Abstract

The dimensionless zero-frequency electronic first hyperpolarizability 31/4βE107/2m3/2(e)3 of an electron in one dimension was maximized by adjusting the shape of a piecewise linear potential. Careful maximizations converged quickly to 0.708951 with increasing numbers of parameters. The Hessian shows that β is strongly sensitive to only two parameters in the potential: sensitivity to additional parameters decreases rapidly. With more than two parameters, a wide range of potentials and an apparently narrower range of wavefunctions have nearly optimal hyperpolarizability. Modulations of the potential to which the unique maximum is insensitive were characterized. Prospects for concise description of the two important constraints on near-optimum potentials are discussed.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
    [CrossRef]
  2. M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85, 1218–1221 (2000).
    [CrossRef]
  3. K. Tripathy, J. Moreno, M. Kuzyk, B. Coe, K. Clays, and A. Kelley, “Why hyperpolarizabilities fall short of the fundamental quantum limits,” J. Chem. Phys. 121, 7932 (2004).
    [CrossRef]
  4. J. Zhou, M. Kuzyk, and D. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31, 2891–2893 (2006).
    [CrossRef]
  5. 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]
  6. B. Champagne and B. Kirtman, “Comment on ‘physical limits on electronic nonlinear molecular susceptibilities,’” Phys. Rev. Lett. 95, 109401 (2005).
    [CrossRef]
  7. 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]
  8. 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]
  9. M. Kuzyk, “Erratum: Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 90, 039902 (2003).
    [CrossRef]
  10. M. Kuzyk, “Compact sum-over-states expression without dipolar terms for calculating nonlinear susceptibilities,” Phys. Rev. A 72, 053819 (2005).
    [CrossRef]
  11. D. S. Watkins and M. G. Kuzyk, “The effect of electron interactions on the universal properties of systems with optimized off-resonant intrinsic hyperpolarizability,” J. Chem. Phys. 134, 094109 (2011).
    [CrossRef]
  12. J. Pérez-Moreno, K. Clays, and M. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
    [CrossRef]
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).
  14. A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, 2005).
  15. G. Wiggers and R. Petschek, “Comment on ‘pushing the hyperpolarizability to the limit,’” Opt. Lett. 32, 942–943 (2007).
    [CrossRef]
  16. G. L. Bretthorst, Bayesian Spectrum Analysis and Parameter Estimation (Springer-Verlag, 1988).
  17. Available online from http://tatherton.phy.tufts.edu/.

2011

D. S. Watkins and M. G. Kuzyk, “The effect of electron interactions on the universal properties of systems with optimized off-resonant intrinsic hyperpolarizability,” J. Chem. Phys. 134, 094109 (2011).
[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]

2008

J. Pérez-Moreno, K. Clays, and M. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
[CrossRef]

2007

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

J. Zhou, M. Kuzyk, and D. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31, 2891–2893 (2006).
[CrossRef]

2005

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

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

2004

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

2003

M. Kuzyk, “Erratum: Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 90, 039902 (2003).
[CrossRef]

2000

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

1995

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

Bretthorst, G. L.

G. L. Bretthorst, Bayesian Spectrum Analysis and Parameter Estimation (Springer-Verlag, 1988).

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.

J. Pérez-Moreno, K. Clays, and M. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
[CrossRef]

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

Coe, B.

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

Dalton, L.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

Fetterman, H.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Ghosn, R.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

Harper, A.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

Jen, A.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

Kelley, A.

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

J. Pérez-Moreno, K. Clays, and M. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
[CrossRef]

J. Zhou, M. Kuzyk, and D. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31, 2891–2893 (2006).
[CrossRef]

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

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

M. Kuzyk, “Erratum: Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 90, 039902 (2003).
[CrossRef]

Kuzyk, M. G.

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]

D. S. Watkins and M. G. Kuzyk, “The effect of electron interactions on the universal properties of systems with optimized off-resonant intrinsic hyperpolarizability,” J. Chem. Phys. 134, 094109 (2011).
[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]

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

Moreno, J.

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

Mustacich, R.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

Pérez-Moreno, J.

