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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 10 — May. 9, 2011
  • pp: 9600–9606
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Time-resolved EUV photoelectron spectroscopy of dissociating I2 by laser harmonics at 80 nm

Mizuho Fushitani, Akitaka Matsuda, and Akiyoshi Hishikawa  »View Author Affiliations


Optics Express, Vol. 19, Issue 10, pp. 9600-9606 (2011)
http://dx.doi.org/10.1364/OE.19.009600


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Abstract

Generation of single-order laser harmonics in extreme ultraviolet (EUV) and the application to the time-resolved photoelectron spectroscopy of I2 are demonstrated. The EUV pulses at 80 nm were generated from Kr as the 5th order harmonics of intense 400 nm laser pulses and then separated from other harmonic orders by a thin indium foil. The pump-probe photoelectron spectroscopy of I2 in the B 3Π(0u+) and B” 1Π(1u) states excited by visible laser pulses at 490 nm showed a rapid increase in the yield of atomic iodine (~400 fs), reflecting the dissociation dynamics evolving simultaneously in the two excited states.

© 2011 OSA

1. Introduction

Laser high-order harmonics have attracted considerable attention in recent years as they can serve as a novel probe for ultrashort dynamics [1

1. P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3(6), 381–387 (2007). [CrossRef]

] in Auger decay [2

2. M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002). [CrossRef] [PubMed]

], coherent electronic excitation [3

3. E. Goulielmakis, Z.-H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010). [CrossRef] [PubMed]

], photoemission from surfaces [4

4. A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449(7165), 1029–1032 (2007). [CrossRef] [PubMed]

] as well as ultrafast chemical reactions of molecules [5

5. L. Nugent-Glandorf, M. Scheer, D. A. Samuels, A. M. Mulhisen, E. R. Grant, X. Yang, V. M. Bierbaum, and S. R. Leone, “Ultrafast time-resolved soft x-ray photoelectron spectroscopy of dissociating Br2.,” Phys. Rev. Lett. 87(19), 193002 (2001). [CrossRef] [PubMed]

8

8. A. Peralta Conde, J. Kruse, O. Faucher, P. Tzallas, E. P. Benis, and D. Charalambidis, “Realization of time-resolved two-vacuum-ultraviolet-photon ionization,” Phys. Rev. A 79(6), 061405 (2009). [CrossRef]

]. High-order harmonics are often obtained as a train of pulses as they are generated every half optical cycle of the driving laser pulse [9

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

].

In time-resolved spectroscopy, isolated harmonic pulses rather than trains are preferred to avoid the complexity in the probing process. Generation of single harmonic pulses has been demonstrated by using carrier-envelop phase-locking [10

10. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008). [CrossRef] [PubMed]

,11

11. H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100(10), 103906 (2008). [CrossRef] [PubMed]

], by which the main burst of high-order harmonics is attained only once in the optical cycles.

An alternative is to transform a pulse train to a single harmonic pulse by spectral filtering. A train of harmonic pulses forms a comb-like spectrum consisting of odd order harmonics. The selection of one of these harmonic orders in the spectrum converts a pulse train into a single pulse at the expense of temporal resolution. For the spectral filtering, grating monochromators [5

5. L. Nugent-Glandorf, M. Scheer, D. A. Samuels, A. M. Mulhisen, E. R. Grant, X. Yang, V. M. Bierbaum, and S. R. Leone, “Ultrafast time-resolved soft x-ray photoelectron spectroscopy of dissociating Br2.,” Phys. Rev. Lett. 87(19), 193002 (2001). [CrossRef] [PubMed]

,6

6. P. Wernet, M. Odelius, K. Godehusen, J. Gaudin, O. Schwarzkopf, and W. Eberhardt, “Real-time evolution of the valence electronic structure in a dissociating molecule,” Phys. Rev. Lett. 103(1), 013001 (2009). [CrossRef] [PubMed]

] have been used in the region of low photon energy ( = 10~30 eV) and, more recently, a zone plate for 30~70 eV [12

12. J. Gaudin, S. Rehbein, P. Guttmann, S. Godé, G. Schneider, P. Wernet, and W. Eberhardt, “Selection of a single femtosecond high-order harmonic using a zone plate based monochromator,” J. Appl. Phys. 104(3), 033112–033114 (2008). [CrossRef]

]. These approaches, however, introduce pulse stretching of the harmonics due to spatial dispersion, unless auxiliary compensation of the dispersion is introduced [13

13. F. Frassetto, P. Villoresi, and L. Poletto, “Beam separator for high-order harmonic radiation in the 3-10 nm spectral region,” J. Opt. Soc. Am. A 25(5), 1104–1114 (2008). [CrossRef]

]. The pulse stretching can be avoided, either by using selective reflection of narrow-band multilayer mirrors such as SiC/Mg [14

14. Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, “Production of doubly charged helium ions by two-photon absorption of an intense sub-10-fs soft x-ray pulse at 42 eV photon energy,” Phys. Rev. Lett. 94(4), 043001 (2005). [CrossRef] [PubMed]

], Sc/Si [15

15. A. Kosuge, T. Sekikawa, X. Zhou, T. Kanai, S. Adachi, and S. Watanabe, “Frequency-resolved optical gating of isolated attosecond pulses in the extreme ultraviolet,” Phys. Rev. Lett. 97(26), 263901 (2006). [CrossRef]

] and Mo/Si [7

7. E. Gagnon, P. Ranitovic, X.-M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, “Soft X-ray-driven femtosecond molecular dynamics,” Science 317(5843), 1374–1378 (2007). [CrossRef] [PubMed]

