Abstract

The off-resonant hyperpolarizability is calculated by using the dipole-free sum-over-states expression from a randomly chosen set of energies and transition dipole moments that are forced to be consistent with the sum rules. The process is repeated so that the distribution of hyperpolarizabilities can be determined. We find this distribution to be a cycloidlike function. In contrast to variational techniques that when applied to the potential energy function yield an intrinsic hyperpolarizability less than 0.71, our Monte Carlo method yields values that approach unity. While many transition dipole moments are large when the calculated hyperpolarizability is near the fundamental limit, only two excited states dominate the hyperpolarizability—consistent with the three-level ansatz. We speculate on the character of the Hamiltonian that is needed to optimize the intrinsic hyperpolarizability.

© 2008 Optical Society of America

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  1. Q. Y. Chen, L. Kuang, Z. Y. Wang, and E. H. Sargent, "Cross-linked C-60 polymer breaches the quantum gap," Nano Lett. 4, 1673-1675 (2004).
    [CrossRef]
  2. B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X.-L. Wu, S. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
    [CrossRef]
  3. S. Kawata, H.-B. Sun, T. Tanaka, and K. Takada, "Finer features for functional microdevices," Nature 412, 697-698 (2001).
    [CrossRef] [PubMed]
  4. A. Karotki, M. Drobizhev, Y. Dzenis, P. N. Taylor, H. L. Anderson, and A. Rebane, "Dramatic enhancement of intrinsic two-photon absorption in a conjugated porphyrin dimer," Phys. Chem. Chem. Phys. 6, 7-10 (2004).
    [CrossRef]
  5. I. Roy, O. T. Y., H. E. Pudavar, E. J. Bergey, A. R. Oseroff, J. Morgan, T. J. Dougherty, and P. N. Prasad, "Ceramic-based nanoparticles entrapping water-insoluble photosensitizing anticancer drugs: a novel drug-carrier system for photodynamic therapy," J. Am. Chem. Soc. 125, 7860-7865 (2003).
    [CrossRef] [PubMed]
  6. M. G. Kuzyk, "Physical limits on electronic nonlinear molecular susceptibilities," Phys. Rev. Lett. 85, 1218-1221 (2000).
    [CrossRef] [PubMed]
  7. M. G. Kuzyk, "Erratum: physical limits on electronic nonlinear molecular susceptibilities," Phys. Rev. Lett. 90, 039902 (2003).
    [CrossRef]
  8. B. Champagne and B. Kirtman, "Comment on 'physical limits on electronic nonlinear molecular susceptibilities'," Phys. Rev. Lett. 95, 109401 (2005).
    [CrossRef] [PubMed]
  9. M. G. Kuzyk, "Reply to comment on 'physical limits on electronic nonlinear molecular susceptibilities'," Phys. Rev. Lett. 95, 109402 (2005).
    [CrossRef]
  10. M. G. Kuzyk, "Compact sum-over-states expression without dipolar terms for calculating nonlinear susceptibilities," Phys. Rev. A 72, 053819 (2005).
    [CrossRef]
  11. M. G. Kuzyk, "Fundamental limits of all nonlinear-optical phenomena that are representable by a second-order susceptibility," J. Chem. Phys. 125, 154108 (2006).
    [CrossRef] [PubMed]
  12. K. Tripathi, P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, "Why hyperpolarizabilities fall short of the fundamental quantum limits," J. Chem. Phys. 121, 7932-7945 (2004).
    [CrossRef]
  13. J. Zhou, M. Kuzyk, and D. S. Watkins, "Pushing the hyperpolarizability to the limit," Opt. Lett. 31, 2891-2893 (2006).
    [CrossRef] [PubMed]
  14. J. Pérez Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk, "Modulated conjugation as a means for attaining a record high intrinsic hyperpolarizability," Opt. Lett. 32, 59-61 (2007).
    [CrossRef]
  15. 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]
  16. M. G. Kuzyk and D. S. Watkins, "The effects of geometry on the hyperpolarizability," J. Chem. Phys. 124, 244104 (2006).
    [CrossRef] [PubMed]
  17. J. Pérez Moreno, I. Asselberghs, Y. Zhao, K. Song, H. Nakanishi, S. Okada, K. Nogi, O.-K. Kim, J. Je, J. Matrai, M. De Mayer, and M. G. Kuzyk, "Combined molecular and supramolecular bottom-up nano-engineering for enhanced nonlinear optical response: experiments, modelling and approaching the fundamental limit," J. Chem. Phys. 126, 074705 (2007).
    [CrossRef] [PubMed]

2007 (3)

J. Pérez Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk, "Modulated conjugation as a means for attaining a record high intrinsic hyperpolarizability," Opt. Lett. 32, 59-61 (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]

