Abstract

We numerically investigate high-order harmonic generation (HHG) in noble gases in the vicinity of fractal structures of metallic rough surfaces described by the restricted solid-on-solid model. The calculated intensity enhancement factors in the range of 103 enables HHG up to the 50-th order with low pump intensity down to tens of GW/cm2. The increased interaction volume of “hot spots” in the case of grazing incidence of s-polarized pump pulses leads to an efficiency of harmonics in the plateau region of about 10−7.

© 2011 Optical Society of America

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References

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  1. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
    [Crossref]
  2. S.-C. Kim, J.-H. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
    [Crossref] [PubMed]
  3. A. Husakou, S.-J. Im, and J. Herrmann, “Theory of plasmon-enhanced high-order harmonic generation in the vicinity of metal nanostructures in noble gases,” Phys. Rev. A 83(4), 043839 (2011).
    [Crossref]
  4. V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal-Dielectric Films (Springer-Verlag, 2000).
  5. M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, “Giant fluctuations of local optical fields in fractal clusters,” Phys. Rev. Lett. 72(15), 2486–2489 (1994).
    [Crossref] [PubMed]
  6. V. M. Shalaev, R. Botet, J. Mercer, and E. B. Stechel, “Optical properties of self-affine thin films,” Phys. Rev. B 54(11), 8235–8242 (1996).
    [Crossref]
  7. E. Y. Poliakov, V. A. Markel, V. M. Shalaev, and R. Botet, “Nonlinear optical phenomena on rough surfaces of metal thin films,” Phys. Rev. B 57(23), 14901–14913 (1998).
    [Crossref]
  8. C. K. Chen, A. R. B. de Castro, and Y. R. Shen, “Surface-enhanced second-harmonic generation,” Phys. Rev. Lett. 46(2), 145–148 (1981).
    [Crossref]
  9. K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surfaces,” Phys. Rev. B 55(12), 7985–7992 (1997).
    [Crossref]
  10. E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95(22), 227402 (2005).
    [Crossref] [PubMed]
  11. P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball, “Ballistic deposition on surfaces,” Phys. Rev. A 34(6), 5091–5103 (1986).
    [Crossref] [PubMed]
  12. J. M. Kim and J. M. Kosterlitz, “Growth in a restricted solid-on-solid model,” Phys. Rev. Lett. 62(19), 2289–2292 (1989).
    [Crossref] [PubMed]
  13. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994).
    [Crossref]
  14. K.-H. Kim, A. Husakou, and J. Herrmann, “Linear and nonlinear optical characteristics of composites containing metal nanoparticles with different sizes and shapes,” Opt. Express 18(7), 7488–7496 (2010).
    [Crossref] [PubMed]
  15. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
    [Crossref] [PubMed]

2011 (1)

A. Husakou, S.-J. Im, and J. Herrmann, “Theory of plasmon-enhanced high-order harmonic generation in the vicinity of metal nanostructures in noble gases,” Phys. Rev. A 83(4), 043839 (2011).
[Crossref]

2010 (1)

2009 (1)

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

2008 (1)

S.-C. Kim, J.-H. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

2005 (1)

E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95(22), 227402 (2005).
[Crossref] [PubMed]

1998 (1)

E. Y. Poliakov, V. A. Markel, V. M. Shalaev, and R. Botet, “Nonlinear optical phenomena on rough surfaces of metal thin films,” Phys. Rev. B 57(23), 14901–14913 (1998).
[Crossref]

1997 (1)

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surfaces,” Phys. Rev. B 55(12), 7985–7992 (1997).
[Crossref]

1996 (1)

V. M. Shalaev, R. Botet, J. Mercer, and E. B. Stechel, “Optical properties of self-affine thin films,” Phys. Rev. B 54(11), 8235–8242 (1996).
[Crossref]

1994 (3)

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, “Giant fluctuations of local optical fields in fractal clusters,” Phys. Rev. Lett. 72(15), 2486–2489 (1994).
[Crossref] [PubMed]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994).
[Crossref]

1989 (1)

