Nonlinear optical effect in the soft x-ray Nonlinear optical effect in the soft x-ray region by two-photon ionizationregion by two-photon ionization

of Heof He++

Kenichi Ishikawa*Kenichi Ishikawa* 、、 Katsumi MidorikawaKatsumi MidorikawaLaser Technology Laboratory, RIKEN, JapanLaser Technology Laboratory, RIKEN, Japan

**Present address : Present address : Department of Quantum Engineering & Systems Department of Quantum Engineering & Systems Science, Graduate School of Engineering, University of TokyoScience, Graduate School of Engineering, University of Tokyo

ishiken@q.t.u-tokyo.ac.jpishiken@q.t.u-tokyo.ac.jp

CLEO/QELS2002 May 22, 2002CLEO/QELS2002 May 22, 2002

Phys. Rev. A Phys. Rev. A 6565, 043405 (2002), 043405 (2002)

CLEO/QELS2002 May 22, 2002 No. 2　

Nonlinear Optical EffectNonlinear Optical Effect

High-order High-order harmonic harmonic generationgeneration

High-order High-order harmonic harmonic generationgeneration

The material response to an applied optical field depends on the field strength in a nonlinear fashion

TheoryTheory 19311931 GGöppert-Mayeröppert-MayerRFRF 19591959 WinterWinterVisibleVisible 19611961 Franken et al. (Franken et al. (2nd-order harmonic2nd-order harmonic))

Kaiser and Garrett (Kaiser and Garrett (Two-photon excitationTwo-photon excitation))VUVVUV 19981998 Kobayashi et al.Kobayashi et al.XUVXUV 20012001 Descamps et al.Descamps et al.

Soft X-raySoft X-ray ？？ ？？

CLEO/QELS2002 May 22, 2002 No. 3　

High-order harmonic generationHigh-order harmonic generation

High-order harmonics of a Ti:Sapphire laserHigh-order harmonics of a Ti:Sapphire laserHigh-order harmonics of a Ti:Sapphire laserHigh-order harmonics of a Ti:Sapphire laser

27HG (27HG (30nm, 41.85eV)30nm, 41.85eV) → → 〜〜 300nJ, < 30fs300nJ, < 30fs27HG (27HG (30nm, 41.85eV)30nm, 41.85eV) → → 〜〜 300nJ, < 30fs300nJ, < 30fs

5×105×101313W/cmW/cm225×105×101313W/cmW/cm22

MotivationMotivation ：： to propose a system for to propose a system for the experimental observation of a the experimental observation of a nonlinear optical effect in a soft x-ray nonlinear optical effect in a soft x-ray range using such an intense pulse.range using such an intense pulse.

MotivationMotivation ：： to propose a system for to propose a system for the experimental observation of a the experimental observation of a nonlinear optical effect in a soft x-ray nonlinear optical effect in a soft x-ray range using such an intense pulse.range using such an intense pulse.

focused to an area of 10focused to an area of 10mm2 2 by a soft x-ray mirror by a soft x-ray mirror

Two-photon ionization of HeTwo-photon ionization of He++Two-photon ionization of HeTwo-photon ionization of He++

40.8 eV40.8 eV

11ss

22pp

HeHe++

27HG27HG

((30nm, 41.85eV)30nm, 41.85eV)

HeHe22++

EE = 0 = 0

CLEO2002 CMQ3

CLEO/QELS2002 May 22, 2002 No. 4　

Three conditionsThree conditions

The two-photon ionization probability must The two-photon ionization probability must bebe

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 simulationNumerical simulation

CLEO/QELS2002 May 22, 2002 No. 5　

ModelModel

Numerical methodNumerical method– Alternating direction implicit (Peaceman-Rachford) methodAlternating direction implicit (Peaceman-Rachford) method

HeHe2+ 2+ YieldYield

– evaluated as the number of electrons absorbed by the evaluated as the number of electrons absorbed by the mask functionmask function at the outer radial boundary. at the outer radial boundary.

Time-dependent Schrodinger equationTime-dependent Schrodinger equation

€

i∂Φ r,t( )

∂t= −

12∇2 −

2r

− zEX t( ) ⎡ ⎣ ⎢

⎤ ⎦ ⎥Φ r,t( )

Field of the soft x-ray pulse

CLEO/QELS2002 May 22, 2002 No. 6　

HeHe2+ 2+ yieldyield vs. vs. Peak intensityPeak intensity

Saturation can be seen at Saturation can be seen at II > 10 > 101313 W/cm W/cm22..– NotNot due to the decrease of the He due to the decrease of the He++ population. population. – Due to the ac-Stark effect. (Haberland 1987)Due to the ac-Stark effect. (Haberland 1987)

The two-photon ionization probability is sufficiently high to be The two-photon ionization probability is sufficiently high to be observed experimentally.observed experimentally.

The yield is approximately proportional to (Intensity)The yield is approximately proportional to (Intensity)22..

gaussian 30fs-27th harmonic pulsegaussian 30fs-27th harmonic pulse

““analytical” means,analytical” means,

= 2.9×10= 2.9×10-52-52 cm cm44ss

CLEO/QELS2002 May 22, 2002 No. 7　

Two-photon ionization probability Two-photon ionization probability ( He( He2+ 2+ Yield )Yield )

The HeThe He2+2+ yield is sufficiently yield is sufficiently high if we use the 27HG. high if we use the 27HG.

