Abstract

We investigate the photoelectron momentum distribution of molecular-ion H2+driven by ultrashort intense circularly polarized laser pulses. Both numerical solutions of the time-dependent Schrödinger equation (TDSE) and a quasiclassical model indicate that the photoelectron holography (PH) with circularly polarized pulses can occur in molecule. It is demonstrated that the interference between the direct electron wave and rescattered electron wave from one core to its neighboring core induces the PH. Moreover, the results of the TDSE predict that there is a tilt angle between the interference pattern of the PH and the direction perpendicular to the molecular axis. Furthermore, the tilt angle is sensitively dependent on the wavelength of the driven circularly polarized pulse, which is confirmed by the quasiclassical calculations. The PH induced by circularly polarized laser pulses provides a tool to resolve the electron dynamics and explore the spatial information of molecular structures.

© 2014 Optical Society of America

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  22. M. Protopapas, C. H. Keitel, P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60(4), 389–486 (1997).
    [CrossRef]
  23. W. F. Yang, X. H. Song, Z. N. Zeng, R. X. Li, Z. Z. Xu, “Quantum path interferences of electron trajectories in two-center molecules,” Opt. Express 18(3), 2558–2565 (2010).
    [CrossRef] [PubMed]
  24. W. F. Yang, X. H. Song, Z. J. Chen, “Phase-dependent above-barrier ionization of excited-state electrons,” Opt. Express 20(11), 12067–12075 (2012).
    [CrossRef] [PubMed]
  25. K. J. Yuan, A. D. Bandrauk, “Angle-dependent molecular above-threshold ionization with ultrashort intense linearly and circularly polarized laser pulses,” Phys. Rev. A 84(1), 013426 (2011).
    [CrossRef]
  26. M. Odenweller, N. Takemoto, A. Vredenborg, K. Cole, K. Pahl, J. Titze, L. P. H. Schmidt, T. Jahnke, R. Dörner, A. Becker, “Strong Field Electron Emission from Fixed in Space H2+ Ions,” Phys. Rev. Lett. 107(14), 143004 (2011).
    [CrossRef] [PubMed]

2012 (3)

X. B. Bian, A. D. Bandrauk, “Attosecond Time-Resolved Imaging of Molecular Structure by Photoelectron Holography,” Phys. Rev. Lett. 108(26), 263003 (2012).
[CrossRef] [PubMed]

Y. Huismans, A. Gijsbertsen, A. S. Smolkowska, J. H. Jungmann, A. Rouzée, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, M. Yu. Ivanov, T. M. Yan, D. Bauer, O. Smirnova, M. J. J. Vrakking, “Scaling Laws for Photoelectron Holography in the Midinfrared Wavelength Regime,” Phys. Rev. Lett. 109(1), 013002 (2012).
[CrossRef] [PubMed]

W. F. Yang, X. H. Song, Z. J. Chen, “Phase-dependent above-barrier ionization of excited-state electrons,” Opt. Express 20(11), 12067–12075 (2012).
[CrossRef] [PubMed]

2011 (5)

X. B. Bian, Y. Huismans, O. Smirnova, K. J. Yuan, M. J. J. Vrakking, A. D. Bandrauk, “Subcycle interference dynamics of time-resolved photoelectron holography with midinfrared laser pulses,” Phys. Rev. A 84(4), 043420 (2011).
[CrossRef]

K. J. Yuan, A. D. Bandrauk, “Angle-dependent molecular above-threshold ionization with ultrashort intense linearly and circularly polarized laser pulses,” Phys. Rev. A 84(1), 013426 (2011).
[CrossRef]

M. Odenweller, N. Takemoto, A. Vredenborg, K. Cole, K. Pahl, J. Titze, L. P. H. Schmidt, T. Jahnke, R. Dörner, A. Becker, “Strong Field Electron Emission from Fixed in Space H2+ Ions,” Phys. Rev. Lett. 107(14), 143004 (2011).
[CrossRef] [PubMed]

