Abstract

The generation of specific high harmonics for an optical two-level system is elucidated. The desired emitted radiation can be induced by a carefully designed excitation pulse, which is found by a multiparameter optimization procedure. The presented mechanism can also be applied to semiconductor structures for which the calculations result in much higher emission frequencies. The optimization procedure is either performed using a genetic algorithm or a rigorous mathematical optimization technique.

© 2012 Optical Society of America

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References

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  1. M. Wegener, Extreme Nonlinear Optics (Springer, 2005).
  2. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).
  3. P. Meystre and M. Sargent, Elements of Quantum Optics, 2nd ed. (Springer, 1991).
  4. T. Tritschler, O. D. Mücke, and M. Wegener, “Extreme nonlinear optics of two-level systems,” Phys. Rev. A 68, 033404 (2003).
    [CrossRef]
  5. D. Golde, T. Meier, and S. W. Koch, “Microscopic analysis of extreme nonlinear optics in semiconductor nanostructures,” J. Opt. Soc. Am. B 23, 2559–2565 (2006).
    [CrossRef]
  6. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
    [CrossRef]
  7. T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
    [CrossRef]
  8. M. Sobczak, A. Plucińska, and R. Parzyński, “Few-cycle light pulses in two-level systems: effects of pulse carrier-envelope offset phase in low-order harmonic spectra,” Laser Phys. 14, 1483–1487 (2004).
  9. X. Song, S. Gong, W. Yang, and Z. Xu, “Propagation of an attosecond pulse in a dense two-level medium,” Phys. Rev. A 70, 013817 (2004).
    [CrossRef]
  10. M. Holthaus and B. Just, “Generalized π pulses,” Phys. Rev. A 49, 1950–1960 (1994).
    [CrossRef]
  11. O. D. Mücke, “Isolated high-order harmonics pulse from two-color-driven Bloch oscillations in bulk semiconductors,” Phys. Rev. B 84, 081202(R) (2011).
    [CrossRef]
  12. D. Golde, T. Meier, and S. W. Koch, “High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations,” Phys. Rev. B 77, 075330 (2008).
    [CrossRef]
  13. C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
    [CrossRef]
  14. E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
    [CrossRef]
  15. S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guérin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
    [CrossRef]
  16. T. Meier, P. Thomas, and S. W. Koch, Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (Springer, 2007).
  17. B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 188, 1969–1975 (1969).
    [CrossRef]
  18. A. E. Eiben and J. E. Smith, Introduction to Evolutionary Computing (Springer, 2003).
  19. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific, 2009).
  20. M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B 38, 3342–3350 (1988).
    [CrossRef]
  21. G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
    [CrossRef]
  22. T. Meier, G. von Plessen, P. Thomas, and S. W. Koch, ”Coherent electric-field effects in semiconductors,” Phys. Rev. Lett. 73, 902–905 (1994).
    [CrossRef]
  23. T. Meier, F. Rossi, P. Thomas, and S. W. Koch, “Dynamic localization in anisotropic Coulomb systems: field induced crossover of the exciton dimension,” Phys. Rev. Lett. 75, 2558–2561 (1995).
    [CrossRef]
  24. A. Wächter and L. T. Biegler, “On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming,” Math. Program. 106, 25–57 (2006).
    [CrossRef]
  25. J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).
  26. R. H. Byrd, P. Lu, and J. Nocedal, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190–1208 (1995).
    [CrossRef]
  27. A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed.(Society for Industrial and Applied Mathematics, 2008).

2011 (1)

O. D. Mücke, “Isolated high-order harmonics pulse from two-color-driven Bloch oscillations in bulk semiconductors,” Phys. Rev. B 84, 081202(R) (2011).
[CrossRef]

2010 (1)

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guérin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

2009 (1)

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

2008 (1)

D. Golde, T. Meier, and S. W. Koch, “High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations,” Phys. Rev. B 77, 075330 (2008).
[CrossRef]

2007 (1)

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef]

2006 (3)

A. Wächter and L. T. Biegler, “On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming,” Math. Program. 106, 25–57 (2006).
[CrossRef]

C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
[CrossRef]

D. Golde, T. Meier, and S. W. Koch, “Microscopic analysis of extreme nonlinear optics in semiconductor nanostructures,” J. Opt. Soc. Am. B 23, 2559–2565 (2006).
[CrossRef]

2004 (2)

M. Sobczak, A. Plucińska, and R. Parzyński, “Few-cycle light pulses in two-level systems: effects of pulse carrier-envelope offset phase in low-order harmonic spectra,” Laser Phys. 14, 1483–1487 (2004).

