Abstract

The dimensionless zero-frequency intrinsic second hyperpolarizability γint=γ/4E105m2(e)4 was optimized for a single electron in a 1D well by adjusting the shape of the potential. Optimized potentials were found to have hyperpolarizabilities in the range 0.15γint0.60; potentials optimizing gamma were arbitrarily close to the lower bound and were within 0.5% of the upper bound. All optimal potentials possess parity symmetry. Analysis of the Hessian of γint around the maximum reveals that effectively only a single parameter, one of those chosen in the piecewise linear representation adopted, is important to obtaining an extremum. Prospects for designing chromophores based on the design principle here elucidated are discussed.

© 2013 Optical Society of America

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References

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  1. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).
  2. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2003).
  3. M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000).
    [CrossRef]
  4. M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. 25, 1183–1185 (2000).
    [CrossRef]
  5. M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
    [CrossRef]
  6. J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
    [CrossRef]
  7. T. Luu, E. Elliott, A. Slepkov, S. Eisler, R. McDonald, F. Hegmann, and R. Tykwinski, “Synthesis, structure, and nonlinear optical properties of diarylpolyynes,” Org. Lett. 7, 51–54 (2005).
    [CrossRef]
  8. J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
    [CrossRef]
  9. B. Champagne and B. Kirtman, “Comment on ‘physical limits on electronic nonlinear molecular susceptibilities’,” Phys. Rev. Lett. 95, 109401 (2005).
    [CrossRef]
  10. B. Champagne and B. Kirtman, “Evaluation of alternative sum-over-states expressions for the first hyperpolarizability of push-pull pi-conjugated systems,” J. Chem. Phys. 125, 024101 (2006).
    [CrossRef]
  11. D. S. Watkins and M. G. Kuzyk, “Universal properties of the optimized off-resonant intrinsic second hyperpolarizability,” J. Opt. Soc. Am. B 29, 1661–1671 (2012).
    [CrossRef]
  12. K. Tripathy, J. P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. Myers Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932(2004).
    [CrossRef]
  13. J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
    [CrossRef]
  14. S. Shafei, M. C. Kuzyk, and M. G. Kuzyk, “Monte Carlo studies of the intrinsic second hyperpolarizability,” J. Opt. Soc. Am. B 27, 1849–1856 (2010).
    [CrossRef]
  15. T. J. Atherton, J. Lesnefsky, G. A. Wiggers, and R. G. Petschek, “Maximizing the hyperpolarizability poorly determines the potential,” J. Opt. Soc. Am. B 29, 513–520 (2012).
  16. M. Kuzyk and C. Dirk, “Effects of centrosymmetry on the nonresonant electronic third-order nonlinear optical susceptibility,” Phys. Rev. A 41, 5098 (1990).
    [CrossRef]
  17. G. Wiggers and R. Petschek, “Comment on ‘pushing the hyperpolarizability to the limit’,” Opt. Lett. 32, 942–943(2007).
    [CrossRef]
  18. S. Shafei and M. G. Kuzyk, “Critical role of the energy spectrum in determining the nonlinear-optical response of a quantum system,” J. Opt. Soc. Am. B 28, 882–891 (2011).
    [CrossRef]

2012

2011

2010

S. Shafei, M. C. Kuzyk, and M. G. Kuzyk, “Monte Carlo studies of the intrinsic second hyperpolarizability,” J. Opt. Soc. Am. B 27, 1849–1856 (2010).
[CrossRef]

J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
[CrossRef]

2008

M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
[CrossRef]

2007

G. Wiggers and R. Petschek, “Comment on ‘pushing the hyperpolarizability to the limit’,” Opt. Lett. 32, 942–943(2007).
[CrossRef]

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[CrossRef]

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[CrossRef]

2006

B. Champagne and B. Kirtman, “Evaluation of alternative sum-over-states expressions for the first hyperpolarizability of push-pull pi-conjugated systems,” J. Chem. Phys. 125, 024101 (2006).
[CrossRef]

2005

B. Champagne and B. Kirtman, “Comment on ‘physical limits on electronic nonlinear molecular susceptibilities’,” Phys. Rev. Lett. 95, 109401 (2005).
[CrossRef]