T. Marchenko, Y. Huismans, K. J. Schafer, M. J. J. Vrakking, “Criteria for the observation of strong-field photoelectron holography,” Phys. Rev. A 84(5), 053427 (2011).
[CrossRef]

Y. Huismans, A. Rouzée, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Yu. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, M. J. J. Vrakking, “Time-Resolved Holography with Photoelectrons,” Science 331(6013), 61–64 (2011).
[CrossRef] [PubMed]

2010 (3)

O. Smirnova, “Spectroscopy: Attosecond prints of electrons,” Nature 466(7307), 700–702 (2010).
[CrossRef] [PubMed]

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, F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010).
[CrossRef] [PubMed]

W. F. Yang, X. H. Song, Z. N. Zeng, R. X. Li, Z. Z. Xu, “Quantum path interferences of electron trajectories in two-center molecules,” Opt. Express 18(3), 2558–2565 (2010).
[CrossRef] [PubMed]

2009 (3)

F. Krausz, M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
[CrossRef]

Y. Zheng, Z. Zeng, P. Zou, L. Zhang, X. Li, P. Liu, R. Li, Z. Xu, “Dynamic chirp control and pulse compression for attosecond high-order harmonic emission,” Phys. Rev. Lett. 103(4), 043904 (2009).
[CrossRef] [PubMed]

W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, X. Liu, J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo, H. Xu, Y. Fu, Y. Cheng, Z. Z. Xu, “Classical aspects in above-threshold ionization with a midinfrared strong laser field,” Phys. Rev. Lett. 103(9), 093001 (2009).
[CrossRef] [PubMed]

2008 (1)

Q. Liao, P. Lu, P. Lan, W. Cao, Y. Li, “Phase dependence of high-order above-threshold ionization in asymmetric molecules,” Phys. Rev. A 77(1), 013408 (2008).
[CrossRef]

2007 (4)

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

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

M. Uiberacker, T. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H.-G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond real-time observation of electron tunnelling in atoms,” Nature 446(7136), 627–632 (2007).
[CrossRef] [PubMed]

W. F. Yang, X. H. Song, S. Q. Gong, Y. Cheng, Z. Z. Xu, “Carrier-Envelope Phase Dependence of Few-Cycle Ultrashort Laser Pulse Propagation in a Polar Molecule Medium,” Phys. Rev. Lett. 99(13), 133602 (2007).
[CrossRef] [PubMed]

2004 (1)

M. Spanner, O. Smirnova, P. B. Corkum, M. Y. Ivanov, “Reading diffraction images in strong field ionization of diatomic molecules,” J. Phys. At. Mol. Opt. Phys. 37(12), L243–L250 (2004).
[CrossRef]

2002 (1)

H. Niikura, F. Légaré, R. Hasbani, A. D. Bandrauk, M. Y. Ivanov, D. M. Villeneuve, P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417(6892), 917–922 (2002).
[CrossRef] [PubMed]

2000 (1)

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

1998 (1)

S. Chelkowski, C. Foisy, A. D. Bandrauk, “Electron-nuclear dynamics of multiphoton H2+ dissociative ionization in intense laser fields,” Phys. Rev. A 57(2), 1176–1185 (1998).
[CrossRef]

1997 (1)

M. Protopapas, C. H. Keitel, P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60(4), 389–486 (1997).
[CrossRef]

1993 (1)

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[CrossRef] [PubMed]

Azzeer, A. M.

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, F. Krausz, “Real-time observation of valence electron motion,” Nature 466(7307), 739–743 (2010).
[CrossRef] [PubMed]

Bakker, J. M.

Y. Huismans, A. Gijsbertsen, A. S. Smolkowska, J. H. Jungmann, A. Rouzée, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, M. Yu. Ivanov, T. M. Yan, D. Bauer, O. Smirnova, M. J. J. Vrakking, “Scaling Laws for Photoelectron Holography in the Midinfrared Wavelength Regime,” Phys. Rev. Lett. 109(1), 013002 (2012).
[CrossRef] [PubMed]

Y. Huismans, A. Rouzée, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Yu. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, M. J. J. Vrakking, “Time-Resolved Holography with Photoelectrons,” Science 331(6013), 61–64 (2011).
[CrossRef] [PubMed]

Bandrauk, A. D.

