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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 15 — Jul. 28, 2014
  • pp: 17657–17665
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Carrier-envelope phase effects on the spatial coherence of high-order harmonics

Hanhu Diao, Yinghui Zheng, Zhinan Zeng, Xiaochun Ge, Chuang Li, Ruxin Li, and Zhizhan Xu  »View Author Affiliations


Optics Express, Vol. 22, Issue 15, pp. 17657-17665 (2014)
http://dx.doi.org/10.1364/OE.22.017657


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Abstract

In this work, we report the Carrier-Envelope Phase Effects on the Spatial Coherence of High-order Harmonics driven by phase-stabilized few-cycle mid-infrared laser pulses. The degree of coherence varies with carrier-envelope phase periodically with a period of π. At the same time, as the harmonic orders increase, the extreme points on the curve of coherence degree vs. carrier-envelope phase shift toward the direction of carrier-envelope phase increasing. Through theoretical analysis, we find that the ionization induced frequency chirp plays an important role in the Carrier-Envelope Phase Effects on the Spatial Coherence. This effect suggests a possible method to optimize the spatial coherence of harmonics by tuning the carrier-envelope phase of the driving field.

© 2014 Optical Society of America

1. Introduction

High-order harmonic generation (HHG), which can be described by a simple semi-classical three-step model [1

1. P. B. Corkum, “Plasma perspective on strong-Field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef] [PubMed]

3

3. C. Gong, J. Jiang, C. Li, L. Song, Z. Zeng, J. Miao, X. Ge, Y. Zheng, R. Li, and Z. Xu, “Phase-matching mechanism for high-photon-energy harmonics of a long trajectory driven by a midinfrared laser,” Phys. Rev. A 85(3), 033410 (2012). [CrossRef]

], is a very outstanding way of generating both temporal and spatial coherent light in the ultraviolet, EUV and soft x-ray regions of the spectrum. Although other mechanisms, such as synchrotron sources, x-ray lasers and conventional free-electron lasers (FEL) can also achieve this goal, they can only provide partially coherent light [4

4. R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297(5580), 376–378 (2002). [PubMed]

]. Recently, many researchers have achieved to get the highly coherent EUV light by using seeded FEL [5

5. E. Allaria, R. Appio, L. Badano, W. A. Barletta, S. Bassanese, S. G. Biedron, A. Borga, E. Busetto, D. Castronovo, P. Cinquegrana, S. Cleva, D. Cocco, M. Cornacchia, P. Craievich, I. Cudin, G. D’Auria, M. Dal Forno, M. B. Danailov, R. De Monte, G. De Ninno, P. Delgiusto, A. Demidovich, S. Di Mitri, B. Diviacco, A. Fabris, R. Fabris, W. Fawley, M. Ferianis, E. Ferrari, S. Ferry, L. Froehlich, P. Furlan, G. Gaio, F. Gelmetti, L. Giannessi, M. Giannini, R. Gobessi, R. Ivanov, E. Karantzoulis, M. Lonza, A. Lutman, B. Mahieu, M. Milloch, S. V. Milton, M. Musardo, I. Nikolov, S. Noe, F. Parmigiani, G. Penco, M. Petronio, L. Pivetta, M. Predonzani, F. Rossi, L. Rumiz, A. Salom, C. Scafuri, C. Serpico, P. Sigalotti, S. Spampinati, C. Spezzani, M. Svandrlik, C. Svetina, S. Tazzari, M. Trovo, R. Umer, A. Vascotto, M. Veronese, R. Visintini, M. Zaccaria, D. Zangrando, and M. Zangrando, “Highly coherent and stable pulses from the FERMI seeded free-electron laser in the extreme ultraviolet,” Nat. Photonics 6(10), 699–704 (2012). [CrossRef]

], and on the soft x-ray spectral band HHG-made highly coherent light can be the preferred seed for FEL. Actually, the temporal and spatial coherence of high harmonics are of great importance not only for applications, since they make it possible for interferometry experiments using amplitude division and wave-front division respectively [6

6. L. L. Déroff, P. Salière, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61(4), 043802 (2000). [CrossRef]

], i.e. Randy et al. used their HHG-made coherent EUV radiation to record the Gabor hologram of a near-field scanning optical microscopy (NSOM) tip and a ~10-μm-diameter water jet [4

4. R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297(5580), 376–378 (2002). [PubMed]

], but also for understanding the corresponding physics process [7

7. P. Sauères, A. L’huillier, P. Antoine, and M. Lewenstein, “Study of the spatial and temporal coherence of high-order harmonics,” Adv. At. Mol. Opt. Phys. 41, 83–142 (1999). [CrossRef]

].

