AG Moderne Optik

Institut fur Physik

Humboldt-Universitat

zu Berlin, Germany

Non-dipole Effects in Multi-Photon Ionization of theHydrogen Atom using Intense Laser Pulses

E. O. Jobunga and A. Saenz

Acknowledgement

Figure 3. PE spectra at a peak electric field strength of (a)25 au (b)35 au for 15 cycle 13 nm laser pulse. Comparison ofrelative contributions of the multipole A · p and A2 components up to the quadrupole term. Black: dipole A · p, Blue:dipole A · p plus dipole A2, Red: dipole plus quadrupole A · p, and magenta: dipole plus quadrupole A · p+A2 terms.

References

1. R. R. Freeman and P. H. Bucksbaum, Investigations of above-threshold ionization using subpicosecond

laser pulses. Journal of Physics B: Atomic, Molecular and Optical Physics, Volume 24, Number 2, p.

325, 1991.

2. M. Førre and A. S. Simonsen, Nondipole ionization dynamics in atoms induced by intense xuv laser

fields. Phys. Rev. A 90, 053411, (2014).

3. B. H. Bransden and C. J. Joachain, Physics of atoms and molecules. Longman Scientific and Technical,

Essex, p.101, (1990).

Introduction and Motivation

The temporal and spatial variation of the laser has a great bearing on above threshold ionization (ATI)

spectrum and the ionization rate [1]. For laser fields with lower intensities and/or longer wavelengths,

the spatial distribution may be ignored and the use of dipole approximation is justified. But for high

intensity and/or very short wavelength laser sources currently available, the excursion radius r of an

atomic, ionic or molecular system in the radiation field may be comparable to the wavelength λ. This

would make the spatial dependence of the laser field non-negligible and consequently, the dipole (Dip)

approximation may not be valid. In this case, the simulation of the electron dynamical processes of the

system needs to include the spatial distribution of the laser field.

In this study, the non-dipole (ND) effects arising from the spatial variation are investigated upto the

quadrupole term of the multipole expansion. We were able to reproduce literature data [2] which were

also evaluated by spectral expansion using B splines. We confirm the existence of significant non-dipole

effects at all intensities corresponding with the peak electric field strengths considered in that work.

0 1 2 3 4 5 6 7 8 9 10Photoelectron Energy (a.u.)

10-3

10-2

10-1

100

dP

/dE

L=10

L=15

L=20

L=25

L=30

(b)

0 40 80 120 160 200 240 280 320 360Photoelectron Energy (a.u.)

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Lo

g10

(d

P/d

E)

Dip_A.p (SBA)

DIP_A.p (TA)

240 245 250 255 260-11

-10

-9(a)

0 0.5 1 1.5 2 2.5 3Photoelectron Energy (a.u.)

10-2

10-1

100

dP

/dE

Dip_A.p (TA)

Dip_A.p (SBA)

Dip_A.p + Dip_A.A

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

Dip_A.p (TA)

Dip_A.p (SBA)

Dip_A.p + Dip_A.A

(a)

0 0.5 1 1.5 2 2.5Photoelectron Energy (a.u.)

10-1

100

dP

/dE

Dip_A.p (TA)

Dip_A.p (SBA)

Dip_A.p + Dip_A.A (SBA)

0 0.02 0.04 0.06 0.08 0.10

2

4

6

Dip_A.p (TA)

Dip_A.p (SBA)

Dip_A.p + Dip_A.A (SBA)

(b)

Figure 2. PE spectrum at an intensity of (a) 1.0 × 1015 Wcm−2 and (b) 1.0 × 10

16 Wcm−2 for 800 nm 10 cyclepulse. Used: rmax = 2000 au, B splines= 4000, Lmax = 50. Ponderomotive shift Up significantly modifies the PEspectrum. Inset: Shows the very low energy structures ”(VLES)” enhanced by this ponderomotive shift.

Figure 1. (a) Comparison of the Taylor approximated(TA) and the sherical Bessel approximated (SBA) dipole PEspectrum at a peak electric field strength of 50 au for 9.11 nm 10 cycle pulse. Inset: Minor spatial modulation ofthe PE spectrum at higher energy regime. (b) Convergence of dipole plus electric quadrupole PE spectrum withangular momentum at a peak electric field strength of 45 au for a 13 nm 15 cycle laser pulse.

0 1 2 3 4 5 6 7 8 9 10

Photoelectron Energy (a.u.)

10-5

10-4

10-3

10-2

10-1

100

101

dP

/dE

Dip_A.p

Dip_A.p + Dip_A.A

Dip_A.p + ND_A.p

(Dip + ND)(A.p+A.A)

E0 = 25 a.u.(a)

0 1 2 3 4 5 6 7 8 9 10Photoelectron Energy (a.u.)

10-2

10-1

100

dP

/dE

E’0 = 25 au

E0 = 25 au

E’0 = 35 au

E0 = 35 au

E’0 = 45 au

E0 = 45 au

(b)

0 1 2 3 4 5 6 7 8 9 10Photoelectron Energy (a.u.)

10-4

10-3

10-2

10-1

100

101

dP

/dE

Dip_A.p

Dip_A.p + Dip_A.A

Dip_A.p + ND_A.p

(Dip + ND) (A.p + A.A)

E0 = 45 a.u.(a)

Figure 4. (a) Same as figure 3 but for a peak electric field strength of 45 au. (b) Multipole PE spectra for theelectric field amplitudes (E0 = 25, 35, 45) considered. Solid lines: ignore the derivative part of the CEP in theelectric field. Broken lines: includes the derivative part of the CEP. See A2(t) in the theory section.