,16

16. M. Fushitani, A. Matsuda, E. J. Takahashi, and A. Hishikawa, “Time-resolved reaction imaging by intense few-cycle laser pulses and laser high-order harmonics,” J. Phys.: Conf. Ser. 185, 012009 (2009). [CrossRef]

] (40~100 eV), or by using optical windows such as MgF2 [8

8. A. Peralta Conde, J. Kruse, O. Faucher, P. Tzallas, E. P. Benis, and D. Charalambidis, “Realization of time-resolved two-vacuum-ultraviolet-photon ionization,” Phys. Rev. A 79(6), 061405 (2009). [CrossRef]

] and LiF [17

17. S. Zamith, V. Blanchet, B. Girard, J. Andersson, S. L. Sorensen, I. Hjelte, O. Bjorneholm, D. Gauyacq, J. Norin, J. Mauritsson, and A. L'Huillier, “The predissociation of highly excited states in acetylene by time-resolved photoelectron spectroscopy,” J. Chem. Phys. 119(7), 3763–3773 (2003). [CrossRef]

] as a low-pass filter (< 10 eV).

In the present study, we employ a thin metal foil as a harmonic order selector. Thin metal foils have been used as narrow-band filters in EUV [18

18. W. R. Hunter, D. W. Angel, and R. Tousey, “Thin films and their uses for the extreme ultraviolet,” Appl. Opt. 4(8), 891–898 (1965). [CrossRef]

,19

19. F. R. Powell, J. F. Lindblom, S. F. Powell, and P. W. Vedder, “Thin film filter performance for extreme ultraviolet and x-ray applications,” Opt. Eng. 29(6), 614–624 (1990). [CrossRef]

] for a variety of applications in astronomy, microscopy and plasma physics. The application to laser high-order harmonics in the 10~25 eV has been reported using thin tin [20

20. E. Skantzakis, P. Tzallas, J. E. Kruse, C. Kalpouzos, O. Faucher, G. D. Tsakiris, and D. Charalambidis, “Tracking autoionizing-wave-packet dynamics at the 1-fs temporal scale,” Phys. Rev. Lett. 105(4), 043902 (2010). [CrossRef] [PubMed]

] and indium [21

21. P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosecond light bunching,” Nature 426(6964), 267–271 (2003). [CrossRef] [PubMed]

,22

22. N. A. Papadogiannis, L. A. A. Nikolopoulos, D. Charalambidis, G. D. Tsakiris, P. Tzallas, and K. Witte, “Two-photon ionization of he through a superposition of higher harmonics,” Phys. Rev. Lett. 90(13), 133902 (2003). [CrossRef] [PubMed]

] foils, where the band widths are too broad to separate a single harmonic order of the fundamental NIR pulse.

Here we demonstrate the generation of single ultrashort EUV pulses at 80 nm by using an intense UV laser at 400 nm (see Fig. 1
Fig. 1 (a) Transmittance of an indium filter with a thickness of 0.1 µm and reflectivity of a gold (Au) mirror for the angle of incidence of 10 degrees, in the photon energy region of 8~23 eV. It should be noted that the peak transmission of the indium foil is substantially lower than an expected value of ~40% [18], because of the degradation of the foil by the exposure to air. The energies of laser high-order harmonics are indicated by stick bars for driving laser pulses at 400 nm (blue, lower) and 800 nm (red, upper). Photoelectron spectra of I2 molecules by laser high-order harmonic pulses without (b) and with (c) the indium filter. The photoelectron peaks are assigned to the ground (2Πg) and excited (2Πu, 2Σg +) states of I2 + by the 5th order harmonics (H5) of the UV (400 nm) pulse as well as by the 7th order harmonics (H7). The sidebands associated with the 3rd (H3) and H5, H3 + UV and H5 + UV respectively, are also identified.
) as a driver pulse. The obtained EUV pulses are particularly important for applications such as time-resolved photoelectron spectroscopy [5

5. L. Nugent-Glandorf, M. Scheer, D. A. Samuels, A. M. Mulhisen, E. R. Grant, X. Yang, V. M. Bierbaum, and S. R. Leone, “Ultrafast time-resolved soft x-ray photoelectron spectroscopy of dissociating Br2.,” Phys. Rev. Lett. 87(19), 193002 (2001). [CrossRef] [PubMed]

,6

6. P. Wernet, M. Odelius, K. Godehusen, J. Gaudin, O. Schwarzkopf, and W. Eberhardt, “Real-time evolution of the valence electronic structure in a dissociating molecule,” Phys. Rev. Lett. 103(1), 013001 (2009). [CrossRef] [PubMed]

,23

23. T. Horio, T. Fuji, Y. Suzuki, and T. Suzuki, “Probing ultrafast internal conversion through conical intersection via time-energy map of photoelectron angular anisotropy,” J. Am. Chem. Soc. 131(30), 10392–10393 (2009). [CrossRef] [PubMed]

], since (i) the photon energy (15.5 eV) is sufficiently high to ionize most atoms/molecules, and consequently, (ii) both fragments and parent molecules can be monitored during the course of chemical reaction. In addition, the application to the photoelectron spectroscopy allows a probing of the changes in the electronic states, which is complementary to structural information gained, e.g., by Coulomb explosion imaging [24

24. A. Hishikawa, A. Matsuda, M. Fushitani, and E. J. Takahashi, “Visualizing recurrently migrating hydrogen in acetylene dication by intense ultrashort laser pulses,” Phys. Rev. Lett. 99(25), 258302 (2007). [CrossRef]

].