Thus, it is essential to get fully understanding of the temporal and spatial coherence of HHG and many groups have made great progress [8

8. T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. G. Tisch, D. D. Meyerhofer, and M. H. R. Hutchinson, “Spatial coherence measurement of soft X-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77(23), 4756–4759 (1996). [CrossRef] [PubMed]

12

12. L. A. Wilson, A. K. Rossall, E. Wagenaars, C. M. Cacho, E. Springate, I. C. E. Turcu, and G. J. Tallents, “Double slit interferometry to measure the EUV refractive indices of solids using high harmonics,” Appl. Opt. 51(12), 2057–2061 (2012). [CrossRef] [PubMed]

]. The research of the spatial coherence of harmonic radiation first focused on the far-field emission profiles, i.e. Peatross et al. [13

13. J. Peatross and D. D. Meyerhofer, “Intensity-dependent atomic-phase effects in high-order harmonic generation,” Phys. Rev. A 52(5), 3976–3987 (1995). [CrossRef] [PubMed]

] and Tisch et al. [14

14. J. W. G. Tisch, R. A. Smith, J. E. Muffett, M. Ciarrocca, J. P. Marangos, and M. H. R. Hutchinson, “Angularly resolved high-order harmonic generation in helium,” Phys. Rev. A 49(1), R28–R31 (1994). [CrossRef] [PubMed]

] observed distorted profiles. However, for fully characterizing the beam quality, the far-field emission profiles are not enough. Then researchers began to study the field property of harmonic radiation to discover the mechanisms that lead to their abilities to interfere. Déroff et al. have studied the influence of gas jet position, backing pressure and diameter of the gas cell on the spatial coherence of harmonic radiation [6

6. L. L. Déroff, P. Salière, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61(4), 043802 (2000). [CrossRef]

]. In all these works, high-order harmonics were obtained by using multi-cycle driving laser pulses, so the Carrier-Envelope Phase (CEP) effects on the spatial coherence of harmonic radiation have never been mentioned, since the random shot-to-shot shifts of CEP have prevented the reproducible guiding of atomic processes. To achieve even high temporal and spatial resolutions on the measurements of atomic and molecular systems, few-cycle and CEP stabilized laser pulse began to be used as the driving field in which case a single attosecond burst can be generated [15

15. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000). [CrossRef]

]. Different from the case of multi-cycle driving laser field, when using few-cycle laser pulses, the temporal evolution of the electric field depends sensitively on the CEP, thus many properties of HHG closely relate to CEP, such as cutoff energy [16

16. A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003). [CrossRef] [PubMed]

, 17

17. L. E. Chipperfield, L. N. Gaier, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Conditions for the reliable production of attosecond pulses using ultra-short laser-generated high harmonics,” J. Mod. Opt. 52, 243–260 (2005). [CrossRef]

] and high-order harmonics conversion efficiency [18

18. G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, V. Strelkov, I. Sola, L. B. Elouga, A. Zaïr, E. Mével, and E. Constant, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80(6), 063837 (2009). [CrossRef]

].

In this paper, by choosing a few-cycle, CEP stabilized laser pulse as the source light of HHG process, we focus on the CEP effects on the spatial coherence of high-order harmonics. To our knowledge, the CEP effects on the spatial coherence of high-order harmonics have never been discussed.

