Chapter 6. Conclusion
6.2 Recommendation of future studies
1. The key to accurately predict the interaction of laser and multi-electron
effective potential pertinent to the real physical system. There are no systematic studies
about how to choose the parameters for a model potential. Systematic study soft
Coulomb potential parameters is strongly recommended to further improve the
applicability of the developed TDSE solver.
2. Current studies do not include the spin effect of electron, spin effect should be
taken into account. Then the model potential will be more general and capable to deal
the spin correlated problem.
References
1. Thomas Brabec and Ferenc Krausz, Rev. Mod. Phys. 72, 545 - 591 (2000).
2. S. X. Hu and L. A. Collins, Phys. Rev. Lett. 94, 123902 (2005).
Corkum, and D. M. Villeneuve, Nature (London) 432, 867 (2004).
7. M. Drescher, M. Hentschel, R. Kienberger, M. Uilevacker, V. Yakovlev, A.
Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz,
Nature London 419, 803 (2002).
8. M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, Th.
Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, J. Electron
Spectrosc. Relat. Phenom. 137, 259 (2004)
9. J. Muth-Böhm, A. Becker , and F. H. M. Faisal, Phys. Rev. Lett. 85, 2280
22. G.L. Kamta and A.D. Bandrauk, Phys. Rev. A 70, 011404 (2004).
23. G. Lagmago Kamta and A. D. Bandrauk, Phys. Rev. A 71, 053407 (2005).
24. G. Lagmago Kamta and A. D. Bandrauk, Phys. Rev. Lett. 94, 203003 (2005).
25. G. Lagmago Kamta and A. D. Bandrauk, Phys. Rev. A 74, 033415 (2006).
26. G. Lagmago Kamta and A. D. Bandrauk, Phys. Rev. A 75, 41401 (2007).
27. Dmitry A. Telnov and Shih-I Chu, Phys. Rev. A 71, 013408 (2005).
28. J. P. Hansen, T. So”revik and L. B. Madsen, Phys. Rev. A 68, 031401 (2003).
38. H. Yu and A.D. Bandrauk, J. Chem. Phys. 102, 1257 (1995).
39. M.V. Ammosov, N.B. Delone, and V.P. Krainov, Zh. Eksp.Teor. Fiz. 91, 2008
(1986).
40. J. H. Posthumus, Rep. Prog. Phys. 67, 623–665 (2004)
41. X. M. Tong, Z. X. Zhao, and C. D. Lin, Phys. Rev. A 66, 033402 (2002).
48. Bauer and Koval , Comput. Phys. Commun. 174, 396 (2006)
49. Kenneth C. Kulander, Phys. Rev. A 36, 2726 - 2738 (1987)
50. L. A. A. Nikolopoulos, T. K. Kjeldsen, and L. B. Madsen Phys. Rev. A 76,
033402 (2007)
51. Calvin Stubbins , Phys. Rev. A 48, 220 - 227 (1993)
52. Jonathan S Parker, Edward S Smyth and K T Taylor, J. Phys. B, At. Mol. Opt.
55. Manohar Awasthi, Yulian V. Vanne, Alejandro Saenz, Alberto Castro and Piero
Decleva, Phys. Rev. A 77, 063403 (2008)
56. M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. Elbert, M. S. Gordon, J. J.
Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. Su et al., J. Comput. Chem.
14, 1347 (1993).
dynamics: the finite volume method (Addison Wesley Longman, Harlow,
1995).
61. Daniel Dundas, J F McCann, Jonathan S Parker and K T Taylor, J. Phys. B: At.
Mol. Opt. Phys. 33, 3261-3276 (2000).
62. George Karypis and Vipin Kumar, SIAM Journal on Scientific Computing, Vol.
20, No. 1, pp. 359 – 392 (1999).
63. Herbert W. Jones and Babak Etemadi, Phys. Rev. A 47 03430 (1993).
64. Mohsen Vafaee, Hassan Sabzyan, Zahra Vafaee, and Ali Katanforoush, Phys.
Rev. A 74, 043416 (2006).
65. Domagoj Pavičić, Kevin F. Lee, D. M. Rayner, P. B. Corkum, and D. M.
Villeneuve Phys. Rev. Lett. 98, 243001 (2007).
Inter-nuclear distance (R)
Cell number of
computational mesh J-D solver (a.u.) Exact (a.u.) [76] Difference (%)
0.2 569,600 -1.9254908379592 -1.9286202 0.1623%
1 569,600 -1.4502284265718 -1.4517863133781 0.1073%
2 569,600 -1.1009589141748 -1.1026342144949 0.1519%
3 608,000 -0.9097227973807 -0.9108961973823 0.1288%
Table 1. Ground state eigen-energy of H2+
molecule in different inter-nuclear distance.