X. B. Bian, A. D. Bandrauk, “Attosecond Time-Resolved Imaging of Molecular Structure by Photoelectron Holography,” Phys. Rev. Lett. 108(26), 263003 (2012).
[CrossRef] [PubMed]

K. J. Yuan, A. D. Bandrauk, “Angle-dependent molecular above-threshold ionization with ultrashort intense linearly and circularly polarized laser pulses,” Phys. Rev. A 84(1), 013426 (2011).
[CrossRef]

X. B. Bian, Y. Huismans, O. Smirnova, K. J. Yuan, M. J. J. Vrakking, A. D. Bandrauk, “Subcycle interference dynamics of time-resolved photoelectron holography with midinfrared laser pulses,” Phys. Rev. A 84(4), 043420 (2011).
[CrossRef]

H. Niikura, F. Légaré, R. Hasbani, A. D. Bandrauk, M. Y. Ivanov, D. M. Villeneuve, P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417(6892), 917–922 (2002).
[CrossRef] [PubMed]

S. Chelkowski, C. Foisy, A. D. Bandrauk, “Electron-nuclear dynamics of multiphoton H2+ dissociative ionization in intense laser fields,” Phys. Rev. A 57(2), 1176–1185 (1998).
[CrossRef]

Bauer, D.

Y. Huismans, A. Gijsbertsen, A. S. Smolkowska, J. H. Jungmann, A. Rouzée, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, M. Yu. Ivanov, T. M. Yan, D. Bauer, O. Smirnova, M. J. J. Vrakking, “Scaling Laws for Photoelectron Holography in the Midinfrared Wavelength Regime,” Phys. Rev. Lett. 109(1), 013002 (2012).
[CrossRef] [PubMed]

Y. Huismans, A. Rouzée, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Yu. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, M. J. J. Vrakking, “Time-Resolved Holography with Photoelectrons,” Science 331(6013), 61–64 (2011).
[CrossRef] [PubMed]

Becker, A.

M. Odenweller, N. Takemoto, A. Vredenborg, K. Cole, K. Pahl, J. Titze, L. P. H. Schmidt, T. Jahnke, R. Dörner, A. Becker, “Strong Field Electron Emission from Fixed in Space H2+ Ions,” Phys. Rev. Lett. 107(14), 143004 (2011).
[CrossRef] [PubMed]

Berden, G.

Y. Huismans, A. Gijsbertsen, A. S. Smolkowska, J. H. Jungmann, A. Rouzée, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, M. Yu. Ivanov, T. M. Yan, D. Bauer, O. Smirnova, M. J. J. Vrakking, “Scaling Laws for Photoelectron Holography in the Midinfrared Wavelength Regime,” Phys. Rev. Lett. 109(1), 013002 (2012).
[CrossRef] [PubMed]

Y. Huismans, A. Rouzée, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Yu. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, M. J. J. Vrakking, “Time-Resolved Holography with Photoelectrons,” Science 331(6013), 61–64 (2011).
[CrossRef] [PubMed]

Bian, X. B.

X. B. Bian, A. D. Bandrauk, “Attosecond Time-Resolved Imaging of Molecular Structure by Photoelectron Holography,” Phys. Rev. Lett. 108(26), 263003 (2012).
[CrossRef] [PubMed]

X. B. Bian, Y. Huismans, O. Smirnova, K. J. Yuan, M. J. J. Vrakking, A. D. Bandrauk, “Subcycle interference dynamics of time-resolved photoelectron holography with midinfrared laser pulses,” Phys. Rev. A 84(4), 043420 (2011).
[CrossRef]

Brabec, T.

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

Cao, W.

Q. Liao, P. Lu, P. Lan, W. Cao, Y. Li, “Phase dependence of high-order above-threshold ionization in asymmetric molecules,” Phys. Rev. A 77(1), 013408 (2008).
[CrossRef]

Cauchy, C.