2. Experiment

In our experiment, the driving laser system (1.0mJ, 12fs, 1.75μm and CEP stabilized) is: firstly, pumped by a commercial Ti:sapphire laser system (40fs, 800nm,1KHz); secondly, amplified by a three-stage optical parametric amplifier (OPA); and thirdly, compressed by an argon-filled hollow fiber and dispersion compensated by a pair of fused silica wedges. The peak intensity of the driving field is about 2.0 × 1014 W/cm2. A steel tube with an inner diameter of 1.9 mm and filled with argon gas is used as the gas cell and it’s movable along the beam propagation direction by a high-precision translation stage for achieving optimal phase match condition. A concave mirror with a focal length of 210 mm is used to focus the laser pulse into the steel tube to generate high-order harmonics. We place a 150-nm-thick aluminum foil after the gas cell to block the lower-order harmonics and the residual mid-IR driving laser. A spectrometer with a flat-field grating (1200 lines/mm) and a soft-x-ray CCD is used to detect the HHG spectra. Note, in our experiment, the focal spot in the gas cell is at a distance of 237mm from the centre of grating. It means the focal spot just replaces the entrance slit of the grating. For measuring the spatial coherence of harmonic signal, we place horizontally a double slit spaced 30 μm center-to-center in front of the grating and stick the slits on the side face of the grating, where the width and length of each slit were 10μm and 3mm respectively. During the experiment, the argon gas pressure is fixed at ~200 Torr. The CEP of the driving laser pulse can be changed relatively from 0 to 2π in steps of π/8 by adjusting the thickness of the wedge. The schematic experiment setup is shown in Fig. 1.
Fig. 1 The schematic diagram of the experiment setup.

3. Results and discussion

The raw measured image of harmonic coherence signal obtained by CCD in the harmonic order region (100th-165th) for CEP = 2π is shown in Fig. 2(a).
Fig. 2 (a) The raw measured interference image of high-order harmonic obtained by CCD (CEP = 2π); (b), (c), (d) and (e) the spatial interference pattern as a function of CEP [-2π; 2π] for harmonic order 103th, 115th, 125th and 150th respectively.
From Fig. 1 one can see obvious interference fringes along the pixel axis (radial direction).

4. Theoretical analysis

In the simulation, we use the saddle-point approximation of the Lewenstein model [19

19. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef] [PubMed]

] to calculate the single-atom response of the time-dependent dipole moment, and use the first-order propagation equation [20

20. M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83(15), 2930–2933 (1999). [CrossRef]

, 21

21. N. Milosevic, A. Scrinzi, and T. Brabec, “Numerical characterization of high harmonic attosecond pulses,” Phys. Rev. Lett. 88(9), 093905 (2002). [CrossRef] [PubMed]

], which is under the slowly evolving wave approximation, to calculate propagation effects of the fundamental and harmonic signals. In addition, we assume the original driving pulse is both temporal and spatial Gaussian with a central wavelength of 1800 nm, the full width at half maximum (FWHM) pulse duration of 12 fs, a peak intensity of 2.0 × 1014 W/cm2 and a waist of 160μm. The gas pressure is fixed at 200Torr, and the center of the gas medium is put 2 mm after the focal point. The CEP is changed from-2πtowith the step of π/8.The far-field on the double-slit is calculated by using of Hankle transform [22

22. M. B. Gaarde, F. Salin, E. Constant, Ph. Balcou, K. J. Schafer, K. C. Kulander, and A. L’huillier, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A 59(2), 1367–1373 (1999). [CrossRef]

]. Figure 3(b) shows the coherence degree γdoc(r1,r2;τ) of four harmonics calculated by Eq. (1) for two slits (r1, r2). Here, we set τ=0for keeping the same condition as the experimental data, in which the visibility of fringes is calculated in the center of the fringes pattern.