J. Pérez Moreno, I. Asselberghs, Y. Zhao, K. Song, H. Nakanishi, S. Okada, K. Nogi, O.-K. Kim, J. Je, J. Matrai, M. De Mayer, and M. G. Kuzyk, "Combined molecular and supramolecular bottom-up nano-engineering for enhanced nonlinear optical response: experiments, modelling and approaching the fundamental limit," J. Chem. Phys. 126, 074705 (2007).
[CrossRef] [PubMed]

2006 (3)

J. Zhou, M. Kuzyk, and D. S. Watkins, "Pushing the hyperpolarizability to the limit," Opt. Lett. 31, 2891-2893 (2006).
[CrossRef] [PubMed]

M. G. Kuzyk and D. S. Watkins, "The effects of geometry on the hyperpolarizability," J. Chem. Phys. 124, 244104 (2006).
[CrossRef] [PubMed]

M. G. Kuzyk, "Fundamental limits of all nonlinear-optical phenomena that are representable by a second-order susceptibility," J. Chem. Phys. 125, 154108 (2006).
[CrossRef] [PubMed]

2005 (3)

B. Champagne and B. Kirtman, "Comment on 'physical limits on electronic nonlinear molecular susceptibilities'," Phys. Rev. Lett. 95, 109401 (2005).
[CrossRef] [PubMed]

M. G. Kuzyk, "Reply to comment on 'physical limits on electronic nonlinear molecular susceptibilities'," Phys. Rev. Lett. 95, 109402 (2005).
[CrossRef]

M. G. Kuzyk, "Compact sum-over-states expression without dipolar terms for calculating nonlinear susceptibilities," Phys. Rev. A 72, 053819 (2005).
[CrossRef]

2004 (3)

Q. Y. Chen, L. Kuang, Z. Y. Wang, and E. H. Sargent, "Cross-linked C-60 polymer breaches the quantum gap," Nano Lett. 4, 1673-1675 (2004).
[CrossRef]

A. Karotki, M. Drobizhev, Y. Dzenis, P. N. Taylor, H. L. Anderson, and A. Rebane, "Dramatic enhancement of intrinsic two-photon absorption in a conjugated porphyrin dimer," Phys. Chem. Chem. Phys. 6, 7-10 (2004).
[CrossRef]

K. Tripathi, P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, "Why hyperpolarizabilities fall short of the fundamental quantum limits," J. Chem. Phys. 121, 7932-7945 (2004).
[CrossRef]

2003 (2)

I. Roy, O. T. Y., H. E. Pudavar, E. J. Bergey, A. R. Oseroff, J. Morgan, T. J. Dougherty, and P. N. Prasad, "Ceramic-based nanoparticles entrapping water-insoluble photosensitizing anticancer drugs: a novel drug-carrier system for photodynamic therapy," J. Am. Chem. Soc. 125, 7860-7865 (2003).
[CrossRef] [PubMed]

M. G. Kuzyk, "Erratum: physical limits on electronic nonlinear molecular susceptibilities," Phys. Rev. Lett. 90, 039902 (2003).
[CrossRef]

2001 (1)

S. Kawata, H.-B. Sun, T. Tanaka, and K. Takada, "Finer features for functional microdevices," Nature 412, 697-698 (2001).
[CrossRef] [PubMed]

2000 (1)

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

1999 (1)

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X.-L. Wu, S. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
[CrossRef]

J. Am. Chem. Soc. (1)

I. Roy, O. T. Y., H. E. Pudavar, E. J. Bergey, A. R. Oseroff, J. Morgan, T. J. Dougherty, and P. N. Prasad, "Ceramic-based nanoparticles entrapping water-insoluble photosensitizing anticancer drugs: a novel drug-carrier system for photodynamic therapy," J. Am. Chem. Soc. 125, 7860-7865 (2003).
[CrossRef] [PubMed]

J. Chem. Phys. (4)

M. G. Kuzyk and D. S. Watkins, "The effects of geometry on the hyperpolarizability," J. Chem. Phys. 124, 244104 (2006).
[CrossRef] [PubMed]

J. Pérez Moreno, I. Asselberghs, Y. Zhao, K. Song, H. Nakanishi, S. Okada, K. Nogi, O.-K. Kim, J. Je, J. Matrai, M. De Mayer, and M. G. Kuzyk, "Combined molecular and supramolecular bottom-up nano-engineering for enhanced nonlinear optical response: experiments, modelling and approaching the fundamental limit," J. Chem. Phys. 126, 074705 (2007).
[CrossRef] [PubMed]

M. G. Kuzyk, "Fundamental limits of all nonlinear-optical phenomena that are representable by a second-order susceptibility," J. Chem. Phys. 125, 154108 (2006).
[CrossRef] [PubMed]

K. Tripathi, P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, "Why hyperpolarizabilities fall short of the fundamental quantum limits," J. Chem. Phys. 121, 7932-7945 (2004).
[CrossRef]