Y. Huismans, A. Gijsbertsen, A. S. Smolkowska, J. H. Jungmann, A. Rouzée, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, M. Yu. Ivanov, T. M. Yan, D. Bauer, O. Smirnova, M. J. J. Vrakking, “Scaling Laws for Photoelectron Holography in the Midinfrared Wavelength Regime,” Phys. Rev. Lett. 109(1), 013002 (2012).
[CrossRef] [PubMed]

Y. Huismans, A. Rouzée, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Yu. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, M. J. J. Vrakking, “Time-Resolved Holography with Photoelectrons,” Science 331(6013), 61–64 (2011).
[CrossRef] [PubMed]

Chelkowski, S.

S. Chelkowski, C. Foisy, A. D. Bandrauk, “Electron-nuclear dynamics of multiphoton H2+ dissociative ionization in intense laser fields,” Phys. Rev. A 57(2), 1176–1185 (1998).
[CrossRef]

Chen, J.

W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, X. Liu, J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo, H. Xu, Y. Fu, Y. Cheng, Z. Z. Xu, “Classical aspects in above-threshold ionization with a midinfrared strong laser field,” Phys. Rev. Lett. 103(9), 093001 (2009).
[CrossRef] [PubMed]

Chen, S. G.

W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, X. Liu, J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo, H. Xu, Y. Fu, Y. Cheng, Z. Z. Xu, “Classical aspects in above-threshold ionization with a midinfrared strong laser field,” Phys. Rev. Lett. 103(9), 093001 (2009).
[CrossRef] [PubMed]

Chen, Z. J.

Cheng, Y.

W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, X. Liu, J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo, H. Xu, Y. Fu, Y. Cheng, Z. Z. Xu, “Classical aspects in above-threshold ionization with a midinfrared strong laser field,” Phys. Rev. Lett. 103(9), 093001 (2009).
[CrossRef] [PubMed]

W. F. Yang, X. H. Song, S. Q. Gong, Y. Cheng, Z. Z. Xu, “Carrier-Envelope Phase Dependence of Few-Cycle Ultrashort Laser Pulse Propagation in a Polar Molecule Medium,” Phys. Rev. Lett. 99(13), 133602 (2007).
[CrossRef] [PubMed]

Cocke, C. L.

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

Cole, K.

M. Odenweller, N. Takemoto, A. Vredenborg, K. Cole, K. Pahl, J. Titze, L. P. H. Schmidt, T. Jahnke, R. Dörner, A. Becker, “Strong Field Electron Emission from Fixed in Space H2+ Ions,” Phys. Rev. Lett. 107(14), 143004 (2011).
[CrossRef] [PubMed]

Corkum, P. B.

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

M. Spanner, O. Smirnova, P. B. Corkum, M. Y. Ivanov, “Reading diffraction images in strong field ionization of diatomic molecules,” J. Phys. At. Mol. Opt. Phys. 37(12), L243–L250 (2004).
[CrossRef]

H. Niikura, F. Légaré, R. Hasbani, A. D. Bandrauk, M. Y. Ivanov, D. M. Villeneuve, P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417(6892), 917–922 (2002).
[CrossRef] [PubMed]

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[CrossRef] [PubMed]

Dörner, R.

M. Odenweller, N. Takemoto, A. Vredenborg, K. Cole, K. Pahl, J. Titze, L. P. H. Schmidt, T. Jahnke, R. Dörner, A. Becker, “Strong Field Electron Emission from Fixed in Space H2+ Ions,” Phys. Rev. Lett. 107(14), 143004 (2011).
[CrossRef] [PubMed]

Drescher, M.

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T. Marchenko, Y. Huismans, K. J. Schafer, M. J. J. Vrakking, “Criteria for the observation of strong-field photoelectron holography,” Phys. Rev. A 84(5), 053427 (2011).
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Y. Huismans, A. Rouzée, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Yu. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, M. J. J. Vrakking, “Time-Resolved Holography with Photoelectrons,” Science 331(6013), 61–64 (2011).
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Y. Huismans, A. Rouzée, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Yu. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, M. J. J. Vrakking, “Time-Resolved Holography with Photoelectrons,” Science 331(6013), 61–64 (2011).
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Figures (3)

Fig. 1
Fig. 1

Photoelectron momentum distribution of H 2 + driven by a circularly polarized laser pulse (a) on linear scale and (b) on logarithmic scale. The parameters of laser electric field are peak intensity I 0 = 6.0 × 10 14 W/cm2, wavelength λ = 400 nm, and duration T p = 17.35 fs. The lines are guides to the eye.