X. Song, S. Gong, W. Yang, and Z. Xu, “Propagation of an attosecond pulse in a dense two-level medium,” Phys. Rev. A 70, 013817 (2004).
[CrossRef]

2003 (1)

T. Tritschler, O. D. Mücke, and M. Wegener, “Extreme nonlinear optics of two-level systems,” Phys. Rev. A 68, 033404 (2003).
[CrossRef]

2000 (1)

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

1999 (1)

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

1995 (2)

T. Meier, F. Rossi, P. Thomas, and S. W. Koch, “Dynamic localization in anisotropic Coulomb systems: field induced crossover of the exciton dimension,” Phys. Rev. Lett. 75, 2558–2561 (1995).
[CrossRef]

R. H. Byrd, P. Lu, and J. Nocedal, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190–1208 (1995).
[CrossRef]

1994 (2)

T. Meier, G. von Plessen, P. Thomas, and S. W. Koch, ”Coherent electric-field effects in semiconductors,” Phys. Rev. Lett. 73, 902–905 (1994).
[CrossRef]

M. Holthaus and B. Just, “Generalized π pulses,” Phys. Rev. A 49, 1950–1960 (1994).
[CrossRef]

1988 (1)

M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B 38, 3342–3350 (1988).
[CrossRef]

1969 (1)

B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 188, 1969–1975 (1969).
[CrossRef]

Ahn, J.

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guérin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Biegler, L. T.

A. Wächter and L. T. Biegler, “On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming,” Math. Program. 106, 25–57 (2006).
[CrossRef]

Brabec, T.

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

Byrd, R. H.

R. H. Byrd, P. Lu, and J. Nocedal, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190–1208 (1995).
[CrossRef]

Cohen, J. L.

C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
[CrossRef]

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Eiben, A. E.

A. E. Eiben and J. E. Smith, Introduction to Evolutionary Computing (Springer, 2003).

Gibbs, H. M.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

Golde, D.

D. Golde, T. Meier, and S. W. Koch, “High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations,” Phys. Rev. B 77, 075330 (2008).
[CrossRef]

D. Golde, T. Meier, and S. W. Koch, “Microscopic analysis of extreme nonlinear optics in semiconductor nanostructures,” J. Opt. Soc. Am. B 23, 2559–2565 (2006).
[CrossRef]

Gong, S.

X. Song, S. Gong, W. Yang, and Z. Xu, “Propagation of an attosecond pulse in a dense two-level medium,” Phys. Rev. A 70, 013817 (2004).
[CrossRef]

Griewank, A.

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed.(Society for Industrial and Applied Mathematics, 2008).

Guérin, S.

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guérin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

Hakobyan, V.

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guérin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

Haug, H.

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific, 2009).

Holthaus, M.

M. Holthaus and B. Just, “Generalized π pulses,” Phys. Rev. A 49, 1950–1960 (1994).
[CrossRef]

Ivanov, M.

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

Jahnke, F.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

Just, B.

M. Holthaus and B. Just, “Generalized π pulses,” Phys. Rev. A 49, 1950–1960 (1994).
[CrossRef]

Khitrova, G.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

Kira, M.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

Koch, S. W.

