider high-energy attosecond pulses produced by loosely focusing IR driving pulses in a gas cell. Since the spot size of the laser focus can be as large as a few hundreds of microns, also the XUV source size can be relatively large and the divergence quite low. We will assume the following design constraints: (i) p1 ≤ 1500 mm, to limit the overall size of the XUV beamline; (ii) q2 = 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
Fig. 3 Optical layout of the beamline for XUV-pump/XUV-probe attosecond measurements: a) beamline; b-c) view of the split-mirror, the two delayed replicas are indicated in red and blue. An example of realization of the splitting mirror is discussed by Shimitzu et al [14].
. The design parameters are resumed in Table 1

Table 1. Design parameters of the XUV-XUV attosecond beamline analyzed in the text

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. The incidence angle on the mirrors is chosen to be 80 deg to have high reflectivity in the 20-80 nm region.

The design procedure is described in the following. The demagnification of section 1 is chosen to be M1 = 10, to fulfill the requirement on total demagnification M > 10. Since p1 and α have been already chosen, the parameters of the mirrors of section 1, i.e. the tangential and sagittal radii, are univocally determined. M2 is then calculated from Eq. (12), therefore also the parameters of the output mirror are determined since q2 is already given. Finally, a ray-tracing optimization of the output aberrations is performed by varying M2 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
Fig. 4 Aberrations at the output of the beamline as a function of the demagnification of section 2, as calculated by the ray-tracing optimization procedure around the analytical solution given by Eq. (12). To calculate the aberrations, a point-like source having the same divergence of the actual source is placed at the input of the beamline. The aberrations are then defined as (σtan2 + σsag2)1/2, where σtan and σsag are the standard deviations of the output spot respectively along the tangential and sagittal directions.
. The analytical solution gives M2 = 1.12, the ray-tracing optimization around the condition calculated by the analytical formula gives M2 = 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

Table 2. Optical parameters of the XUV-XUV attosecond beamline as determined by the optimization

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. 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.

The ray-traced spots and the beam profile in the intermediate focus and at the output are shown in Fig. 5
Fig. 5 Spot size in the intermediate focus (a) and at the output (b) of the XUV-XUV beamline.
, Fig. 6
Fig. 6 Spot profile in the intermediate focus (i.e. at the output of the section 1): a) tangential; b) sagittal. The coma aberration is clearly evident particularly in the tangential direction.
and Fig. 7
Fig. 7 Spot profile in the output plane (i.e. at the output of the section 2): a) tangential; b) sagittal. The coma aberration is almost totally canceled in both directions.
. The spot in the intermediate focus suffers from a high residual coma, particularly in the tangential direction, which is almost completely compensated for by the mirror of section 2. The beam size in the intermediate focus is 20 μm FWHM and is reduced to 11 μm FWHM in the output plane, well within the design requirements. Beyond the noticeable reduction of the spot size, the space available at the output is increased from 150 mm at the output of section 1 to 600 mm at the output of section 2. This confirms the advantages of the double-section configuration to provide both a micro-focused image almost completely coma-free and a long exit arm suitable to accommodate an additional experimental chamber for applications.

4. Temporal pulse distortion

Since the proposed configurations are specifically designed for micro-focusing of attosecond pulses, the effects of the residual aberrations on the variations of optical path length across the beam and the corresponding temporal distortion have to be carefully evaluated. We have used a geometric approach, since the changes in path lengths in the focal spot are simulated by ray-tracing. The temporal dispersion of the optical paths in the intermediate focus is 700 as FWHM, corresponding to 0.21-μm-FWHM spatial dispersion on the optical paths: this is due to the large coma at the output of section 1 and cannot be accepted for attosecond resolution. The dispersion at the output of the beamline as a function of the demagnification of section 2 is shown in Fig. 8
Fig. 8 Temporal dispersion of the optical paths at the output of the beamline due to the residual aberrations. The dispersion in the intermediate focus is 700 as FWHM.
, as calculated by ray-tracing simulations around the analytical solution.

Due to the compensation of the aberrations, the temporal dispersion at the output is considerably reduced with respect to the intermediate focus. The minimum temporal dispersion is 165 as FWHM, corresponding to 0.05-μm-FWHM spatial dispersion, and it is achieved for the optimum solution as already calculated in Fig. 4. As already pointed out, the optimum numerical solution is slightly better than the analytical one since it minimizes the global aberrations of the optical system, not only the coma. The temporal dispersion of the analytical solution is slightly higher than the numerical one, being the former 230 as FWHM.

5. Effects of the slope errors

Figures 9(a)
Fig. 9 Aberrations (a) and temporal dispersion (b) as a function of the slope errors of the three mirrors of the beamline discussed in the text.
and 9(b) show the calculated aberration and temporal dispersion, respectively, as a function of the slope errors of the three mirrors used in the proposed beamline. The calculations have been performed by ray-tracing simulations. The effect of the mirrors of section 1 is almost negligible for slope errors as high as 5 μrad rms, definitively higher than the small values of slope errors that are expected on the reduced illuminated area (about 15 mm × 3 mm) of the first two mirrors. On the contrary, the influence of the output mirror is not negligible even for small slope errors, since the output arm has been chosen to be long. In this case, the slope errors give almost negligible effects below 1 μrad rms, which is the optical quality that has to be assured for the third mirror. Since the illuminated area is also quite small, about 55 mm × 10 mm, the fulfillment of such requirement is well within the capabilities of manufacturers.