By using the same parameters as those in the experiment, the simulated results reproduce our measurements well by comparing Fig. 3(a) with Fig. 3(b). It is well known that ionization plays a big role during the process of HHG and early researches have reported many ionization connected effects during the HHG, like quantum path interference [23

23. M. Holler, A. Zaïr, F. Schapper, T. Auguste, E. Cormier, A. Wyatt, A. Monmayrant, I. A. Walmsley, L. Gallmann, P. Salières, and U. Keller, “Ionization effects on spectral signatures of quantum-path interference in high-harmonic generation,” Opt. Express 17(7), 5716–5722 (2009). [CrossRef] [PubMed]

], phase matching in HHG [24

24. T. Popmintchev, M. C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009). [CrossRef] [PubMed]

]. In addition, other groups have mentioned that ionization and atomic dipole phase may induce the degrading of spatial coherence for multi-cycle driving laser pulses [6

6. L. L. Déroff, P. Salière, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61(4), 043802 (2000). [CrossRef]

, 25

25. T. Ditmire, J. W. G. Tisch, E. T. Gumbrell, R. A. Smith, D. D. Meyerhofer, and M. H. R. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B 65(3), 313–328 (1997). [CrossRef]

]. To find out the reasons in our experiment, we’ve calculated the frequency chirp of harmonics in each slits (r1 and r2) to check if the phase of the harmonic radiation is CEP-related, before they pass through the double-slit.

Figure 4 shows the frequency chirps of the four selected harmonic orders in two slits r1 (blue solid line) and r2 (black dashed line), where the frequency chirp is obtained byωc(r,t)=ϕq(r,t)/t and ϕq(r,t) is the phase of harmonic field for order q.
Fig. 4 Frequency chirps for four sampling harmonic orders for CEP value at γdocmaximum point and minimum point are shown in (a) and (b), respectively; in figure (a) and (b) the four sub-graphs, from the top to the bottom, are for harmonic order 100th, 115th, 128th and 150th, respectively. In these Figures, blue solid line, black dashed line are for slit position r1 and r2, respectively; red dot-dash line marks the central frequency location for each harmonic order.
Figure 4(a) is at the CEP values where the degree of coherence reaches the maximum point, while Fig. 4(b) is for the minimum point. The locations of the central frequency of each harmonic have been marked as red dot-dash line in Fig. 4. We can find that the difference of the frequency chirps in the two slits are much smaller in Fig. 4(a) than that in Fig. 4(b), and that is to say when the harmonic signals in the two slits are coherent with each other well, the frequency chirps’ difference should be small. Note that this comparison is made for the same harmonic order, and when the γdocreaches the maxima, the coincidence of the frequency chirps curves at two spatial positions is better than that when the γdocreaches the minima. In Fig. 4(a) for harmonic orders 128th and 150th, the frequency chirps of harmonics at two spatial positions do not coincide well. This is because the spatial coherence for 128th and 150th are poor in the whole CEP range, compared with the other two harmonic orders of 100th and 115th, as shown in Fig. 3(b). Besides, for harmonic order 150th, Δchirp vs. CEP is plotted in Fig. 5(c), where Δchirpis given by Δchirp=t1t2|ωc(r1,t)ωc(r2,t)|, and the summation time range is from −1 to 3 periods which is judged by the corresponding amplitude of electric field of harmonics (see Fig. 5(a) and Fig. 5(b)).
Fig. 5 Sub-graphs (a) and (b) show the amplitude of electric field of harmonic order 150th as a function of time in two different CEP values φmin and φmax, respectively. φminand φmax are of CEP values at γdocminimum and maximum point for harmonic order 150th, respectively. Red solid line and blue dashed line are for slit position r1 and r2, respectively. (c) Shows Δchirp(red dot-dash line, left scale) and γdoc (blue solid line, right scale) as a function of CEP for harmonic order 150th. The two curves on the sub-graph (c) show the negative correlation relationship.
Figure 5(c) shows that Δchirp and γdoc has the negative correlation relationship. It means that the phase difference, especially the frequency chirp difference of harmonics in different spatial positions has a significant association with the CEP dependence of spatial coherence.

