1. Nonlinear optical effect in the soft x-ray region by two-photon ionization of He + Nonlinear optical effect in the soft x-ray region by two-photon ionization of He + Kenichi Ishikawa* 、 Katsumi Midorikawa Laser Technology Laboratory, RIKEN, Japan * Present address : Department of Quantum Engineering & Systems Science, Graduate School of Engineering, University of Tokyo ishiken@q.t.u-tokyo.ac.jp CLEO/QELS2002 May 22, 2002 Phys. Rev. A 65, 043405 (2002)

2. CLEO/QELS2002 May 22, 2002 No. 1 Nonlinear Optical Effect High-order harmonic generation The material response to an applied optical field depends on the field strength in a nonlinear fashion Theory1931 Göppert-Mayer RF 1959 Winter Visible1961 Franken et al. ( 2nd-order harmonic ) Kaiser and Garrett ( Two-photon excitation ) VUV1998 Kobayashi et al. XUV2001 Descamps et al. Soft X-ray ？ ？

3. CLEO/QELS2002 May 22, 2002 No. 2 High-order harmonic generation High-order harmonics of a Ti:Sapphire laser 27HG (30nm, 41.85eV) → 〜 300nJ, < 30fs 5×10 13 W/cm 2 Motivation ： to propose a system for the experimental observation of a nonlinear optical effect in a soft x-ray range using such an intense pulse. focused to an area of 10  m 2 by a soft x-ray mirror Two-photon ionization of He + 40.8 eV 1s1s1s1s 2p2p2p2p He + 27HG (30nm, 41.85eV) He 2 + E = 0 CLEO2002 CMQ3

4. CLEO/QELS2002 May 22, 2002 No. 3 Three conditions The two-photon ionization probability must be High. High. Quadratic in soft x-ray intensity. Quadratic in soft x-ray intensity. Linear in pulse duration. Linear in pulse duration. Not evident Not evident at high intensity and ultrashort pulse duration. Numerical simulation

5. CLEO/QELS2002 May 22, 2002 No. 4 Model Numerical method Numerical method –Alternating direction implicit (Peaceman-Rachford) method He 2+ Yield He 2+ Yield –evaluated as the number of electrons absorbed by the mask function at the outer radial boundary. Time-dependent Schrodinger equation Field of the soft x-ray pulse

6. CLEO/QELS2002 May 22, 2002 No. 5 He 2+ yield vs. Peak intensity Saturation can be seen at I > 10 13 W/cm 2. Saturation can be seen at I > 10 13 W/cm 2. –Not due to the decrease of the He + population. –Due to the ac-Stark effect. (Haberland 1987) The two-photon ionization probability is sufficiently high to be observed experimentally. The two-photon ionization probability is sufficiently high to be observed experimentally. The yield is approximately proportional to (Intensity) 2. The yield is approximately proportional to (Intensity) 2. gaussian 30fs-27th harmonic pulse “analytical” means,  = 2.9×10 -52 cm 4 s

7. CLEO/QELS2002 May 22, 2002 No. 6 Two-photon ionization probability ( He 2+ Yield ) The He 2+ yield is sufficiently high if we use the 27HG. The He 2+ yield is sufficiently high if we use the 27HG. –5.6×10 -5 at I = 2×10 13 W/cm 2 –3.3×10 -4 at I = 5×10 13 W/cm 2 Ac-Stark effect → Saturation Ac-Stark effect → Saturation ↓Nevertheless … ↓Nevertheless … The yield is approximately quadratic in intensity for the case of the 27HG. The yield is approximately quadratic in intensity for the case of the 27HG. Two-photon ionization probability by a Gaussian pulse with a duration (FWHM) of 30 fs

8. CLEO/QELS2002 May 22, 2002 No. 7 Three conditions The two-photon ionization probability must be High. High. Quadratic in soft x-ray intensity. Quadratic in soft x-ray intensity. Linear in pulse duration. Linear in pulse duration. OK

9. CLEO/QELS2002 May 22, 2002 No. 8 He 2+ yield vs. Pulse duration If the pulse duration is > 5 fs, the yield is proportional to the duration. If the pulse duration is > 5 fs, the yield is proportional to the duration. –Application such as the pulse duration measurement by autocorrelation If the pulse duration is < 5 fs, the linearity breaks down. If the pulse duration is < 5 fs, the linearity breaks down. –Excitation of the 2p level due to energy uncertainty. of a gaussian 27th harmonic pulse with a peak intensity of 5×10 13 W/cm 2.

10. CLEO/QELS2002 May 22, 2002 No. 9 Three conditions The two-photon ionization probability IS High! High! Quadratic in soft x-ray intensity! Quadratic in soft x-ray intensity! Linear in pulse duration! (> 5fs) Linear in pulse duration! (> 5fs) Attractive candidate for experimental observation of a nonlinear optical effect in the soft x-ray domain.

11. CLEO/QELS2002 May 22, 2002 No. 10 Two-photon ionization by a double pulse 27HG → The final He 2+ yield is NOT twice as large as that after the first pulse. 27HG → The final He 2+ yield is NOT twice as large as that after the first pulse. Double pulse : Intensity = 5×10 13 W/cm 2 、 Pulse duration = 1 fs Quantum effect The final yield depends on Time interval between the two pulses Time interval between the two pulses Relative phase of the two pulses Relative phase of the two pulses Quantum effect The final yield depends on Time interval between the two pulses Time interval between the two pulses Relative phase of the two pulses Relative phase of the two pulses

12. CLEO/QELS2002 May 22, 2002 No. 11 Dependence on the relative phase Single pulse Single pulse –independent of  1 Double pulse Double pulse –The yield oscillates with    . Electric field of the double pulse Pulse width = 1 fs, Interval = 4.14824 fs Single pulse: F 2 (t)=0          

13. CLEO/QELS2002 May 22, 2002 No. 12 Dependence on the time interval The period (0.1 fs) of the dipole moment oscillation corresponds to the energy difference between 1s and 2p. Electric field of the double pulse Pulse width = 1 fs Dipole moment after single pulse (He + : superposition of 1s and 2p) The relation between the phase of the dipole moment induced by the 1st pulse and the phase of the 2nd pulse causes an oscillation in the He 2+ yield. 0.1 fs

14. CLEO/QELS2002 May 22, 2002 No. 13 Conclusions The two-photon ionization of He + by a 27th harmonic pulse of a Ti:Sapphire laser is an attractive candidate for experimental observation of a nonlinear optical effect in the soft x-ray domain. The two-photon ionization of He + by a 27th harmonic pulse of a Ti:Sapphire laser is an attractive candidate for experimental observation of a nonlinear optical effect in the soft x-ray domain. –High probability –Yield quadratic in pulse intensity –Yield linear in pulse duration (> 5 fs) When the pulse duration is < 5 fs, the He 2 + yield is NOT linear in pulse duration. The yield by a double pulse oscillates with the pulse interval. When the pulse duration is < 5 fs, the He 2 + yield is NOT linear in pulse duration. The yield by a double pulse oscillates with the pulse interval. Phys. Rev. A 65, 043405 (2002)