D. Golde, T. Meier, and S. W. Koch, “High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations,” Phys. Rev. B 77, 075330 (2008).
[CrossRef]

D. Golde, T. Meier, and S. W. Koch, “Microscopic analysis of extreme nonlinear optics in semiconductor nanostructures,” J. Opt. Soc. Am. B 23, 2559–2565 (2006).
[CrossRef]

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

T. Meier, F. Rossi, P. Thomas, and S. W. Koch, “Dynamic localization in anisotropic Coulomb systems: field induced crossover of the exciton dimension,” Phys. Rev. Lett. 75, 2558–2561 (1995).
[CrossRef]

T. Meier, G. von Plessen, P. Thomas, and S. W. Koch, ”Coherent electric-field effects in semiconductors,” Phys. Rev. Lett. 73, 902–905 (1994).
[CrossRef]

M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B 38, 3342–3350 (1988).
[CrossRef]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific, 2009).

T. Meier, P. Thomas, and S. W. Koch, Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (Springer, 2007).

Krausz, F.

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

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

Lee, S.

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guérin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

Lim, J.

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guérin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

Lindberg, M.

M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B 38, 3342–3350 (1988).
[CrossRef]

Lu, P.

R. H. Byrd, P. Lu, and J. Nocedal, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190–1208 (1995).
[CrossRef]

Meier, T.

D. Golde, T. Meier, and S. W. Koch, “High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations,” Phys. Rev. B 77, 075330 (2008).
[CrossRef]

D. Golde, T. Meier, and S. W. Koch, “Microscopic analysis of extreme nonlinear optics in semiconductor nanostructures,” J. Opt. Soc. Am. B 23, 2559–2565 (2006).
[CrossRef]

T. Meier, F. Rossi, P. Thomas, and S. W. Koch, “Dynamic localization in anisotropic Coulomb systems: field induced crossover of the exciton dimension,” Phys. Rev. Lett. 75, 2558–2561 (1995).
[CrossRef]

T. Meier, G. von Plessen, P. Thomas, and S. W. Koch, ”Coherent electric-field effects in semiconductors,” Phys. Rev. Lett. 73, 902–905 (1994).
[CrossRef]

T. Meier, P. Thomas, and S. W. Koch, Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (Springer, 2007).

Menzel-Jones, C.

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef]

Meystre, P.

P. Meystre and M. Sargent, Elements of Quantum Optics, 2nd ed. (Springer, 1991).

Milner, V.

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef]

Mollow, B. R.

B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 188, 1969–1975 (1969).
[CrossRef]

Mücke, O. D.

O. D. Mücke, “Isolated high-order harmonics pulse from two-color-driven Bloch oscillations in bulk semiconductors,” Phys. Rev. B 84, 081202(R) (2011).
[CrossRef]

T. Tritschler, O. D. Mücke, and M. Wegener, “Extreme nonlinear optics of two-level systems,” Phys. Rev. A 68, 033404 (2003).
[CrossRef]

Nocedal, J.

R. H. Byrd, P. Lu, and J. Nocedal, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190–1208 (1995).
[CrossRef]

J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

Parzynski, R.

M. Sobczak, A. Plucińska, and R. Parzyński, “Few-cycle light pulses in two-level systems: effects of pulse carrier-envelope offset phase in low-order harmonic spectra,” Laser Phys. 14, 1483–1487 (2004).

Plucinska, A.

M. Sobczak, A. Plucińska, and R. Parzyński, “Few-cycle light pulses in two-level systems: effects of pulse carrier-envelope offset phase in low-order harmonic spectra,” Laser Phys. 14, 1483–1487 (2004).

Rossi, F.

T. Meier, F. Rossi, P. Thomas, and S. W. Koch, “Dynamic localization in anisotropic Coulomb systems: field induced crossover of the exciton dimension,” Phys. Rev. Lett. 75, 2558–2561 (1995).
[CrossRef]

Sargent, M.

P. Meystre and M. Sargent, Elements of Quantum Optics, 2nd ed. (Springer, 1991).

Shapiro, E. A.

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef]

Shapiro, M.

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef]

Smith, J. E.

A. E. Eiben and J. E. Smith, Introduction to Evolutionary Computing (Springer, 2003).

Sobczak, M.

M. Sobczak, A. Plucińska, and R. Parzyński, “Few-cycle light pulses in two-level systems: effects of pulse carrier-envelope offset phase in low-order harmonic spectra,” Laser Phys. 14, 1483–1487 (2004).

Song, X.