The time-resolved photoelectron spectroscopy has been demonstrated by using laser high-order harmonics for Br2 in the C 1Πu state [5

5. L. Nugent-Glandorf, M. Scheer, D. A. Samuels, A. M. Mulhisen, E. R. Grant, X. Yang, V. M. Bierbaum, and S. R. Leone, “Ultrafast time-resolved soft x-ray photoelectron spectroscopy of dissociating Br2.,” Phys. Rev. Lett. 87(19), 193002 (2001). [CrossRef] [PubMed]

,6

6. P. Wernet, M. Odelius, K. Godehusen, J. Gaudin, O. Schwarzkopf, and W. Eberhardt, “Real-time evolution of the valence electronic structure in a dissociating molecule,” Phys. Rev. Lett. 103(1), 013001 (2009). [CrossRef] [PubMed]

]. Here we apply the obtained EUV pulse to the dissociation dynamics of I2 in the B 3Π(0u +) and B” 1Π(1u) states to demonstrate the monitoring of the nuclear wavepacket evolving simultaneously on the two electronic states.

2. Experimental

The experimental setup for generation and characterization of the laser high-order harmonics has been described previously [16

16. M. Fushitani, A. Matsuda, E. J. Takahashi, and A. Hishikawa, “Time-resolved reaction imaging by intense few-cycle laser pulses and laser high-order harmonics,” J. Phys.: Conf. Ser. 185, 012009 (2009). [CrossRef]

]. The output from a Ti:Sapphire laser system (800 nm, ~35 fs, 1 kHz, 2 mJ/pulse) was frequency-doubled by a BBO crystal with a thickness of 0.5 mm. After separation from the fundamental pulses by several reflections on dielectric multilayer mirrors (TLM1-400, CVI Melles Griot Inc.), the obtained UV laser pulses (400 nm, 180 μJ/pulse) were focused with a plano-convex lens (f = 500 mm) to Kr gas (~5.3 × 103 Pa) in a cylindrical cell (inner diameter 10 mm) placed in a high vacuum chamber (< 10−6 Pa). The laser field intensity inside the cell was estimated to be 1 × 1014 W/cm2.

The generated harmonics were introduced into a beam line for the order-selection and the focusing of single-order harmonics. The beam line was equipped with a pair of plane and concave (r = 1000 mm) gold mirrors placed at an incidence angle of ~5 degrees and a 0.1 µm-thick indium metal foil supported by nickel mesh. To evaluate a performance as a harmonic selector, optical properties of these materials were measured at BL7B of the UVSOR facility at Institute for Molecular Science. The synchrotron radiation, dispersed by the G2 (600 lines/mm) or G3 (300 lines/mm) gratings, was detected as a reference light by a Si photodiode (AXUV100, IRD Inc.) in the photon energy range of 2~30 eV. The transmission through the indium foil or the reflection by the gold mirror was measured with the same detector.

Iodine molecules (99.8%, Wako Pure Chemical Industries Ltd.) were introduced into the detection chamber by a carrier gas (He) through a needle nozzle (inner dia. 750 μm) kept at ~340 K. The partial pressure of I2 molecules at the exit of the nozzle was estimated to be 1 Pa. Ultrashort visible laser pulses at 490 nm (30 μJ/pulse) to excite I2 molecules to the low-lying electronically excited states [25

25. J. Tellinghuisen, “Transition strengths in the visible-infrared absorption spectrum of I2,” J. Chem. Phys. 76(10), 4736–4744 (1982). [CrossRef]

] were generated by an optical parametric amplifier (TOPAS-C, Light Conversion Ltd.) pumped by the rest of the output from the regenerative amplifier. The temporal width of the visible laser pulse was determined to be 90(4) fs by an autocorrelator (Mini, APE GmbH). The visible laser pulse was loosely focused by a plano-convex lens with f = 800 mm and introduced into the interaction region with an angle of 4 mrad with respect to the probing harmonic pulse. The intensity of the visible laser pulse was optimized by an iris to avoid extensive multiphoton ionization of I2 molecules. A typical photoelectron detection rate was about 0.4 per laser shot at the chamber pressure of ~10−4 Pa.

Photoelectron spectroscopy was performed by using a magnetic bottle-type photoelectron spectrometer [26

26. Y. Hikosaka, M. Fushitani, A. Matsuda, C.-M. Tseng, A. Hishikawa, E. Shigemasa, M. Nagasono, K. Tono, T. Togashi, H. Ohashi, H. Kimura, Y. Senba, M. Yabashi, and T. Ishikawa, “Multiphoton double ionization of Ar in intense extreme ultraviolet laser fields studied by shot-by-shot photoelectron spectroscopy,” Phys. Rev. Lett. 105(13), 133001 (2010). [CrossRef]

]. The typical energy resolution ΔE was EE = 50 for the photoelectron energy of E < 200 eV [26

26. Y. Hikosaka, M. Fushitani, A. Matsuda, C.-M. Tseng, A. Hishikawa, E. Shigemasa, M. Nagasono, K. Tono, T. Togashi, H. Ohashi, H. Kimura, Y. Senba, M. Yabashi, and T. Ishikawa, “Multiphoton double ionization of Ar in intense extreme ultraviolet laser fields studied by shot-by-shot photoelectron spectroscopy,” Phys. Rev. Lett. 105(13), 133001 (2010). [CrossRef]

]. A retarding voltage of 3.0 V was applied to the flight tube to achieve a higher energy resolution in time-resolved measurements. The photoelectron spectra exhibited sidebands of molecular iodine peaks when the time delay between the visible and EUV pulses was close to zero (Δt ~0), while the peaks associated with the atomic iodine appeared at a longer delay. The former provided the cross-correlation trace between the visible and EUV pulses, while the latter provided the pump-probe trace of the dissociating I2 molecules, as presented below.