J. M. Kim and J. M. Kosterlitz, “Growth in a restricted solid-on-solid model,” Phys. Rev. Lett. 62(19), 2289–2292 (1989).
[Crossref] [PubMed]

1986 (1)

P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball, “Ballistic deposition on surfaces,” Phys. Rev. A 34(6), 5091–5103 (1986).
[Crossref] [PubMed]

1981 (1)

C. K. Chen, A. R. B. de Castro, and Y. R. Shen, “Surface-enhanced second-harmonic generation,” Phys. Rev. Lett. 46(2), 145–148 (1981).
[Crossref]

Aktsipetrov, O. A.

E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95(22), 227402 (2005).
[Crossref] [PubMed]

Bader, M. A.

E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95(22), 227402 (2005).
[Crossref] [PubMed]

Balcou, P.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

Ball, R. C.

P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball, “Ballistic deposition on surfaces,” Phys. Rev. A 34(6), 5091–5103 (1986).
[Crossref] [PubMed]

Botet, R.

E. Y. Poliakov, V. A. Markel, V. M. Shalaev, and R. Botet, “Nonlinear optical phenomena on rough surfaces of metal thin films,” Phys. Rev. B 57(23), 14901–14913 (1998).
[Crossref]

V. M. Shalaev, R. Botet, J. Mercer, and E. B. Stechel, “Optical properties of self-affine thin films,” Phys. Rev. B 54(11), 8235–8242 (1996).
[Crossref]

Chen, C. K.

C. K. Chen, A. R. B. de Castro, and Y. R. Shen, “Surface-enhanced second-harmonic generation,” Phys. Rev. Lett. 46(2), 145–148 (1981).
[Crossref]

Corkum, P. B.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

de Castro, A. R. B.

C. K. Chen, A. R. B. de Castro, and Y. R. Shen, “Surface-enhanced second-harmonic generation,” Phys. Rev. Lett. 46(2), 145–148 (1981).
[Crossref]

Draine, B. T.

Elovikov, S. S.

E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95(22), 227402 (2005).
[Crossref] [PubMed]

Flatau, P. J.

George, T. F.

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, “Giant fluctuations of local optical fields in fractal clusters,” Phys. Rev. Lett. 72(15), 2486–2489 (1994).
[Crossref] [PubMed]

Herrmann, J.

A. Husakou, S.-J. Im, and J. Herrmann, “Theory of plasmon-enhanced high-order harmonic generation in the vicinity of metal nanostructures in noble gases,” Phys. Rev. A 83(4), 043839 (2011).
[Crossref]

K.-H. Kim, A. Husakou, and J. Herrmann, “Linear and nonlinear optical characteristics of composites containing metal nanoparticles with different sizes and shapes,” Opt. Express 18(7), 7488–7496 (2010).
[Crossref] [PubMed]

Husakou, A.

A. Husakou, S.-J. Im, and J. Herrmann, “Theory of plasmon-enhanced high-order harmonic generation in the vicinity of metal nanostructures in noble gases,” Phys. Rev. A 83(4), 043839 (2011).
[Crossref]

K.-H. Kim, A. Husakou, and J. Herrmann, “Linear and nonlinear optical characteristics of composites containing metal nanoparticles with different sizes and shapes,” Opt. Express 18(7), 7488–7496 (2010).
[Crossref] [PubMed]

Im, S.-J.

A. Husakou, S.-J. Im, and J. Herrmann, “Theory of plasmon-enhanced high-order harmonic generation in the vicinity of metal nanostructures in noble gases,” Phys. Rev. A 83(4), 043839 (2011).
[Crossref]

Ivanov, M.

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

Ivanov, M. Y.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

Jin, J.-H.

S.-C. Kim, J.-H. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Kim, E. M.

E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95(22), 227402 (2005).
[Crossref] [PubMed]

Kim, J. M.

J. M. Kim and J. M. Kosterlitz, “Growth in a restricted solid-on-solid model,” Phys. Rev. Lett. 62(19), 2289–2292 (1989).
[Crossref] [PubMed]

Kim, K.-H.