X. Song, S. Gong, W. Yang, and Z. Xu, “Propagation of an attosecond pulse in a dense two-level medium,” Phys. Rev. A 70, 013817 (2004).
[CrossRef]

Thomas, P.

T. Meier, F. Rossi, P. Thomas, and S. W. Koch, “Dynamic localization in anisotropic Coulomb systems: field induced crossover of the exciton dimension,” Phys. Rev. Lett. 75, 2558–2561 (1995).
[CrossRef]

T. Meier, G. von Plessen, P. Thomas, and S. W. Koch, ”Coherent electric-field effects in semiconductors,” Phys. Rev. Lett. 73, 902–905 (1994).
[CrossRef]

T. Meier, P. Thomas, and S. W. Koch, Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (Springer, 2007).

Trallero-Herrero, C.

C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
[CrossRef]

Tritschler, T.

T. Tritschler, O. D. Mücke, and M. Wegener, “Extreme nonlinear optics of two-level systems,” Phys. Rev. A 68, 033404 (2003).
[CrossRef]

von Plessen, G.

T. Meier, G. von Plessen, P. Thomas, and S. W. Koch, ”Coherent electric-field effects in semiconductors,” Phys. Rev. Lett. 73, 902–905 (1994).
[CrossRef]

Wächter, A.

A. Wächter and L. T. Biegler, “On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming,” Math. Program. 106, 25–57 (2006).
[CrossRef]

Walther, A.

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed.(Society for Industrial and Applied Mathematics, 2008).

Wegener, M.

T. Tritschler, O. D. Mücke, and M. Wegener, “Extreme nonlinear optics of two-level systems,” Phys. Rev. A 68, 033404 (2003).
[CrossRef]

M. Wegener, Extreme Nonlinear Optics (Springer, 2005).

Weinacht, T.

C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
[CrossRef]

Wright, S.

J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

Xu, Z.

X. Song, S. Gong, W. Yang, and Z. Xu, “Propagation of an attosecond pulse in a dense two-level medium,” Phys. Rev. A 70, 013817 (2004).
[CrossRef]

Yang, W.

X. Song, S. Gong, W. Yang, and Z. Xu, “Propagation of an attosecond pulse in a dense two-level medium,” Phys. Rev. A 70, 013817 (2004).
[CrossRef]

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

Laser Phys. (1)

M. Sobczak, A. Plucińska, and R. Parzyński, “Few-cycle light pulses in two-level systems: effects of pulse carrier-envelope offset phase in low-order harmonic spectra,” Laser Phys. 14, 1483–1487 (2004).

Math. Program. (1)

A. Wächter and L. T. Biegler, “On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming,” Math. Program. 106, 25–57 (2006).
[CrossRef]

Phys. Rev. (1)

B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 188, 1969–1975 (1969).
[CrossRef]

Phys. Rev. A (4)

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guérin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

X. Song, S. Gong, W. Yang, and Z. Xu, “Propagation of an attosecond pulse in a dense two-level medium,” Phys. Rev. A 70, 013817 (2004).
[CrossRef]

M. Holthaus and B. Just, “Generalized π pulses,” Phys. Rev. A 49, 1950–1960 (1994).
[CrossRef]

T. Tritschler, O. D. Mücke, and M. Wegener, “Extreme nonlinear optics of two-level systems,” Phys. Rev. A 68, 033404 (2003).
[CrossRef]

Phys. Rev. B (3)

O. D. Mücke, “Isolated high-order harmonics pulse from two-color-driven Bloch oscillations in bulk semiconductors,” Phys. Rev. B 84, 081202(R) (2011).
[CrossRef]

D. Golde, T. Meier, and S. W. Koch, “High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations,” Phys. Rev. B 77, 075330 (2008).
[CrossRef]

M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B 38, 3342–3350 (1988).
[CrossRef]

Phys. Rev. Lett. (4)

T. Meier, G. von Plessen, P. Thomas, and S. W. Koch, ”Coherent electric-field effects in semiconductors,” Phys. Rev. Lett. 73, 902–905 (1994).
[CrossRef]

T. Meier, F. Rossi, P. Thomas, and S. W. Koch, “Dynamic localization in anisotropic Coulomb systems: field induced crossover of the exciton dimension,” Phys. Rev. Lett. 75, 2558–2561 (1995).
[CrossRef]

C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
[CrossRef]

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef]

Rev. Mod. Phys. (3)

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

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

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

SIAM J. Sci. Comput. (1)

R. H. Byrd, P. Lu, and J. Nocedal, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190–1208 (1995).
[CrossRef]

Other (8)

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed.(Society for Industrial and Applied Mathematics, 2008).