T. Luu, E. Elliott, A. Slepkov, S. Eisler, R. McDonald, F. Hegmann, and R. Tykwinski, “Synthesis, structure, and nonlinear optical properties of diarylpolyynes,” Org. Lett. 7, 51–54 (2005).
[CrossRef]

2004

K. Tripathy, J. P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. Myers Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932(2004).
[CrossRef]

2000

M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000).
[CrossRef]

M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. 25, 1183–1185 (2000).
[CrossRef]

1990

M. Kuzyk and C. Dirk, “Effects of centrosymmetry on the nonresonant electronic third-order nonlinear optical susceptibility,” Phys. Rev. A 41, 5098 (1990).
[CrossRef]

Atherton, T. J.

Barlow, S.

J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
[CrossRef]

Biaggio, I.

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Bredas, J.-L.

J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
[CrossRef]

Bures, F.

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[CrossRef]

Champagne, B.

B. Champagne and B. Kirtman, “Evaluation of alternative sum-over-states expressions for the first hyperpolarizability of push-pull pi-conjugated systems,” J. Chem. Phys. 125, 024101 (2006).
[CrossRef]

B. Champagne and B. Kirtman, “Comment on ‘physical limits on electronic nonlinear molecular susceptibilities’,” Phys. Rev. Lett. 95, 109401 (2005).
[CrossRef]

Clays, K.

K. Tripathy, J. P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. Myers Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932(2004).
[CrossRef]

Coe, B. J.

K. Tripathy, J. P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. Myers Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932(2004).
[CrossRef]

Cogdell, R.

M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
[CrossRef]

Diederich, F.

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[CrossRef]

Dirk, C.

M. Kuzyk and C. Dirk, “Effects of centrosymmetry on the nonresonant electronic third-order nonlinear optical susceptibility,” Phys. Rev. A 41, 5098 (1990).
[CrossRef]

Eisler, S.

T. Luu, E. Elliott, A. Slepkov, S. Eisler, R. McDonald, F. Hegmann, and R. Tykwinski, “Synthesis, structure, and nonlinear optical properties of diarylpolyynes,” Org. Lett. 7, 51–54 (2005).
[CrossRef]

Elliott, E.

T. Luu, E. Elliott, A. Slepkov, S. Eisler, R. McDonald, F. Hegmann, and R. Tykwinski, “Synthesis, structure, and nonlinear optical properties of diarylpolyynes,” Org. Lett. 7, 51–54 (2005).
[CrossRef]

Fujiwara, M.

M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
[CrossRef]

Gall, A.

M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
[CrossRef]

Hales, J. M.

J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
[CrossRef]

Hashimoto, H.

M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
[CrossRef]

Hegmann, F.

T. Luu, E. Elliott, A. Slepkov, S. Eisler, R. McDonald, F. Hegmann, and R. Tykwinski, “Synthesis, structure, and nonlinear optical properties of diarylpolyynes,” Org. Lett. 7, 51–54 (2005).
[CrossRef]

Kirtman, B.

B. Champagne and B. Kirtman, “Evaluation of alternative sum-over-states expressions for the first hyperpolarizability of push-pull pi-conjugated systems,” J. Chem. Phys. 125, 024101 (2006).
[CrossRef]

B. Champagne and B. Kirtman, “Comment on ‘physical limits on electronic nonlinear molecular susceptibilities’,” Phys. Rev. Lett. 95, 109401 (2005).
[CrossRef]

Kuzyk, M.

M. Kuzyk and C. Dirk, “Effects of centrosymmetry on the nonresonant electronic third-order nonlinear optical susceptibility,” Phys. Rev. A 41, 5098 (1990).
[CrossRef]

Kuzyk, M. C.

Kuzyk, M. G.