The eigen-energy are calculated by J-D solver and compared to exact value listed in reference 76. These calculated eigen-energy are different from exact values [76] by less than 0.2%.
Number of processors
Computation time (s)
Communication
time (s) Total time(s) Efficiency (%)
2 528.071 1.380444 529.4514 100
4 264.4042 2.596778 267.001 99.1478
8 132.1166 4.322556 136.4391 97.0124
16 68.39622 2.587222 70.98344 93.235
32 33.80311 0.985111 34.78822 95.1205
64 17.212 1.773333 18.98533 87.1481
128 9.512111 1.399333 10.91144 75.8165
Table 2. Timing breakdown of a typical parallel simulation for 400 timesteps (2,545,548 cells, time step size=0.01 a.u., laser intensity = 1014 W/cm2, and wave length
=1064nm.).
Case 1 Case 2 Angle of incidence
(degree) 0 90
Laser intensity (1014
W/cm2) 1 1
Wave length (nm) 1,064 1,064
Cell number 7,366,758 7,366,758
Simulation domain size
(a.u.) X<|96|, Y<|96|, Z<|112| X<|96|, Y<|96|, Z<|112|
Time step size (a.u.) 0.01 0.01
Inter nuclei distance
(a.u.) 9 9
Pulse cycles 25 (88.589 fs) 25 (88.589 fs)
Table 3. Simulation conditions for weaker laser incidence onto a H2+
molecule at χ=0°
and 90°.
Case A Case B Case C Case D
Cell number 13,461,224 13,461,224 13,461,224 13,461,224 Simulation
different angles of incidence (χ=0°, 30°, 60° and 90°).
Fitting type Ionization energy &
molecular orbital shape
Atom type N
Equation used for Yukawa like soft-coulomb potential
2-6b
Zeff 0.5
Zcore 7.0
α 1.35
β 0.51
Orbital energy -0.5748(5th MO)
Table 5. Fitting parameters of Yukawa like soft-coulomb potential for N2 molecule.
Fitting type Ionization energy &
molecular orbital shape
Atom type O
Equation used for Yukawa like soft-coulomb potential
2-6b
Zeff 0.5
Zcore 15.0
α 1.36
β 0.94
Orbital energy -0.4435(7th MO)
Table 6. Fitting parameters of Yukawa like soft-coulomb potential for O2 molecule.
Fitting type Ionization energy &
molecular orbital shape
Atom type C O
Equation used for Yukawa like soft-coulomb potential
2-6a 2-6b
Zeff X 0.5
Zcore 4 7
α 1.5 0.57
β 1.0 0.54
Orbital energy -0.5511(8th MO)
Table 7. Fitting parameters of Yukawa like soft-coulomb potential for CO2 molecule.
N2 O2 CO2
Cell number 2,153,704 1,163,158 1,902,752
Simulation domain
Table 8. Simulation conditions for laser incidence onto the N2, O2 and CO2 molecule at
different angles of incidence.
Figure 1. Sketch of the typical finite-element grid system projected in
two-dimensional space.
(x
0, y
0, z
0)
(x
1, y
1, z
1)
(x
1, y
1, z
0) (x
1, y
0, z
0)
(x
0, y
1, z
0) (x
0, y
1, z
1)
(x
1, y
0, z
1)
(x
0, y
0, z
1)
1
2 3
4 5
6 7
8
Figure 2. Local number and coordinates of the finite-element grid system.
Figure 3. Sketch of the typical finite-volume grid system projected in
two-dimensional space.
Figure 4. (a)Typical grid system for 3D TDSE simulation (a slice through the
midplane).
Figure 4. (b) Typical slice of domain decomposition through midplane (16
processors).
(c)
Figure 4. (c) Typical surface domain decomposition (16 processors).
0 10 20 30 40
Figure 5. Total electron probability and total energy variance of H2+
molecule
without laser incident, the internuclear distance is 9 au.
0 5 10 15 20 25
Time (pulse cycle)
-0.08 -0.04 0 0.04 0.08
L a s e r S tr e n g th
Figure 6. Applied electric field to the H2+
molecule along the H-H axis as a function
of time. Laser intensity = 1014 W/cm2, and wave length =1064nm.
0 2 4 6 8 10 12 14 16
Figure 7. Comparison of the ionization rates as a function of inter-nucleous distance,
obtained by the present parallelized 3D TDSE solver and previous
2D-axisymmetric TDSE solver for an aligned sub-femto-second linearly
polarized laser pulse interacting with a H2+
molecule (power
intensity=1014W/cm2, wave length=1064nm, pulse duration=25 cycles).
20 40 60 80 100 120 140
Figure 8. Parallel efficiency of the present parallelized 3D TDSE solver as a
function of the number of processors. Case 1: 2.54M cell grid and case2:
14.8M cell grid. Speedup is normalized by 2 processor data.
Figure 9. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=0 a.u. (0.00 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=0°).