The slope errors are the ultimate factor limiting the temporal resolution also in the case of Cartesian surfaces, which give ideally an aberration-free image. Let us analyze the response of a beamline with the same geometrical characteristics of that analyzed in this work, i.e. same demagnification, arms and angles, but based on the use of two parabolic mirrors in section 1 and an elliptical mirror in section 2. In the case of ideal mirror surfaces, the configuration gives high demagnification and it is aberration-free with no temporal dispersion. The effects of the slope errors are shown in Fig. 10
Fig. 10 Aberrations (a) and temporal dispersion (b) as a function of the slope errors of the three Cartesian mirrors of the beamline with the same geometric characteristics of that discussed in the text.
.

The effect of the mirrors of section 1 is almost negligible below 150 as FWHM for slope errors as high as 6 μrad rms. The effect of the output elliptical mirror is almost the same as the toroidal one for slope errors higher than 3.5 μrad rms and lower below 3 μrad rms. A response below 150 as FWHM requires slope errors lower than 2 μrad rms on the output mirror. Generally, since the contribution of the residual aberrations is negligible for Cartesian surfaces, the constraints on the slope errors are slightly more relaxed than those on toroidal mirrors.

6. Conclusions

Micro-focusing of XUV attosecond pulses can be achieved through the design of grazing-incidence toroidal mirrors, combined to compensate for the coma. The configurations discussed in this work consist of two sections: the first one provides a demagnified image of the source that is free from defocusing but with a large coma aberration, the second one consists of a relay mirror that is mounted in Z-shaped geometry with respect to the first one, in order to compensate for the coma introduced by the first section. Furthermore, the use of two mirrors in the first section, the former to collimate the beam and the latter to focus it, provides an intermediate region where the XUV beam is collimated, thus offering the possibility to insert a plane split-mirror for the generation of two XUV pulse replicas with variable temporal delay, essential for pump-probe applications. Finally, the configuration is able to provide a long exit arm to have enough space to accommodate the experimental chamber, although in a table-top compact environment. The use of toroidal mirrors is a much cheaper solution with respect to Cartesian surfaces for experiments requiring a temporal resolution of few hundreds of attoseconds.

Acknowledgments

The authors would like to thank Prof. Giuseppe Tondello for the useful discussions on the aberration theory.

The research leading to the results presented in this paper has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. 227355–ELYCHE. We acknowledge financial support from MC-RTN ATTOFEL (FP7-238362).

References and links

1.

P. Jaeglè, Coherent Sources of XUV Radiation (Springer, 2006).

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]

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 328(5986), 1658–1662 (2010). [CrossRef] [PubMed]

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 302(5650), 1540–1543 (2003). [CrossRef] [PubMed]

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 7360, 736003, 736003-12 (2009). [CrossRef]

8.

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]

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. 75(11), 4413 (2004). [CrossRef]

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. 28(12), 1049–1051 (2003). [CrossRef] [PubMed]

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. 45(3), 573–577 (2006). [CrossRef] [PubMed]

12.

C. Bourassin-Bouchet, S. de Rossi, F. Delmotte, and P. Chavel, “Spatiotemporal distortions of attosecond pulses,” J. Opt. Soc. Am. A 27(6), 1395–1403 (2010). [CrossRef] [PubMed]

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 19(18), 17357–17371 (2011). [CrossRef] [PubMed]

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]

16.

D. E. Aspnes, “Imaging performance of mirror pairs for grazing-incidence applications: a comparison,” Appl. Opt. 21(14), 2642–2646 (1982). [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]

22.

W. J. Smith, Modern Lens Design (McGraw Hill, 2005).

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]

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

  1. P. Jaeglè, Coherent Sources of XUV Radiation (Springer, 2006).
  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. Photonics5(11), 655–663 (2011). [CrossRef]
  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,” Science328(5986), 1658–1662 (2010). [CrossRef] [PubMed]
  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,” Science302(5650), 1540–1543 (2003). [CrossRef] [PubMed]
  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. SPIE7360, 736003, 736003-12 (2009). [CrossRef]
  8. 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]
  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.75(11), 4413 (2004). [CrossRef]
  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.28(12), 1049–1051 (2003). [CrossRef] [PubMed]
  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.45(3), 573–577 (2006). [CrossRef] [PubMed]
  12. 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]
  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. Express19(18), 17357–17371 (2011). [CrossRef] [PubMed]
  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. A75(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]
  16. D. E. Aspnes, “Imaging performance of mirror pairs for grazing-incidence applications: a comparison,” Appl. Opt.21(14), 2642–2646 (1982). [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]
  22. W. J. Smith, Modern Lens Design (McGraw Hill, 2005).
  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. Express20(4), 4525–4536 (2012). [CrossRef] [PubMed]

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