3. Results and discussion

3.1 Generation and characterization of EUV pulses at 80 nm

Figure 1(a) shows the reflectivity of the gold mirrors and the transmittance of the indium filter. The transmittance curve shows a peak around 15 eV, which coincides well with the photon energy of H5. The transmittance of H5 through the indium filter is estimated to be 3%, while those for the 3rd (H3) and 7th (H7) order harmonics is 1 × 10−4 and 5 × 10−5, respectively. Since the transmission of the fundamental UV pulses (not shown) is at most 0.1%, the insertion of the indium filter in the beam path significantly reduces both the neighboring harmonics and the fundamental. The spectral filtering can be used for the driving laser pulses at 800 nm [21

21. P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosecond light bunching,” Nature 426(6964), 267–271 (2003). [CrossRef] [PubMed]

,22

22. N. A. Papadogiannis, L. A. A. Nikolopoulos, D. Charalambidis, G. D. Tsakiris, P. Tzallas, and K. Witte, “Two-photon ionization of he through a superposition of higher harmonics,” Phys. Rev. Lett. 90(13), 133902 (2003). [CrossRef] [PubMed]

], though a more careful tuning of the fundamental wavelength is required (see Fig. 1(a)).

The photoelectron spectra of I2 molecules obtained by the high-order harmonics are shown in Figs. 1(b) and 1(c). The spectrum recorded without the indium filter (Fig. 1(b)) exhibits a series of peaks in the energy regions of 2~6 and 8~12 eV. The main features of the former are associated with the photoelectrons to the ground (2Πg) and excited (2Πu, 2Σg +) states of I2 + [27

27. A. B. Cornford, D. C. Frost, C. A. McDowell, J. L. Ragle, and I. A. Stenhouse, “Photoelectron spectra of the halogens,” J. Chem. Phys. 54(6), 2651–2657 (1971). [CrossRef]

] by H5, while the latter series is due to H7.

Additional peaks in the spectrum in Fig. 1(b) are the sidebands of these main features [2

2. M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002). [CrossRef] [PubMed]

] generated by the fundamental UV pulse. The peak observed at 9.2 eV as well as the lower energy component of the peak at 8.6 eV are due to the photoelectrons by H5, up-shifted in energy by the simultaneous absorption of the fundamental pulse (H5 + UV). Similarly, the peaks at 3.0 and 2.4 eV are associated with H3 + UV. The contributions from the down-shifting components, i.e., H5−UV and H7−UV, are hardly visible.

When the indium filter was inserted in the beam line, the photoelectron spectrum showed a simple peak structure consisting of five peaks associated with the ionization to the ground (2Πg) and excited (2Πu, 2Σg +) states of I2 + by H5 (Fig. 1(c)). This clearly shows that the contributions from the fundamental and the neighboring order harmonics (H3 and H7) are negligibly small, demonstrating the selective transmission of H5 by the indium filter. The spectral width of the EUV pulse is estimated to be 120 meV from a separate photoelectron spectrum of the 2P3/2 peak of Xe.

The pulse duration of the EUV pulse thus obtained was estimated by a cross-correlation measurement with the pump laser pulse at 490 nm. When the visible laser pulse is overlapped with the EUV pulse at a time delay of Δt = 0 fs, the electron released by the ionization by H5 further absorbs a 490 nm photon to show the peak energy shift of 2.53 eV. The associated sideband appears at 8.7 and 8.0 eV for the X 2Π3/2g and X 2Π1/2g peaks, respectively (Fig. 2(a)
Fig. 2 (a) Photoelectron spectrum of the X 2Πg spin-orbit doublet of I2 + at the zero time delay (Δt = 0 fs) between visible (490 nm) and EUV (80 nm) pulses. The corresponding sidebands (SB) by the visible pulse (2.53 eV) are observed at 8.7 and 8.0 eV. The broad continuum observed as a background in the electron energy region below 7 eV in Fig. 1(c) is ascribed to ionization to electronically excited states of I2 + [27,28]. (b) Cross-correlation traces between the EUV and visible pulses, obtained from the SB intensities of the X2Π3/2g and X2Π1/2g peaks recorded as a function of Δt (2Π3/2g: open squares and dotted curve, 2Π1/2g: solid squares and solid curve). The full width at half maximum is estimated to be 151 fs in average from a least-square fitting to a Gaussian function.
). The cross-correlation traces obtained from the intensities of these sidebands are shown in Fig. 2(b). From the determined width (151(10) fs) and the pulse duration of the 490 nm pulse (90(4) fs), the pulse width of H5 is estimated to be 121(13) fs, by assuming a Gaussian envelope for each pulse. This pulse width is substantially longer than that expected for the Fourier-transform limit, 15 fs, due possibly to the chirping of the UV laser pulse. A significant improvement in the temporal resolution should be possible by using a thinner BBO crystal or a non-collinear phase matching [29

29. T. Kanai, X. Zhou, T. Sekikawa, S. Watanabe, and T. Togashi, “Generation of subterawatt sub-10-fs blue pulses at 1-5 kHz by broadband frequency doubling,” Opt. Lett. 28(16), 1484–1486 (2003). [CrossRef] [PubMed]

].