Kim, S.-C.

S.-C. Kim, J.-H. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Kim, S.-W.

S.-C. Kim, J.-H. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Kim, Y.

S.-C. Kim, J.-H. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Kim, Y.-J.

S.-C. Kim, J.-H. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Kosterlitz, J. M.

J. M. Kim and J. M. Kosterlitz, “Growth in a restricted solid-on-solid model,” Phys. Rev. Lett. 62(19), 2289–2292 (1989).
[Crossref] [PubMed]

Krausz, F.

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

L’Huillier, A.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

Lewenstein, M.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

Markel, V. A.

E. Y. Poliakov, V. A. Markel, V. M. Shalaev, and R. Botet, “Nonlinear optical phenomena on rough surfaces of metal thin films,” Phys. Rev. B 57(23), 14901–14913 (1998).
[Crossref]

Marowsky, G.

E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95(22), 227402 (2005).
[Crossref] [PubMed]

Meakin, P.

P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball, “Ballistic deposition on surfaces,” Phys. Rev. A 34(6), 5091–5103 (1986).
[Crossref] [PubMed]

Mercer, J.

V. M. Shalaev, R. Botet, J. Mercer, and E. B. Stechel, “Optical properties of self-affine thin films,” Phys. Rev. B 54(11), 8235–8242 (1996).
[Crossref]

Muratov, L. S.

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, “Giant fluctuations of local optical fields in fractal clusters,” Phys. Rev. Lett. 72(15), 2486–2489 (1994).
[Crossref] [PubMed]

Murzina, T. V.

E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95(22), 227402 (2005).
[Crossref] [PubMed]

Nikulin, A. A.

E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95(22), 227402 (2005).
[Crossref] [PubMed]

O’Donnell, K. A.

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surfaces,” Phys. Rev. B 55(12), 7985–7992 (1997).
[Crossref]

Pandey, L. N.

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, “Giant fluctuations of local optical fields in fractal clusters,” Phys. Rev. Lett. 72(15), 2486–2489 (1994).
[Crossref] [PubMed]

Park, I.-Y.

S.-C. Kim, J.-H. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Poliakov, E. Y.

E. Y. Poliakov, V. A. Markel, V. M. Shalaev, and R. Botet, “Nonlinear optical phenomena on rough surfaces of metal thin films,” Phys. Rev. B 57(23), 14901–14913 (1998).
[Crossref]

Ramanlal, P.

P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball, “Ballistic deposition on surfaces,” Phys. Rev. A 34(6), 5091–5103 (1986).
[Crossref] [PubMed]

Sander, L. M.

P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball, “Ballistic deposition on surfaces,” Phys. Rev. A 34(6), 5091–5103 (1986).
[Crossref] [PubMed]

Shalaev, V. M.

E. Y. Poliakov, V. A. Markel, V. M. Shalaev, and R. Botet, “Nonlinear optical phenomena on rough surfaces of metal thin films,” Phys. Rev. B 57(23), 14901–14913 (1998).
[Crossref]

V. M. Shalaev, R. Botet, J. Mercer, and E. B. Stechel, “Optical properties of self-affine thin films,” Phys. Rev. B 54(11), 8235–8242 (1996).
[Crossref]

V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal-Dielectric Films (Springer-Verlag, 2000).

Shen, Y. R.

C. K. Chen, A. R. B. de Castro, and Y. R. Shen, “Surface-enhanced second-harmonic generation,” Phys. Rev. Lett. 46(2), 145–148 (1981).
[Crossref]

Stechel, E. B.

V. M. Shalaev, R. Botet, J. Mercer, and E. B. Stechel, “Optical properties of self-affine thin films,” Phys. Rev. B 54(11), 8235–8242 (1996).
[Crossref]

Stockman, M. I.

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, “Giant fluctuations of local optical fields in fractal clusters,” Phys. Rev. Lett. 72(15), 2486–2489 (1994).
[Crossref] [PubMed]

Torre, R.

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surfaces,” Phys. Rev. B 55(12), 7985–7992 (1997).
[Crossref]

West, C. S.