By considering multi-trajectories, for different radial positions, CEPs and harmonics, firstly, we do a time-frequency analysis [26

26. X. M. Tong and S. I. Chu, “Probing the spectral and temporal structures of high-order harmonic generation in intense laser pulses,” Phys. Rev. A 61(2), 021802 (2000). [CrossRef]

] to get the contributionAq(r,φCEP;t) from each trajectory (the trajectories are grouped by different return time t) and normalize it by Aa¯(r,φCEP;t)=Aq(r,φCEP;t)/t1t2Aq(r,φCEP;t); secondly, we calculate the temporal distribution of electron density ne(r,φCEP;t) by using ADK model [27

27. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

]; finally, we obtain the difference of multi-trajectories weighted electron densities for harmonic order q in two radial position rand rby equation:
Δneq(φCEP)=t1t2[Aq¯(r,φCEP;t)×ne(r,φCEP;t)Aq¯(r,φCEP;t)×ne(r,φCEP;t)]
(3)
where the time range t1~t2 is from −1 to 3 periods of the driving pulse which is the same time range as for calculatingγdoc. Note that we calculate the electron densities by using the electric field at the export of the gas cell. The reasons are shown as follows. On the one hand, in our simulation, the length of gas cell is 1.9 mm, which is much smaller than the Rayleigh length of the laser beam of about 20 mm, so the variation of the electric field (e.g. Gouy phase) is very small during its propagation in the gas cell. On the other hand, we are merely interested in the relative positions of the extreme points on the Δnecurves for different harmonic orders, which will be shown in the latter discussion, and we pay little attention to the absolute value ofΔne. It means that the absolute value of the electron densities influence our theoretical analysis little. Although the electron distribution on the time axis will not change too much, the propagation of harmonic field is still important. So we calculate the electron densities by using the electric field at the export of the gas jet, where we can get the harmonic signals after propagating in the gas medium.

Figure 6 shows difference of electron densities on two radial positions in the near filed as a function of CEP for four selected harmonic orders, black solid line for 100th, blue dashed line for 115th, green dot-dashed line for 128th and red dotted line for 150th.
Fig. 6 Difference of electron densities on two radial position in the near filed as a function of CEP for four selected harmonic orders, black solid line for 100th, blue dashed line for 115th, green dot-dashed line for 128th and red dotted line for 150th.
It can be seen that the difference of electron densities oscillates with CEP with a period of π, the same as the spatial coherence. It means that the electron densities on different radial positions vary with CEP in the process of HHG. By comparing Fig. 6 and Fig. 3(b), it can also be found that when harmonic orders increase, difference of electron densities will increase while the intensity of γdoc curves will decrease. The larger difference of electron densities will lead to the larger difference of phases, thus lead to the worse spatial coherence. Therefore, the increase of the difference of electron densities with harmonic order will lead to the decreasing ofγdoc. This can also prove that the electron density on different radial position greatly impact on the spatial coherent. Note that for the same harmonic, the CEP values for the extreme points on the γdoc curve do not coincide with the ones on the curve of difference of electron densities. Because, γdocis calculated in the far field after the Hankle transformation while difference of electron densities is calculated in the near field, which will lead to the difference of spatial positions. From above discussion, the electron density or the ionization induced frequency chirp indeed has a close relationship with the CEP effects of the spatial coherence.

5. Conclusions

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants No. 11127901, No. 61221064, No. 11134010, No. 11227902, No. 11222439, and No. 11274325), the 973 Project (Grant No. 2011CB808103), and Shanghai Commission of Science and Technology (Grant No. 12QA1403700).

References and links

1.

P. B. Corkum, “Plasma perspective on strong-Field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef] [PubMed]

2.

C. Liu and M. Nisoli, “Control of the polarization of isolated attosecond pulses in atoms with nonvanishing angular quantum number,” Phys. Rev. A 85(1), 013418 (2012). [CrossRef]

3.

C. Gong, J. Jiang, C. Li, L. Song, Z. Zeng, J. Miao, X. Ge, Y. Zheng, R. Li, and Z. Xu, “Phase-matching mechanism for high-photon-energy harmonics of a long trajectory driven by a midinfrared laser,” Phys. Rev. A 85(3), 033410 (2012). [CrossRef]

4.

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297(5580), 376–378 (2002). [PubMed]

5.