3.2 Time-resolved EUV photoelectron spectroscopy of I2 molecules

We applied the ultrashort EUV pulses to the time-resolved photoelectron spectroscopy of I2 in electronically excited states. The experimental scheme and the obtained results are presented in Fig. 3
Fig. 3 (a) The photoelectron spectrum of I2 recorded at a pump-probe time delay of Δt = 880 fs (solid, black). The corresponding spectrum recorded without the pump pulse (dotted, black). The difference spectrum of the pump-probe and probe only spectra (red). The difference spectrum at Δt = −240 fs (blue). (b) Intensity of the atomic iodine peak at 5.1 eV in the difference spectra plotted as a function of the pump-probe time delay Δt (square, blue). The net values obtained after the subtraction of the sideband intensity of the 2Σg + peak (diamond, black) are also plotted (circle, red). The results of wavepacket simulations for the B 3Π(0 u +) and B” 1Π(1u) states and their sum are also shown by black, gray and red curves, respectively. The individual curves for these states obtained by scaling results in early or late saturation (broken curves). (c) Selected potential energy curves of I2 and I2 + molecules [28]. The solid arrows indicate the pump (490 nm) and probe (80 nm) pulses introduced with a time delay of Δt. The nuclear wavepacket at Δt = 120 fs is shown for each excited state. The shaded area in green for R ≥ 8.1 au corresponds to the atomic iodine signal in the time-resolved photoelectron spectra in Fig. 3(b).
. The visible pump pulse at 490 nm creates nuclear wavepackets in both the B 3Π(0u +) and B” 1Π(1u) states with a ratio of 1.7: 1, according to their absorption cross sections of 1.1 and 0.65 Mb from the ground 1Σg + state at 490 nm [25

25. J. Tellinghuisen, “Transition strengths in the visible-infrared absorption spectrum of I2,” J. Chem. Phys. 76(10), 4736–4744 (1982). [CrossRef]

]. Since these wavepackets are launched above their dissociation thresholds, the I2 molecules dissociate to the asymptotes: I(2P3/2) + I*(2P1/2) for the B 3Π(0u +) state and I(2P3/2) + I(2P3/2) for the B” 1Π(1u) state. The wavepacket evolving simultaneously in these two excited states were probed with the ultrashort EUV pulse by monitoring photoelectron signals from molecular and atomic iodine.

From the cross sections and the pump pulse energy, the population in the B 3Π(0u +) and B” 1Π(1u) excited states after the excitation at 490 nm was estimated to be 5 and 3%, respectively. In order to clearly detect the dynamics in the excited states, the difference between the photoelectron spectra with and without the pump pulse was recorded by blocking every other pulse for the pump by an optical chopper.

Figure 3(a) compares the photoelectron spectrum recorded at a pump-probe time delay of Δt = 880 fs, to the spectrum recorded without the pump pulse. In addition to the X 2Πg and A 2Πu peaks of I2 +, two new peaks are observed at 5.1 and 4.3 eV. These additional peaks, which become clearly visible in the difference spectra in Fig. 3(a), are assigned to the photoelectrons from the atomic iodine produced by the pump pulse. The dips observed in the difference spectra at 5.5 and 6.1 eV are attributed to the depletion of the ground I2 molecules by the pump pulse. None of these features are observed at a negative time delay of Δt = −240 fs, as expected, showing that the dissociation dynamics is indeed triggered by the pump pulse.

In order to see how the dissociation dynamics of I2 in the B 3Π(0u +) and B” 1Π(1u) states proceed, the integrated intensity of the atomic iodine peak at 5.1 eV in the difference spectrum is plotted as a function of the pump-probe time delay. At each time delay, the photoelectron spectra are accumulated for 0.6 million laser shots with and without the pump pulse. The obtained plots in Fig. 3(b) show that the peak intensity monotonically increases as the time delay increases, indicating that the excited I2 molecules rapidly reach the dissociation asymptotes within ~400 fs. It should be noted that the sideband of the 2Σg + peak (see Fig. 1(c)) accidentally overlaps with the photoelectron peak at 5.1 eV, thus it contributes to the signal at Δt ~0 fs. The curve obtained after the subtraction of the sideband component is shown in Fig. 3(b). The contribution from the sideband was estimated from the intensity ratio of the sidebands of the X 2Π3/2g and X 2Π1/2g peaks to the original peaks (see Fig. 2(a)).

To understand the observed time dependence of the yield of atomic iodine, a wavepacket calculation on the dissociation dynamics of I2 in the excited states was performed, independently for the B 3Π(0u +) and B” 1Π(1u) states, using the split-operator method [30

30. M. D. Feit, J. A. Fleck Jr, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47(3), 412–433 (1982). [CrossRef]

]. Because of the finite band width, the photoelectron peak at 5.1 eV contained the contribution from molecular iodine at a large internuclear distance as well as that from atomic iodine. Therefore, the peak intensity was calculated by integrating the probability densities of |Ψ(R,t)|2 for R ≥ 8.1 au (see Fig. 3(c)), where the differences between the ionic and neutral potential energies fell within the energy width (0.24 eV) of the photoelectron peak at 5.1 eV. The transition dipole moments for the ionization were assumed to be the same for these states and constant as a function of the internuclear distance.

The results of simulation for the respective electronic states are shown in Fig. 3(b) after the convolution with the instrumental temporal resolution (151 fs). The appearance of the iodine atom for the B 3Π(0u +) state is found to be delayed by ~100 fs from that for the B” 1Π(1u) state, due to the smaller kinetic energy in the asymptotic limit. For a comparison with the experimental results, the contributions from these two states should be added with appropriate weights.