Conclusion

It is observed that:

• ponderomotive shift enhances (very) low energy structures energy spectrum at higher intensities .

• both A · p and A2 terms have comparable beyond dipole corrections as long as Up is not ignored.

• as intensity increases the non-dipole effects distort the ATI structures.

• non-dipole effects persist even in the short pulse limit though in this limit the beating pattern becomes

visible.

• the published results of reference [2] were in excellent agreement with our dipole spectrum and quite

a good agreement in the non-dipole spectrum for the interactions compared.

0 2 4 6 8 10Photoelectron Energy (a.u.)

10-5

10-4

10-3

10-2

10-1

100

Dip_A.p

(Dip + ND) (A.p + A.A)

0 2 4 6 8 1010

-4

10-3

10-2

10-1

100

0 2 4 6 8 1010

-4

10-3

10-2

10-1

100

10 Osc. 5 Osc.

15 Osc.

dP

/dE

(b) (c)

(a)

0 1 2 3 4 50

0.2

0.4

0.6

0.8 Dip_A.p

Dip_A.p + ND_A.A

Ref_Dip

Ref_ND

0 1 2 3 4 50

0.2

0.4

0.6

0.8

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0 1 2 3 4 50

0.2

0.4

0.6

0.8

Dip_A.p + ND_A.A

Photoelectron Energy (a.u.)

dP

/dE

E0 = 25 a.u. E

0 = 35 a.u.

E0 = 45 a.u. E

0 = 45 a.u.

(a) (b)

(c) (d)

Dip Rmax

= 200 a.u.

ND Rmax

= 125 a.u.

ND Rmax

= 200 a.u.

Figure 5. Comparison of the present 13 nm dipole A · p

and the dipole plus quadrupole A2 term with reference[2] results. The reference underestimates the impact ofnon-dipole effects.

Figure 6. Variation of the present 13 nm dipole A · p

and the dipole plus quadrupole A · p + A2 PE spectrawith pulse duration for E0 = 25 au. The non-dipoleeffects still quite significant even for short cycle pulses.

Theory and Method• Non-relativistic time-dependent Schrodinger

equation (TDSE) of an atom interacting with a

classical electromagnetic field

i∂Ψ(r, t)

∂t= [H0 + Hint(r, t)]Ψ(r, t)

• Field-free Hamiltonian H0 and the interaction

potential Hint(r, t) in radiation gauge:

H0 =1

2p2 +V(r)

Hint(r, t) = −qA · p+1

2q2A2

• Vector potential

A(r, t) = A0 f(t) sin (ωt− k · r+ δ) z

= A1(t) cos(k · r)− A2(t) sin(k · r) z

= A(t) cos(k · r) +E(t)

ωsin(k · r) z

• Time-dependent potential functions

A1(t) = A0 f(t) sin(ωt+ δ)

A2(t) ≈ A0 f(t) cos(ωt+ δ)

= A0 [f(t)

ωsin(ωt+ δ) + f(t) cos(ωt+ δ)]

• Spatial dependence in Taylor (TA) multipole

expansion

eik·r = cos(k · r) + i sin(k · r)

=∞∑

l=0

(ik · r)l

l!

• Spatial dependence in spherical Bessel functions

(SBA) multipole expansion jl(kr) [3]

eik·r = 4π

∞∑

l=0

+l∑

ml=−l

iljl(kr)Ym∗l (k)Y m

l (r)

cos(k · r) = 4π∞∑

l,m=0,2,···

il jl(kr)Ym∗l (k)Y m

l (r)

sin(k · r) = 4π∞∑

l,m=1,3,···

i(l−1) jl(kr)Ym∗l (k)Y m

l (r)

• Ansatz spectrally expanded in field-free states

Ψ(r, t) =∑

α={n,l,m}

Cα(t)φα(r)

φα(r) = Rnα,lα(r)Ynα,lα(r)

Rnα,lα(r) =∑

i

ciBi(r)

r

• TDSE expanded as a set of coupled first-order

ordinary differential equations

iCα(t) = Cα(t)Eα +∑

α′

Cα′(t)Vαα′(t)

Vαα′(t) = 〈φα| Hint(r, t) |φα′〉

• Ionization yield is defined as

Yion =∑

α=all continuum states

| Cα(t = τ) |2

where τ is the total duration of the laser pulse.

0 1 2 3 4 5 6 7 8 9 10Photoelectron Energy (a.u.)

10-4

10-3

10-2

10-1

100

101

dP

/dE

Dip_A.p

Dip_A.p + Dip_A.A

Dip_A.p + ND_A.p

(Dip + ND) (A.p + A.A)

E0 = 35 a.u.(b)

Results

• System Parameters : Hydrogen atom, Initial state 1s.

• Laser Parameters: Wavelength = 9.11, 13, and 800nm.

Peak electric field strengths = 25, 35, 45, and50 au for short wavelengths, ∼ 1 au for 800nm.

Pulse duration = 5, 10, 15 optical cycles, linear polarization in the z direction.

• Numerical Parameters: Box radius rmax = 200 au, B splines= 600, Gauge= Radiation,

Angular momenta Lmax,Mmax = 30, Pulse envelope = Cos2, Approx. order = quadrupole.

• Definitions: Taylor Approximation (TA), Spherical Bessel functions Approximation (SBA). Unless

specified SBA is used throughout in our calculations.