D. S. Watkins and M. G. Kuzyk, “Universal properties of the optimized off-resonant intrinsic second hyperpolarizability,” J. Opt. Soc. Am. B 29, 1661–1671 (2012).
[CrossRef]

S. Shafei and M. G. Kuzyk, “Critical role of the energy spectrum in determining the nonlinear-optical response of a quantum system,” J. Opt. Soc. Am. B 28, 882–891 (2011).
[CrossRef]

S. Shafei, M. C. Kuzyk, and M. G. Kuzyk, “Monte Carlo studies of the intrinsic second hyperpolarizability,” J. Opt. Soc. Am. B 27, 1849–1856 (2010).
[CrossRef]

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[CrossRef]

K. Tripathy, J. P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. Myers Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932(2004).
[CrossRef]

M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000).
[CrossRef]

M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. 25, 1183–1185 (2000).
[CrossRef]

Lesnefsky, J.

Luu, T.

T. Luu, E. Elliott, A. Slepkov, S. Eisler, R. McDonald, F. Hegmann, and R. Tykwinski, “Synthesis, structure, and nonlinear optical properties of diarylpolyynes,” Org. Lett. 7, 51–54 (2005).
[CrossRef]

Marder, S. R.

J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
[CrossRef]

Matichak, J.

J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
[CrossRef]

May, J. C.

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[CrossRef]

McDonald, R.

T. Luu, E. Elliott, A. Slepkov, S. Eisler, R. McDonald, F. Hegmann, and R. Tykwinski, “Synthesis, structure, and nonlinear optical properties of diarylpolyynes,” Org. Lett. 7, 51–54 (2005).
[CrossRef]

Moreno, J. P.

K. Tripathy, J. P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. Myers Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932(2004).
[CrossRef]

Myers Kelley, A.

K. Tripathy, J. P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. Myers Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932(2004).
[CrossRef]

Ohira, S.

J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
[CrossRef]

Perry, J. W.

J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
[CrossRef]

Petschek, R.

Petschek, R. G.

Robert, B.

M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
[CrossRef]

Shafei, S.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2003).

Slepkov, A.

T. Luu, E. Elliott, A. Slepkov, S. Eisler, R. McDonald, F. Hegmann, and R. Tykwinski, “Synthesis, structure, and nonlinear optical properties of diarylpolyynes,” Org. Lett. 7, 51–54 (2005).
[CrossRef]

Sugisaki, M.

M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
[CrossRef]

Szafruga, U. B.

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[CrossRef]

Tripathy, K.

K. Tripathy, J. P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. Myers Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932(2004).
[CrossRef]

Tykwinski, R.

T. Luu, E. Elliott, A. Slepkov, S. Eisler, R. McDonald, F. Hegmann, and R. Tykwinski, “Synthesis, structure, and nonlinear optical properties of diarylpolyynes,” Org. Lett. 7, 51–54 (2005).
[CrossRef]

Watkins, D. S.

D. S. Watkins and M. G. Kuzyk, “Universal properties of the optimized off-resonant intrinsic second hyperpolarizability,” J. Opt. Soc. Am. B 29, 1661–1671 (2012).
[CrossRef]

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[CrossRef]

Wiggers, G.

Wiggers, G. A.

Yamauchi, K.

M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
[CrossRef]

Yanagi, K.

M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
[CrossRef]

Yesudas, K.

J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
[CrossRef]

Zhou, J.

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[CrossRef]

Appl. Phys. Lett.

J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90, 251106 (2007).
[CrossRef]

J. Chem. Phys.

B. Champagne and B. Kirtman, “Evaluation of alternative sum-over-states expressions for the first hyperpolarizability of push-pull pi-conjugated systems,” J. Chem. Phys. 125, 024101 (2006).
[CrossRef]

K. Tripathy, J. P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. Myers Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932(2004).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Org. Lett.

T. Luu, E. Elliott, A. Slepkov, S. Eisler, R. McDonald, F. Hegmann, and R. Tykwinski, “Synthesis, structure, and nonlinear optical properties of diarylpolyynes,” Org. Lett. 7, 51–54 (2005).
[CrossRef]

Phys. Rev. A

J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[CrossRef]

M. Kuzyk and C. Dirk, “Effects of centrosymmetry on the nonresonant electronic third-order nonlinear optical susceptibility,” Phys. Rev. A 41, 5098 (1990).
[CrossRef]

Phys. Rev. B

M. Fujiwara, K. Yamauchi, M. Sugisaki, K. Yanagi, A. Gall, B. Robert, R. Cogdell, and H. Hashimoto, “Large third-order optical nonlinearity realized in symmetric nonpolar carotenoids,” Phys. Rev. B 78, 161101 (2008).
[CrossRef]

Phys. Rev. Lett.