As shown in Fig. 3(a), the peak at 5.1 eV can be formed by two different ionization processes from the ground state, I(2P3/2) → I+(3P2) + e, and from the excited state, I*(2P1/2) → I+(3P1) + e. It is found in Fig. 3(a), however, that the photoelectron peaks associated with the ionization from the excited iodine atom I*(2P1/2) to the other spin-orbit components, I+(3P0) and I+(3P2), are missing in the photoelectron spectra, indicating that the contribution from the I*(2P1/2) fragment to the observed photoelectron peak at 5.1 eV is also negligible.

Under the assumption that the atomic peak reflects the yields of the I(2P3/2) fragment only, the two curves in Fig. 3(b) for the B 3Π(0u +) and B” 1Π(1u) states are combined with a weight ratio of 1.7: 2. The total yield curve thus calculated shows a good agreement with the experimental data obtained after the subtraction of the cross-correlation component in Fig. 3(b). It is also clear from Fig. 3(b) that the experimental results can only be reproduced by including contributions from both electronic states in the dissociation process, since scaling of the individual curves results in early or late saturation (see broken curves in Fig. 3(b)) Therefore, the nuclear dynamics evolving on the two different excited states are probed by the present time-resolved EUV photoelectron spectroscopy.

4. Conclusion

Generation of single-order EUV harmonic pulses at 80 nm is demonstrated by using intense laser pulses at 400 nm and an indium metal thin filter. The obtained ultrashort EUV pulses were applied to time-resolved measurements of dissociating I2 molecules in the B 3Π(0u +) and B” 1Π(1u) states. The results clearly show a rapid increase in the yield of atomic iodine, reflecting the ultrafast dissociation dynamics in the two excited states. The simple and robust method for the single EUV pulse generation demonstrated in the present study allows a real-time probing of ultrafast dynamics evolving simultaneously in two different electronic states and will extend the applicability of time-resolved photoelectron spectroscopy to a variety of targets.

Acknowledgments

The authors thank Y. Hikosaka, E. Shigemasa, and K. Kuchitsu for their valuable comments, and Equipment Develop Center and UVSOR at Institute for Molecular Science for the development of the beam line. This work is supported by the PRESTO fund for “Evolution of Light Generation and Manipulation” from Japan Science and Technology Agency.

References and links

1.

P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3(6), 381–387 (2007). [CrossRef]

2.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002). [CrossRef] [PubMed]

3.

E. Goulielmakis, Z.-H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010). [CrossRef] [PubMed]

4.

A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449(7165), 1029–1032 (2007). [CrossRef] [PubMed]

5.

L. Nugent-Glandorf, M. Scheer, D. A. Samuels, A. M. Mulhisen, E. R. Grant, X. Yang, V. M. Bierbaum, and S. R. Leone, “Ultrafast time-resolved soft x-ray photoelectron spectroscopy of dissociating Br2.,” Phys. Rev. Lett. 87(19), 193002 (2001). [CrossRef] [PubMed]

6.

P. Wernet, M. Odelius, K. Godehusen, J. Gaudin, O. Schwarzkopf, and W. Eberhardt, “Real-time evolution of the valence electronic structure in a dissociating molecule,” Phys. Rev. Lett. 103(1), 013001 (2009). [CrossRef] [PubMed]

7.

E. Gagnon, P. Ranitovic, X.-M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, “Soft X-ray-driven femtosecond molecular dynamics,” Science 317(5843), 1374–1378 (2007). [CrossRef] [PubMed]

8.

A. Peralta Conde, J. Kruse, O. Faucher, P. Tzallas, E. P. Benis, and D. Charalambidis, “Realization of time-resolved two-vacuum-ultraviolet-photon ionization,” Phys. Rev. A 79(6), 061405 (2009). [CrossRef]

9.

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

10.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008). [CrossRef] [PubMed]

11.

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100(10), 103906 (2008). [CrossRef] [PubMed]

12.

J. Gaudin, S. Rehbein, P. Guttmann, S. Godé, G. Schneider, P. Wernet, and W. Eberhardt, “Selection of a single femtosecond high-order harmonic using a zone plate based monochromator,” J. Appl. Phys. 104(3), 033112–033114 (2008). [CrossRef]

13.

F. Frassetto, P. Villoresi, and L. Poletto, “Beam separator for high-order harmonic radiation in the 3-10 nm spectral region,” J. Opt. Soc. Am. A 25(5), 1104–1114 (2008). [CrossRef]

14.

Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, “Production of doubly charged helium ions by two-photon absorption of an intense sub-10-fs soft x-ray pulse at 42 eV photon energy,” Phys. Rev. Lett. 94(4), 043001 (2005). [CrossRef] [PubMed]

15.

A. Kosuge, T. Sekikawa, X. Zhou, T. Kanai, S. Adachi, and S. Watanabe, “Frequency-resolved optical gating of isolated attosecond pulses in the extreme ultraviolet,” Phys. Rev. Lett. 97(26), 263901 (2006). [CrossRef]

16.

M. Fushitani, A. Matsuda, E. J. Takahashi, and A. Hishikawa, “Time-resolved reaction imaging by intense few-cycle laser pulses and laser high-order harmonics,” J. Phys.: Conf. Ser. 185, 012009 (2009). [CrossRef]

17.

S. Zamith, V. Blanchet, B. Girard, J. Andersson, S. L. Sorensen, I. Hjelte, O. Bjorneholm, D. Gauyacq, J. Norin, J. Mauritsson, and A. L'Huillier, “The predissociation of highly excited states in acetylene by time-resolved photoelectron spectroscopy,” J. Chem. Phys. 119(7), 3763–3773 (2003). [CrossRef]

18.