E. Allaria, R. Appio, L. Badano, W. A. Barletta, S. Bassanese, S. G. Biedron, A. Borga, E. Busetto, D. Castronovo, P. Cinquegrana, S. Cleva, D. Cocco, M. Cornacchia, P. Craievich, I. Cudin, G. D’Auria, M. Dal Forno, M. B. Danailov, R. De Monte, G. De Ninno, P. Delgiusto, A. Demidovich, S. Di Mitri, B. Diviacco, A. Fabris, R. Fabris, W. Fawley, M. Ferianis, E. Ferrari, S. Ferry, L. Froehlich, P. Furlan, G. Gaio, F. Gelmetti, L. Giannessi, M. Giannini, R. Gobessi, R. Ivanov, E. Karantzoulis, M. Lonza, A. Lutman, B. Mahieu, M. Milloch, S. V. Milton, M. Musardo, I. Nikolov, S. Noe, F. Parmigiani, G. Penco, M. Petronio, L. Pivetta, M. Predonzani, F. Rossi, L. Rumiz, A. Salom, C. Scafuri, C. Serpico, P. Sigalotti, S. Spampinati, C. Spezzani, M. Svandrlik, C. Svetina, S. Tazzari, M. Trovo, R. Umer, A. Vascotto, M. Veronese, R. Visintini, M. Zaccaria, D. Zangrando, and M. Zangrando, “Highly coherent and stable pulses from the FERMI seeded free-electron laser in the extreme ultraviolet,” Nat. Photonics 6(10), 699–704 (2012). [CrossRef]

6.

L. L. Déroff, P. Salière, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61(4), 043802 (2000). [CrossRef]

7.

P. Sauères, A. L’huillier, P. Antoine, and M. Lewenstein, “Study of the spatial and temporal coherence of high-order harmonics,” Adv. At. Mol. Opt. Phys. 41, 83–142 (1999). [CrossRef]

8.

T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. G. Tisch, D. D. Meyerhofer, and M. H. R. Hutchinson, “Spatial coherence measurement of soft X-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77(23), 4756–4759 (1996). [CrossRef] [PubMed]

9.

M. Bellini, C. Lyngå, A. Tozzi, M. B. Gaarde, T. W. Hänsch, A. L’Huillier, and C.-G. Wahlström, “Temporal coherence of ultrashort high-order harmonic pulses,” Phys. Rev. Lett. 81(2), 297–300 (1998). [CrossRef]

10.

C. Lyngå, M. B. Gaarde, C. Delfin, M. Bellini, T. W. Hänsch, A. L’Huillier, and C.-G. Wahlström, “Temporal coherence of high-order harmonics,” Phys. Rev. A 60(6), 4823–4830 (1999). [CrossRef]

11.

M. C. Chen, M. R. Gerrity, S. Backus, T. Popmintchev, X. Zhou, P. Arpin, X. Zhang, H. C. Kapteyn, and M. M. Murnane, “Spatially coherent, phase matched, high-order harmonic EUV beams at 50 kHz,” Opt. Express 17(20), 17376–17383 (2009). [CrossRef] [PubMed]

12.

L. A. Wilson, A. K. Rossall, E. Wagenaars, C. M. Cacho, E. Springate, I. C. E. Turcu, and G. J. Tallents, “Double slit interferometry to measure the EUV refractive indices of solids using high harmonics,” Appl. Opt. 51(12), 2057–2061 (2012). [CrossRef] [PubMed]

13.

J. Peatross and D. D. Meyerhofer, “Intensity-dependent atomic-phase effects in high-order harmonic generation,” Phys. Rev. A 52(5), 3976–3987 (1995). [CrossRef] [PubMed]

14.

J. W. G. Tisch, R. A. Smith, J. E. Muffett, M. Ciarrocca, J. P. Marangos, and M. H. R. Hutchinson, “Angularly resolved high-order harmonic generation in helium,” Phys. Rev. A 49(1), R28–R31 (1994). [CrossRef] [PubMed]

15.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000). [CrossRef]

16.

A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003). [CrossRef] [PubMed]

17.

L. E. Chipperfield, L. N. Gaier, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Conditions for the reliable production of attosecond pulses using ultra-short laser-generated high harmonics,” J. Mod. Opt. 52, 243–260 (2005). [CrossRef]

18.

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, V. Strelkov, I. Sola, L. B. Elouga, A. Zaïr, E. Mével, and E. Constant, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80(6), 063837 (2009). [CrossRef]

19.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef] [PubMed]

20.