Nano Lett. (1)

Q. Y. Chen, L. Kuang, Z. Y. Wang, and E. H. Sargent, "Cross-linked C-60 polymer breaches the quantum gap," Nano Lett. 4, 1673-1675 (2004).
[CrossRef]

Nature (2)

B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X.-L. Wu, S. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
[CrossRef]

S. Kawata, H.-B. Sun, T. Tanaka, and K. Takada, "Finer features for functional microdevices," Nature 412, 697-698 (2001).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Chem. Chem. Phys. (1)

A. Karotki, M. Drobizhev, Y. Dzenis, P. N. Taylor, H. L. Anderson, and A. Rebane, "Dramatic enhancement of intrinsic two-photon absorption in a conjugated porphyrin dimer," Phys. Chem. Chem. Phys. 6, 7-10 (2004).
[CrossRef]

Phys. Rev. A (2)

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. G. Kuzyk, "Compact sum-over-states expression without dipolar terms for calculating nonlinear susceptibilities," Phys. Rev. A 72, 053819 (2005).
[CrossRef]

Phys. Rev. Lett. (4)

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

M. G. Kuzyk, "Erratum: physical limits on electronic nonlinear molecular susceptibilities," Phys. Rev. Lett. 90, 039902 (2003).
[CrossRef]

B. Champagne and B. Kirtman, "Comment on 'physical limits on electronic nonlinear molecular susceptibilities'," Phys. Rev. Lett. 95, 109401 (2005).
[CrossRef] [PubMed]

M. G. Kuzyk, "Reply to comment on 'physical limits on electronic nonlinear molecular susceptibilities'," Phys. Rev. Lett. 95, 109402 (2005).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Distribution of calculated hyperpolarizabilities for various energy weightings. Error bars are calculated from the square root of the frequency of observing a particular value of β INT .

Fig. 2
Fig. 2

Matrix elements that yield the largest value of the intrinsic hyperpolarizability ( β int = 0.9569 ) after 100,000 tries for a 10-state model.

Fig. 3
Fig. 3

Fractional contribution, β int n , m , for a 10-state model with β int = 0.9569 (using matrix elements from Fig. 2).

Fig. 4
Fig. 4

(a) Matrix elements that yield the largest value of the intrinsic hyperpolarizability ( β int = 0.9441 ) after 100,000 tries for a 40-state model; (b) fractional contribution, β int n , m .

Fig. 5
Fig. 5

Fractional contribution, β int n , m , for a 40-state model when the intrinsic hyperpolarizability is (a) 0.033 and (b) 0.31.

Tables (1)

Tables Icon

Table 1 States Whose Contributions Are at Least 10% of the Largest Contribution for the 40-State Model (Fig. 4)

Equations (21)

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n = 0 [ e n 1 2 ( e m + e p ) ] ξ m n ξ n p = δ m p ,
n = 2 N ( e n e 0 ) ξ m 0 2 = 1 ( e 1 e 0 ) ξ 10 2 ,
ξ 20 2 ( 1 e 1 ξ 10 2 ) e 2 ,
ξ 02 = ξ 20 = r ( 1 e 1 ξ 10 2 ) e 2 .
ξ 03 = ξ 30 = r ( 1 e 1 ξ 10 2 e 2 ξ 20 2 ) e 3 ,
ξ 12 = ξ 21 = s ( 1 + e 1 ξ 10 2 ) ( e 2 e 1 ) .
ξ 13 = ξ 31 = s [ 1 + e 1 ξ 10 2 ( e 2 e 1 ) ξ 21 2 ] ( e 3 e 1 ) ,
β int = β β MAX = ( 3 4 ) 3 4 i j ξ 0 i ξ i j ξ j 0 ( 1 e i e j 2 e j e i e i 2 ) ,
F = A ( 1 β INT 1 n ) n ,
β int n , m = ( 3 4 ) 3 4 ξ 0 n ξ n m ξ m 0 [ 1 e n e m 2 e m e n e n 2 ] .
β INT = f ( E ) G ( X ) ,
f ( E ) = ( 1 E ) 3 2 ( E 2 + 3 2 E + 1 ) ,
G ( X ) = 3 4 X 3 2 ( 1 X 4 ) .
n = 0 N 1 e n p ξ n p ξ p n = 1 ,
n = 0 N e n p ξ n p ξ p n = 1 .
ξ N , p ξ p , N e N , p = 0 .
ξ N , p = ξ p , N = 0 .
n = 0 N e n , N 1 ξ N 1 , n ξ n , N 1 = 1 ,
e N , N 1 ξ N , N 1 ξ N 1 , N = 1 + n = 0 N 1 e N 1 , n ξ N 1 , n ξ n , N 1 .
H = 1 2 m ( p e c A ) 2 + ϕ ( r ) ,
H = e 2 m c ( p A + p A ) + H 0 ,

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