W. R. Hunter, D. W. Angel, and R. Tousey, “Thin films and their uses for the extreme ultraviolet,” Appl. Opt. 4(8), 891–898 (1965). [CrossRef]

19.

F. R. Powell, J. F. Lindblom, S. F. Powell, and P. W. Vedder, “Thin film filter performance for extreme ultraviolet and x-ray applications,” Opt. Eng. 29(6), 614–624 (1990). [CrossRef]

20.

E. Skantzakis, P. Tzallas, J. E. Kruse, C. Kalpouzos, O. Faucher, G. D. Tsakiris, and D. Charalambidis, “Tracking autoionizing-wave-packet dynamics at the 1-fs temporal scale,” Phys. Rev. Lett. 105(4), 043902 (2010). [CrossRef] [PubMed]

21.

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosecond light bunching,” Nature 426(6964), 267–271 (2003). [CrossRef] [PubMed]

22.

N. A. Papadogiannis, L. A. A. Nikolopoulos, D. Charalambidis, G. D. Tsakiris, P. Tzallas, and K. Witte, “Two-photon ionization of he through a superposition of higher harmonics,” Phys. Rev. Lett. 90(13), 133902 (2003). [CrossRef] [PubMed]

23.

T. Horio, T. Fuji, Y. Suzuki, and T. Suzuki, “Probing ultrafast internal conversion through conical intersection via time-energy map of photoelectron angular anisotropy,” J. Am. Chem. Soc. 131(30), 10392–10393 (2009). [CrossRef] [PubMed]

24.

A. Hishikawa, A. Matsuda, M. Fushitani, and E. J. Takahashi, “Visualizing recurrently migrating hydrogen in acetylene dication by intense ultrashort laser pulses,” Phys. Rev. Lett. 99(25), 258302 (2007). [CrossRef]

25.

J. Tellinghuisen, “Transition strengths in the visible-infrared absorption spectrum of I2,” J. Chem. Phys. 76(10), 4736–4744 (1982). [CrossRef]

26.

Y. Hikosaka, M. Fushitani, A. Matsuda, C.-M. Tseng, A. Hishikawa, E. Shigemasa, M. Nagasono, K. Tono, T. Togashi, H. Ohashi, H. Kimura, Y. Senba, M. Yabashi, and T. Ishikawa, “Multiphoton double ionization of Ar in intense extreme ultraviolet laser fields studied by shot-by-shot photoelectron spectroscopy,” Phys. Rev. Lett. 105(13), 133001 (2010). [CrossRef]

27.

A. B. Cornford, D. C. Frost, C. A. McDowell, J. L. Ragle, and I. A. Stenhouse, “Photoelectron spectra of the halogens,” J. Chem. Phys. 54(6), 2651–2657 (1971). [CrossRef]

28.

W. A. de Jong, L. Visscher, and W. C. Nieuwpoort, “Relativistic and correlated calculations on the ground, excited, and ionized states of iodine,” J. Chem. Phys. 107(21), 9046–9058 (1997). [CrossRef]

29.

T. Kanai, X. Zhou, T. Sekikawa, S. Watanabe, and T. Togashi, “Generation of subterawatt sub-10-fs blue pulses at 1-5 kHz by broadband frequency doubling,” Opt. Lett. 28(16), 1484–1486 (2003). [CrossRef] [PubMed]

30.

M. D. Feit, J. A. Fleck Jr, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47(3), 412–433 (1982). [CrossRef]

OCIS Codes
(300.6500) Spectroscopy : Spectroscopy, time-resolved
(320.7110) Ultrafast optics : Ultrafast nonlinear optics

ToC Category:
Spectroscopy

History
Original Manuscript: March 18, 2011
Manuscript Accepted: April 18, 2011
Published: May 2, 2011

Citation
Mizuho Fushitani, Akitaka Matsuda, and Akiyoshi Hishikawa, "Time-resolved EUV photoelectron spectroscopy of dissociating I2 by laser harmonics at 80 nm," Opt. Express 19, 9600-9606 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9600