M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83(15), 2930–2933 (1999). [CrossRef]

21.

N. Milosevic, A. Scrinzi, and T. Brabec, “Numerical characterization of high harmonic attosecond pulses,” Phys. Rev. Lett. 88(9), 093905 (2002). [CrossRef] [PubMed]

22.

M. B. Gaarde, F. Salin, E. Constant, Ph. Balcou, K. J. Schafer, K. C. Kulander, and A. L’huillier, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A 59(2), 1367–1373 (1999). [CrossRef]

23.

M. Holler, A. Zaïr, F. Schapper, T. Auguste, E. Cormier, A. Wyatt, A. Monmayrant, I. A. Walmsley, L. Gallmann, P. Salières, and U. Keller, “Ionization effects on spectral signatures of quantum-path interference in high-harmonic generation,” Opt. Express 17(7), 5716–5722 (2009). [CrossRef] [PubMed]

24.

T. Popmintchev, M. C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009). [CrossRef] [PubMed]

25.

T. Ditmire, J. W. G. Tisch, E. T. Gumbrell, R. A. Smith, D. D. Meyerhofer, and M. H. R. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B 65(3), 313–328 (1997). [CrossRef]

26.

X. M. Tong and S. I. Chu, “Probing the spectral and temporal structures of high-order harmonic generation in intense laser pulses,” Phys. Rev. A 61(2), 021802 (2000). [CrossRef]

27.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(320.2250) Ultrafast optics : Femtosecond phenomena
(320.7110) Ultrafast optics : Ultrafast nonlinear optics
(020.2649) Atomic and molecular physics : Strong field laser physics

ToC Category:
Ultrafast Optics

History
Original Manuscript: April 18, 2014
Revised Manuscript: June 21, 2014
Manuscript Accepted: July 7, 2014
Published: July 14, 2014

Citation
Hanhu Diao, Yinghui Zheng, Zhinan Zeng, Xiaochun Ge, Chuang Li, Ruxin Li, and Zhizhan Xu, "Carrier-envelope phase effects on the spatial coherence of high-order harmonics," Opt. Express 22, 17657-17665 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-17657


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References

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  17. L. E. Chipperfield, L. N. Gaier, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Conditions for the reliable production of attosecond pulses using ultra-short laser-generated high harmonics,” J. Mod. Opt.52, 243–260 (2005). [CrossRef]
  18. G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, V. Strelkov, I. Sola, L. B. Elouga, A. Zaïr, E. Mével, and E. Constant, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A80(6), 063837 (2009). [CrossRef]
  19. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A49(3), 2117–2132 (1994). [CrossRef] [PubMed]
  20. M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett.83(15), 2930–2933 (1999). [CrossRef]
  21. N. Milosevic, A. Scrinzi, and T. Brabec, “Numerical characterization of high harmonic attosecond pulses,” Phys. Rev. Lett.88(9), 093905 (2002). [CrossRef] [PubMed]
  22. M. B. Gaarde, F. Salin, E. Constant, Ph. Balcou, K. J. Schafer, K. C. Kulander, and A. L’huillier, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A59(2), 1367–1373 (1999). [CrossRef]
  23. M. Holler, A. Zaïr, F. Schapper, T. Auguste, E. Cormier, A. Wyatt, A. Monmayrant, I. A. Walmsley, L. Gallmann, P. Salières, and U. Keller, “Ionization effects on spectral signatures of quantum-path interference in high-harmonic generation,” Opt. Express17(7), 5716–5722 (2009). [CrossRef] [PubMed]
  24. T. Popmintchev, M. C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A.106(26), 10516–10521 (2009). [CrossRef] [PubMed]
  25. T. Ditmire, J. W. G. Tisch, E. T. Gumbrell, R. A. Smith, D. D. Meyerhofer, and M. H. R. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B65(3), 313–328 (1997). [CrossRef]
  26. X. M. Tong and S. I. Chu, “Probing the spectral and temporal structures of high-order harmonic generation in intense laser pulses,” Phys. Rev. A61(2), 021802 (2000). [CrossRef]
  27. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP64, 1191–1194 (1986).

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