J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

T. Meier, P. Thomas, and S. W. Koch, Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (Springer, 2007).

A. E. Eiben and J. E. Smith, Introduction to Evolutionary Computing (Springer, 2003).

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific, 2009).

M. Wegener, Extreme Nonlinear Optics (Springer, 2005).

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

P. Meystre and M. Sargent, Elements of Quantum Optics, 2nd ed. (Springer, 1991).

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Figures (7)

Fig. 1.
Fig. 1.

Intensity of the emitted radiation of a two-level system for different excitation pulses. The color encoding is logarithmic and covers 8 orders of magnitude; see color bar in (a). Results for (a) a resonant Gaussian pulse with Δt=100fs, (b) a resonant Gaussian pulse with Δt=10fs, and (c)–(e) optimal pulses to generate emissions at 3ω0, 2.5ω0, and 8ω0, respectively, for ωR=2.5ω0 as described in the text.

Fig. 2.
Fig. 2.

Cuts through excitation energies of ωR=2.5ω0: (a) refers to Fig. 1(c), (b) refers to Fig. 1(d), and (c) refers to Fig. 1(e). Displayed in gray is the cut through Fig. 1(b).

Fig. 3.
Fig. 3.

Spectra of the input pulses that generate the emission patterns in Figs. 1(c)(e). Pulse (a) maximizes at 3ω0, (b) at 2.5ω0, and (c) at 8ω0. Additionally, the insets show the respective temporal shape of the pulse in a 200fs window.

Fig. 4.
Fig. 4.

Radiated emission of a semiconductor after excitation with a pulse (a) of standard Gaussian shape, (b) that has been optimized for 20ω0, and (c) for 40ω0 for an input pulse of ωR=2.5ω0.

Fig. 5.
Fig. 5.

(a) Cut through Fig. 4(b), and (b) cut through Fig. 4(c) for ωR=2.5ω0. Displayed in gray is the emission for a standard Gaussian excitation.

Fig. 6.
Fig. 6.

Result obtained with the derivative-based optimization for ωopt=2.5ω0. The situation corresponds to Fig. 1(d).

Fig. 7.
Fig. 7.

Decreasing gradient norm of the derivative-based optimization.

Equations (18)

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H^=H^0+H^L-M=ϵ0c^0c^0+ϵ1c^1c^1-E(t)·(d*c^0c^1+h.c.).
tp=i(ϵ0-ϵ1)p+iE(t)·d(n0-n1),
tn0=2Im[E(t)·dp*],
tn1=-2Im[E(t)·dp*].
P(t)=2Re[dp]
Irad(ω)=|ω2P(ω)|2.
Θ(t)=-tdtdE¯(t).
E[{Ai},{ϕi},{ti},{Δti}](t)=iAiexp(-(t-tiΔti)2)cos(ωit+ϕi),
{{Ai},{ϕi},{ti},{Δti}}Irad(ωopt),
|E(t)|2dt=const.
tpk=-i(ϵke+ϵkh)+i(1-nke-nkh)E(t)d+eE(t)·kpk,
tnke=-2Im[E(t)dpk*]+eE(t)·knke,
tnkh=-2Im[E(t)dpk*]+eE(t)·knkh.
Irad(ω)=|ω2P(ω)+iωJ(ω)|2.
maxxXIrad(ω(x))such that|E(t)|2dt=const,
ϕ¯ϕiϕ¯,t¯ϕit¯,Δt̲ϕiΔt¯,
maxxXf(g(h(x))),
f(g(h(x)))=f(g(h(x)))·g(h(x))=f(g(h(x)))g(h(x))h(x).

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