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References

  1. P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3(6), 381–387 (2007). [CrossRef]
  2. M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419(6909), 803–807 (2002). [CrossRef] [PubMed]
  3. E. Goulielmakis, Z.-H. Loh, A. Wirth, R. Santra, N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F. Kling, S. R. Leone, and F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010). [CrossRef] [PubMed]
  4. A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449(7165), 1029–1032 (2007). [CrossRef] [PubMed]
  5. L. Nugent-Glandorf, M. Scheer, D. A. Samuels, A. M. Mulhisen, E. R. Grant, X. Yang, V. M. Bierbaum, and S. R. Leone, “Ultrafast time-resolved soft x-ray photoelectron spectroscopy of dissociating Br2.,” Phys. Rev. Lett. 87(19), 193002 (2001). [CrossRef] [PubMed]
  6. P. Wernet, M. Odelius, K. Godehusen, J. Gaudin, O. Schwarzkopf, and W. Eberhardt, “Real-time evolution of the valence electronic structure in a dissociating molecule,” Phys. Rev. Lett. 103(1), 013001 (2009). [CrossRef] [PubMed]
  7. E. Gagnon, P. Ranitovic, X.-M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, “Soft X-ray-driven femtosecond molecular dynamics,” Science 317(5843), 1374–1378 (2007). [CrossRef] [PubMed]
  8. A. Peralta Conde, J. Kruse, O. Faucher, P. Tzallas, E. P. Benis, and D. Charalambidis, “Realization of time-resolved two-vacuum-ultraviolet-photon ionization,” Phys. Rev. A 79(6), 061405 (2009). [CrossRef]
  9. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000). [CrossRef]
  10. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008). [CrossRef] [PubMed]
  11. H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100(10), 103906 (2008). [CrossRef] [PubMed]
  12. J. Gaudin, S. Rehbein, P. Guttmann, S. Godé, G. Schneider, P. Wernet, and W. Eberhardt, “Selection of a single femtosecond high-order harmonic using a zone plate based monochromator,” J. Appl. Phys. 104(3), 033112–033114 (2008). [CrossRef]
  13. F. Frassetto, P. Villoresi, and L. Poletto, “Beam separator for high-order harmonic radiation in the 3-10 nm spectral region,” J. Opt. Soc. Am. A 25(5), 1104–1114 (2008). [CrossRef]
  14. Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, “Production of doubly charged helium ions by two-photon absorption of an intense sub-10-fs soft x-ray pulse at 42 eV photon energy,” Phys. Rev. Lett. 94(4), 043001 (2005). [CrossRef] [PubMed]
  15. A. Kosuge, T. Sekikawa, X. Zhou, T. Kanai, S. Adachi, and S. Watanabe, “Frequency-resolved optical gating of isolated attosecond pulses in the extreme ultraviolet,” Phys. Rev. Lett. 97(26), 263901 (2006). [CrossRef]
  16. M. Fushitani, A. Matsuda, E. J. Takahashi, and A. Hishikawa, “Time-resolved reaction imaging by intense few-cycle laser pulses and laser high-order harmonics,” J. Phys.: Conf. Ser. 185, 012009 (2009). [CrossRef]
  17. S. Zamith, V. Blanchet, B. Girard, J. Andersson, S. L. Sorensen, I. Hjelte, O. Bjorneholm, D. Gauyacq, J. Norin, J. Mauritsson, and A. L'Huillier, “The predissociation of highly excited states in acetylene by time-resolved photoelectron spectroscopy,” J. Chem. Phys. 119(7), 3763–3773 (2003). [CrossRef]
  18. W. R. Hunter, D. W. Angel, and R. Tousey, “Thin films and their uses for the extreme ultraviolet,” Appl. Opt. 4(8), 891–898 (1965). [CrossRef]
  19. F. R. Powell, J. F. Lindblom, S. F. Powell, and P. W. Vedder, “Thin film filter performance for extreme ultraviolet and x-ray applications,” Opt. Eng. 29(6), 614–624 (1990). [CrossRef]
  20. E. Skantzakis, P. Tzallas, J. E. Kruse, C. Kalpouzos, O. Faucher, G. D. Tsakiris, and D. Charalambidis, “Tracking autoionizing-wave-packet dynamics at the 1-fs temporal scale,” Phys. Rev. Lett. 105(4), 043902 (2010). [CrossRef] [PubMed]
  21. P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosecond light bunching,” Nature 426(6964), 267–271 (2003). [CrossRef] [PubMed]
  22. N. A. Papadogiannis, L. A. A. Nikolopoulos, D. Charalambidis, G. D. Tsakiris, P. Tzallas, and K. Witte, “Two-photon ionization of he through a superposition of higher harmonics,” Phys. Rev. Lett. 90(13), 133902 (2003). [CrossRef] [PubMed]
  23. T. Horio, T. Fuji, Y. Suzuki, and T. Suzuki, “Probing ultrafast internal conversion through conical intersection via time-energy map of photoelectron angular anisotropy,” J. Am. Chem. Soc. 131(30), 10392–10393 (2009). [CrossRef] [PubMed]
  24. A. Hishikawa, A. Matsuda, M. Fushitani, and E. J. Takahashi, “Visualizing recurrently migrating hydrogen in acetylene dication by intense ultrashort laser pulses,” Phys. Rev. Lett. 99(25), 258302 (2007). [CrossRef]
  25. J. Tellinghuisen, “Transition strengths in the visible-infrared absorption spectrum of I2,” J. Chem. Phys. 76(10), 4736–4744 (1982). [CrossRef]
  26. Y. Hikosaka, M. Fushitani, A. Matsuda, C.-M. Tseng, A. Hishikawa, E. Shigemasa, M. Nagasono, K. Tono, T. Togashi, H. Ohashi, H. Kimura, Y. Senba, M. Yabashi, and T. Ishikawa, “Multiphoton double ionization of Ar in intense extreme ultraviolet laser fields studied by shot-by-shot photoelectron spectroscopy,” Phys. Rev. Lett. 105(13), 133001 (2010). [CrossRef]
  27. A. B. Cornford, D. C. Frost, C. A. McDowell, J. L. Ragle, and I. A. Stenhouse, “Photoelectron spectra of the halogens,” J. Chem. Phys. 54(6), 2651–2657 (1971). [CrossRef]
  28. W. A. de Jong, L. Visscher, and W. C. Nieuwpoort, “Relativistic and correlated calculations on the ground, excited, and ionized states of iodine,” J. Chem. Phys. 107(21), 9046–9058 (1997). [CrossRef]
  29. T. Kanai, X. Zhou, T. Sekikawa, S. Watanabe, and T. Togashi, “Generation of subterawatt sub-10-fs blue pulses at 1-5 kHz by broadband frequency doubling,” Opt. Lett. 28(16), 1484–1486 (2003). [CrossRef] [PubMed]
  30. M. D. Feit, J. A. Fleck, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47(3), 412–433 (1982). [CrossRef]

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