Figure 10. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=450 a.u. (3.07 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=0°).
Figure 11. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=810 a.u. (5.53 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=0°).
Figure 12. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=900 a.u. (6.14 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=0°).
Figure 13. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=2250 a.u. (15.36 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=0°).
Figure 14. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=3600 a.u. (24.57 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=0°).
Figure 15. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=0 a.u. (0.00 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=90°).
Figure 16. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=450 a.u. (3.07 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=90°)
Figure 17. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=810 a.u. (5.53 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=90°)
Figure 18. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=900 a.u. (6.14 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=90°).
Figure 19. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=2250 a.u. (15.36 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=90°).
Figure 20. Typical snapshots of the electron probability distribution over the
axisymmetric plane for a normally incident sub-femto-second linearly
polarized laser pulse interacting when t=3600 a.u. (24.57 cycle) with a H2+
molecule (R=9) (power intensity=1014W/cm2, wave length=1064nm, pulse
duration=25 cycles, angle of incidence=90°).
1x10-8
Figure 21. Harmonic spectra of H2+
for different orientation angles: χ=0~90°. Laser
intensity = 5*1014 W/cm2, and wave length = 800nm. Internuclear distance
=2.0 a.u.
Figure 22. Slice contour topology of Yukawa like soft-coulomb potential for N2
molecule on x=0 plane. The maximum value of the potential is about 4.16.
Figure 23. The 3D iso-surface contour of initial wave function for laser-N2
molecule interaction. The black dots are the positions of the nuclear of N2
molecule. This is a σg type orbital. The orbital is symmetric to molecule
center and molecule axis.
Figure 24. The ionization yield to laser incidence angle in every 15∘for N2
molecule. Laser intensity = 1.5*1014 W/cm2, and wave length = 820nm.
Internuclear distance =2.075 a.u.
Figure 25. Ionization signal S(α) converted from measured ionization yield as a
function of the angleα between the polarization axes of the aligning and the
ionizing beams for N2, and The peak laser intensities is1.5*1014 W/cm2.
This data is from reference [65]. Red solid line and orange dash dotted line
are converted form experimental data by different method, green dotted line
is from MO-ADK calculation.
(a)
(b) Figure 26 Electron probability density distributions of N2 molecule under different
laser incidence angle on x=0 plane at t=5 optical cycle (~13.78 fs). (a) initial
(b)laser incidence angle of χ=0∘, 30∘, 60∘and 90∘.
Figure 27. The slice contour topology of Yukawa like soft-potential for O2
molecule on x=0 plane. The maximum value of the potential is about 7.4.
Figure 28. The 3D iso-surface contour of initial wave function for laser-O2
molecule interaction. The black dots are the positions of the nuclear of O2
molecule. This is a πg type orbital. The orbital is symmetric to molecule
center and anti-symmetric to molecule axis.
Figure 29. The ionization yield to laser incidence angle in every 15∘for O2
molecule. Laser intensity = 1.3*1014 W/cm2, and wave length = 820nm.
Internuclear distance =2.28 a.u.
Figure 30. Ionization signal S(α) converted from measured ionization yield as a
function of the angleα between the polarization axes of the aligning and the
ionizing beams for O2, and The peak laser intensities is1.3*1014 W/cm2.
This data is from reference [65], Red solid line and orange dash dotted line
are converted form experimental data by different method, green dotted line
is from MO-ADK calculation.
(a)
(b)
Figure 31. 3D iso-surface contour of electron probability density distributions of O2
molecule under different laser incidence angle at t=10 optical cycle (~27.57
Figure 32. Slice contour topology of Yukawa like soft-coulomb potential for CO2
molecule on x=0 plane. The maximum value of the potential is about 5.66.
Figure 33. The 3D iso-surface contour of initial wave function for laser-CO2
molecule interaction. The black dots are the positions of the nuclear of CO2
molecule. This is a πg type orbital. The orbital is symmetric to molecule
center and anti-symmetric to molecule axis.
Figure 34. The ionization yield to laser incidence angle in every 15∘for CO2
molecule. Laser intensity = 1.3*1014 W/cm2, and wave length = 820nm.
Internuclear distance =2.28 a.u.
Figure 35. Ionization signal S(α) converted from measured ionization yield as a
function of the angleα between the polarization axes of the aligning and the
ionizing beams for CO2, and The peak laser intensities is1.3*1014 W/cm2.
This data is from reference [65]. Red solid line and orange dash dotted line
are converted form experimental data by different method, green dotted line
is from MO-ADK calculation.
(a)
(b) Figure 36. 3D iso-surface contour of electron probability density distributions of
CO2 molecule under different laser incidence angle at t=10 optical cycle
(~27.57 fs). (a) initial (b)laser incidence angle of χ=0∘, 30∘, 60∘and
90∘.