M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000).
[CrossRef]

B. Champagne and B. Kirtman, “Comment on ‘physical limits on electronic nonlinear molecular susceptibilities’,” Phys. Rev. Lett. 95, 109401 (2005).
[CrossRef]

Science

J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Bredas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327, 1485–1488 (2010).
[CrossRef]

Other

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2003).

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Figures (6)

Fig. 1.
Fig. 1.

Optimized potentials for the upper bound of γint with (a) no enforced symmetry and (b) P symmetry. The energies of the ground and first excited state are indicated by horizontal lines; the corresponding wavefunctions are also displayed. The plots have been rescaled to facilitate comparison by ensuring E1E0=1 and x0=0 while preserving γint.

Fig. 2.
Fig. 2.

Optimized potentials for the lower bound of γint with (a) no enforced symmetry and (b) P symmetry.

Fig. 3.
Fig. 3.

Five parameter potentials with no enforced symmetry optimized using two-parameter P-symmetric optima as the starting point. (a) Upper and (b) lower bounds.

Fig. 4.
Fig. 4.

Visualization of the variation of γint as a function of the parameters x1 and A2 of a two-parameter P-symmetric potential for the (a) upper bound and (b) lower bound. The region of parameter space where γint is within 2% of the maximum value is highlighted in (a) and indicated by an arrow. γint is also shown in the (E,X) parameter space for the (c) upper bound and (d) lower bound.

Fig. 5.
Fig. 5.

Plot of γint, E=E10/E20, and X=x10/x10max as a function of α for the for the potential V=|x|αδ(x).

Fig. 6.
Fig. 6.

Examples of constructing variations in the potential away from a given point in parameter space. (a) Original unrescaled potential at minimum γint (solid blue line) and a perturbation (in the X1 direction) of this potential (dashed black line). (b) The difference between the optimum potential and the perturbation. (c) Rescaled versions of the potentials in (a) along with (d) their difference.

Tables (2)

Tables Icon

Table 1. List of Parameters of Optimized Potentialsa

Tables Icon

Table 2. Second Hyperpolarizabilities and Physical Parameters X=x01/x01max and E=E01/E02 for the Optimized Potentials Obtained in this Work

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

P=αE+βEE+γEEE+O(E4),
(em)4N2E105γ4(em)4N2E105γ0max,
γint=γ/γ0max,
V(x)={A0x+B0x<x0Anx+Bnxn1<x<xn,n{1,N1}ANx+BNx>xN1,
V(x)={Anx+Bnxn1<x<xn,n{1,N1}ANx+BNx>xN1
[12d2dx2+(An+ϵ)x+Bn)]ψn=Eψn
ψn(x)=CnAi[23(BnE+x(An+ϵ))(An+ϵ)2/3]+DnBi[23(BnE+x(An+ϵ))(An+ϵ)2/3].
W·u=0,
detW=0
γ16d4E0dϵ4|ϵ=0;
ddϵdetW=Tr(adjW·dWdϵ),
dWdϵ=Wϵ+WEdEdϵ.
d4Edϵ4=Tr[(d3dϵ3adjW)·dWdϵ+3(d2dϵ2adjW)·d2Wdϵ2+3(ddϵadjW)·d3Wdϵ3+adjW·W]Tr(adjW·WE),
W=d4Wdϵ4WEd4Edϵ4.
Hij=2PiPjγint,
x¯=(xx0)/(E1E0)1/2,V¯(x¯,{P})=(V(x¯,{P})E0)/(E1E0)
ΔVj(x)=V¯(x¯,{Pi+αvij})α|α=0,
V(x)=|x|αδ(x),
H=(2A222A2X12X1A22X12)γint=(1.54×1064.70×1044.70×1040.165).
(0.0028430.999996),(0.9999960.002843).

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