Fig. 2
Fig. 2

Photoelectron momentum distribution of atomic-like reference [(a) and (b)] and of H 2 + [(c)]. The driven lasers are a linearly polarized pulse [(a) and (c)], and a circularly polarized laser pulse [(b)]. The parameters of laser electric field are same with those of Fig. 1. The results are plotted on a logarithmic scale.

Fig. 3
Fig. 3

Comparison of photoelectron momentum distribution of the TDSE simulations [(a)-(c)] and interference fringes ~cos(ΔΦ) of the quasiclassical calculations [(d)-(f)]. The wavelengths of the laser are 320 nm [(a) and (d)], 420 nm [(b) and (e)], and 520 nm [(c) and (f)]. In the quasiclassical calculations, only the photoelectron scattered by the neighboring core is considered as the signal wave of the PH. The phase difference Δ Φ = 0 t c [ v x 2 ( t ' , ϕ ) + v y 2 ( t ' , ϕ ) 2 ] d t ' 0 t r [ v x 2 ( t ' , ϕ ' ) + v y 2 ( t ' , ϕ ' ) 2 ] d t ' I p ( ϕ ϕ ' ) ω leads to interference fringes ~cos(ΔΦ).

Equations (16)

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i t Ψ(r,t)=( p 2 2 +pA(t)+V(r) )Ψ(r,t)
V(x,y|R)= Z (x+R/ 2 ) 2 + y 2 +a + Z (xR/ 2 ) 2 + y 2 +a ,
Ψ( t i )=Ψ( t i )[1 F s ( r c )]+Ψ( t i ) F s ( r c )= Ψ I ( t i )+ Ψ II ( t i )
C(p, t i )= Ψ II ( t i ) e i[pA( t i )]r 2π d 2 r,
Ψ II (, t i )= C ¯ (p, t i ) e ipr 2π d 2 p,
dP(p) d E n = 2 E n | i C ¯ ( p, t i ) |.
A(t)= e ^ x E x0 ω f(t)sin(ωt)+ e ^ y E y0 ω f(t)cos(ωt)
v x (t,ϕ)= 0 t E x (t') dt'= E 0 ω [sin(ωt+ϕ)sin(ϕ)]
v y (t,ϕ)= 0 t E y (t') dt'= E 0 ω [cos(ωt+ϕ)cos(ϕ)],
x(t,ϕ)= 0 t v x (t')dt' = E 0 ω 2 [cos(ωt+ϕ)cos(ϕ)+ωsin(ϕ)t] x 0
y(t,φ)= 0 t v y (t')dt' = E 0 ω 2 [sin(ωt+ϕ)sin(ϕ)ωcos(ϕ)t] y 0 ,
v xf (t,ϕ)= t c t E x (t') dt' = E 0 ω [sin(ωt+ϕ)sin(ω t c +ϕ)]+v( t c ,ϕ)cos( θ c )
v yf (t,ϕ)= t c t E y (t') dt' = E 0 ω [cos(ωt+ϕ)cos(ω t c +ϕ)]+v( t c ,ϕ)sin( θ c ).
P xf = E 0 ω sin(ω t c +ϕ)+v( t c ,ϕ)cos( θ c ).
P yf = E 0 ω cos(ω t c +ϕ)+v( t c ,ϕ)sin( θ c ).
ΔΦ= 0 t c [ v x 2 ( t',ϕ )+ v y 2 ( t',ϕ ) 2 ] dt' 0 t r [ v x 2 ( t',ϕ' )+ v y 2 ( t',ϕ' ) 2 ] dt' I p ( ϕϕ' ) ω .

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