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surfaces,” Phys. Rev. B 55(12), 7985–7992 (1997).
[Crossref]

J. Opt. Soc. Am. A (1)

Nature (1)

S.-C. Kim, J.-H. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Opt. Express (1)

Phys. Rev. A (3)

P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball, “Ballistic deposition on surfaces,” Phys. Rev. A 34(6), 5091–5103 (1986).
[Crossref] [PubMed]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

A. Husakou, S.-J. Im, and J. Herrmann, “Theory of plasmon-enhanced high-order harmonic generation in the vicinity of metal nanostructures in noble gases,” Phys. Rev. A 83(4), 043839 (2011).
[Crossref]

Phys. Rev. B (3)

V. M. Shalaev, R. Botet, J. Mercer, and E. B. Stechel, “Optical properties of self-affine thin films,” Phys. Rev. B 54(11), 8235–8242 (1996).
[Crossref]

E. Y. Poliakov, V. A. Markel, V. M. Shalaev, and R. Botet, “Nonlinear optical phenomena on rough surfaces of metal thin films,” Phys. Rev. B 57(23), 14901–14913 (1998).
[Crossref]

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of second-harmonic generation from randomly rough metal surfaces,” Phys. Rev. B 55(12), 7985–7992 (1997).
[Crossref]

Phys. Rev. Lett. (4)

E. M. Kim, S. S. Elovikov, T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, M. A. Bader, and G. Marowsky, “Surface-enhanced optical third-harmonic generation in Ag island films,” Phys. Rev. Lett. 95(22), 227402 (2005).
[Crossref] [PubMed]

C. K. Chen, A. R. B. de Castro, and Y. R. Shen, “Surface-enhanced second-harmonic generation,” Phys. Rev. Lett. 46(2), 145–148 (1981).
[Crossref]

M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, “Giant fluctuations of local optical fields in fractal clusters,” Phys. Rev. Lett. 72(15), 2486–2489 (1994).
[Crossref] [PubMed]

J. M. Kim and J. M. Kosterlitz, “Growth in a restricted solid-on-solid model,” Phys. Rev. Lett. 62(19), 2289–2292 (1989).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

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

Other (1)

V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal-Dielectric Films (Springer-Verlag, 2000).

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Figures (5)

Fig. 1
Fig. 1 (Color online) Structure of the rough surface and enhanced field distribution. The surface structure is illustrated by yellow color and relative magnitudes and directions of enhanced field are shown by red cones. The blue cone shows the polarization of incident field at 800 nm. Voxel size of 1 nm and 50 nm × 50 nm samples were considered.

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Fig. 2
Fig. 2 (Color online) Maximum intensity enhancement factor in different cross-sections for different wavelengths: 700 nm (a)–(c); 800 nm (d)–(f); 900 nm (g)–(i).

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Fig. 3
Fig. 3 (Color online) Incident pulse (blue dotted line) and enhanced pulse (green line) profiles on the rough Ag surface for one of the realizations of random surface. Maximum field enhancement is more than 40 which corresponds to the instant intensity enhancement of ∼ 2000.

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Fig. 4
Fig. 4 (Color online) The HHG efficiency ηav (N) as a function of harmonic orders N averaged over an ensemble of 20 random samples (thick red line) and for some of the individual samples (thin gray lines).

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Fig. 5
Fig. 5 (Color online) The dependence of HHG efficiency on the incident angle for s- (blue curve) and p- (red curve) polarizations of the pump beam. HHG efficiency spectra for incident angles 0 °, 89 ° and 89.9 ° for s-polarization are shown in (a), (b) and (c), respectively. In (d), the HHG efficiencies in the range of plateau are shown.

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Equations (1)

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x ′ ( t ) = i ∫ 0 ∞ d τ ( π ɛ + i τ / 2 ) 1.5 d x * [ p s t ( t , τ ) − A ( t ) ] × d x [ p s t ( t , τ ) − A ( t − τ ) ] E enh ( t − τ ) × exp [ − i S s t ( t , τ ) ] + c . c .

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