– 5.6×105.6×10-5-5 at at II = 2×10 = 2×101313W/cmW/cm22

– 3.3×103.3×10-4-4 at at II = 5×10 = 5×101313W/cmW/cm22

Ac-Stark effect → SaturationAc-Stark effect → Saturation

　　　　　　　　↓↓ Nevertheless…Nevertheless… The yield is approximately The yield is approximately

quadratic in intensity for the quadratic in intensity for the case of the 27HG.case of the 27HG.

Two-photon ionization probability by a Gaussian pulse with a duration (FWHM) of 30 fsTwo-photon ionization probability by a Gaussian pulse with a duration (FWHM) of 30 fs

CLEO/QELS2002 May 22, 2002 No. 8　

Three conditionsThree conditions

The two-photon ionization probability must The two-photon ionization probability must bebe

High.High. Quadratic in soft x-ray intensity. Quadratic in soft x-ray intensity. Linear in pulse duration.Linear in pulse duration.

OK

OK

CLEO/QELS2002 May 22, 2002 No. 9　

HeHe2+ 2+ yieldyield vs. vs. Pulse durationPulse duration

If the pulse duration is > 5 fs, the yield is proportional to the If the pulse duration is > 5 fs, the yield is proportional to the duration.duration.– Application such as the pulse duration measurement by Application such as the pulse duration measurement by

autocorrelationautocorrelation 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 2Excitation of the 2pp level due to energy uncertainty. level due to energy uncertainty.

of a gaussian 27th harmonic pulse with aof a gaussian 27th harmonic pulse with apeak intensity of 5×10peak intensity of 5×101313 W/cm W/cm22..

CLEO/QELS2002 May 22, 2002 No. 10　

Three conditionsThree conditions

The two-photon ionization probability ISThe 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)

AAttractive candidate for experimental ttractive candidate for experimental observation of a nonlinear optical effect in observation of a nonlinear optical effect in the soft x-ray domain.the soft x-ray domain.

CLEO/QELS2002 May 22, 2002 No. 11　

Two-photon ionization Two-photon ionization by a double pulseby a double pulse

27HG 27HG → The final He→ The final He2+2+ yield is yield is NOTNOT twice as twice as large as that after the first pulse. large as that after the first pulse.

Double pulseDouble pulse : : IntensityIntensity = 5×10 = 5×101313 W/cm W/cm22 、、 Pulse durationPulse duration = 1 fs = 1 fs

Quantum effectQuantum effectThe final yield depends onThe final yield depends on Time interval between the two pulsesTime interval between the two pulses Relative phase of the two pulsesRelative phase of the two pulses

Quantum effectQuantum effectThe final yield depends onThe final yield depends on Time interval between the two pulsesTime interval between the two pulses Relative phase of the two pulsesRelative phase of the two pulses

CLEO/QELS2002 May 22, 2002 No. 12　

Dependence on the relative phaseDependence on the relative phase

Single pulseSingle pulse– independent of independent of 11

Double pulseDouble pulse– The yield oscillates The yield oscillates

with with ..

Electric field of the double pulse

€

EX(t)=F1(t)sinωX +φ1( )+F2(t)sinωX +φ2( )

€

ωX =41.85 eV (27HG)

Pulse width = 1 fs, Interval = 4.14824 fs

Single pulse: F2(t)=0

CLEO/QELS2002 May 22, 2002 No. 13　

Dependence on the time intervalDependence on the time interval

The period (0.1 fs) of the dipole The period (0.1 fs) of the dipole moment oscillation corresponds to moment oscillation corresponds to the energy difference between 1s the energy difference between 1s and 2p.and 2p.

€

EX(t)=F1(t)sinωXt( )+F1(t−Δt)sinωX (t−Δt)( )

Electric field of the double pulse

Pulse width = 1 fs

€

ωX =41.85 eV (27HG)

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 He2+ yield.

€

hωd = 40.8 eV ⇒ ωd = 62.0 fs-1 ⇒ Td = 0.101 fs

0.1 fs 0.1 fs

CLEO/QELS2002 May 22, 2002 No. 14　

ConclusionsConclusions

The two-photon ionization of The two-photon ionization of HeHe++ by a 27th by a 27th harmonic pulse of a Ti:Sapphire laser is aharmonic pulse of a Ti:Sapphire laser is an n attractive candidate for experimental observation attractive candidate for experimental observation of of a nonlinear optical effect in the soft x-ray a nonlinear optical effect in the soft x-ray domaindomain..

– High probabilityHigh probability– Yield quadratic in pulse intensityYield quadratic in pulse intensity– Yield linear in pulse duration (> 5 fs)Yield linear in pulse duration (> 5 fs)

When the pulse duration is < 5 fs, the When the pulse duration is < 5 fs, the HeHe22++ yield is yield is NOTNOT linear in pulse duration. The yield by a double linear in pulse duration. The yield by a double pulse oscillates with the pulse interval.pulse oscillates with the pulse interval.

Phys. Rev. A Phys. Rev. A 6565, 043405 (2002), 043405 (2002)