J. Pérez-Moreno, K. Clays, and M. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
[CrossRef]

Petschek, R.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Shafei, S.

Shea, K.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

Shi, Y.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

Steier, W.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[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]

Tarantola, A.

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, 2005).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Tripathy, K.

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

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Watkins, D.

Watkins, D. S.

D. S. Watkins and M. G. Kuzyk, “The effect of electron interactions on the universal properties of systems with optimized off-resonant intrinsic hyperpolarizability,” J. Chem. Phys. 134, 094109 (2011).
[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.

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]

J. Zhou, M. Kuzyk, and D. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31, 2891–2893 (2006).
[CrossRef]

Ziari, M.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

Chem. Mater.

L. Dalton, A. Harper, R. Ghosn, W. Steier, M. Ziari, H. Fetterman, Y. Shi, R. Mustacich, A. Jen, and K. Shea, “Synthesis and processing of improved organic second-order nonlinear optical materials for applications in photonics,” Chem. Mater. 7, 1060–1081 (1995).
[CrossRef]

J. Chem. Phys.

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

D. S. Watkins and M. G. Kuzyk, “The effect of electron interactions on the universal properties of systems with optimized off-resonant intrinsic hyperpolarizability,” J. Chem. Phys. 134, 094109 (2011).
[CrossRef]

J. Pérez-Moreno, K. Clays, and M. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
[CrossRef]

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]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. A

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

Phys. Rev. Lett.

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

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

M. Kuzyk, “Erratum: Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 90, 039902 (2003).
[CrossRef]

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, 2005).

G. L. Bretthorst, Bayesian Spectrum Analysis and Parameter Estimation (Springer-Verlag, 1988).

Available online from http://tatherton.phy.tufts.edu/.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

(a) Ground and (b) first excited normalized wavefunctions corresponding to (c) optimized potentials for N=2 to N=7 parameters. Physically irrelevant positions and energies have been scaled out, consistent with Eq. (20). The hyperpolarizabilities are noted in (a) and the ground and first excited energy levels are indicated in (c).

Fig. 2.
Fig. 2.

Eigenvalues of the Hessian of βint evaluated at its maximum plotted for different numbers of free parameters N. The dashed lines are intended as a visual aid only.

Fig. 3.
Fig. 3.

Variations of the potential and ground state wavefunction associated with eigenvectors of the Hessian matrix of βint evaluated at the maximum; these have been adjusted so that each variation simultaneously leaves E1E0=1 and x=0. The corresponding eigenvalue of the Hessian is indicated alongside each plot. Each variation is calculated in two measures: the numerically natural measure (solid lines), and the measure of the ground state wavefunction (dashed lines).

Tables (2)

Tables Icon

Table 1. For Different Numbers of Parameters in the Potential: (from left) Optimized Intrinsic Hyperpolarizabilities , the Ratio of Eigenvalues of the Hessian Matrix of the Intrinsic Hyperpolarizability, Oscillator Strengths Between Various States, the Ratio E of the Energy Level Spacings

Tables Icon

Table 2. Eigenvalues of Hessian for N=7 parameters in Each of Three Measures: (from left) the Numerically Natural Measure Induced by the Parametrization and then the Measures ρ0=|ψ0|2 and ρ10=|ψ0|2+|ψ1|2

Equations (26)

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

βmax=34(em)3N3/2E107/2.
βint=β/βmax
xxE1/2,V(x)V(x)E.
V(x)={A0x+B0x<x0Anx+Bnxn1<x<xnANx+BNx>xN1,n{1,2,,N1}
Bn=m=1n1(AmAm+1)xm,n>1.
[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,
ddϵdetW=Tr(adjW·dWdϵ),
dWdϵ=Wϵ+WEdEdϵ,
dEdϵ=Tr(adjW·Wϵ)Tr(adjW·WE).
d2Edϵ2=Tr[ddϵ(adjW)·dWdϵ+adjW·W]Tr(adjW·WE),
W=(2Wϵ2+dEdϵ[22WEϵ+dEdϵ2WE2])
d3Edϵ3=Tr(d2dϵ2(adjW)·dWdϵ+2ddϵ(adjW)·d2Wdϵ2+adjW·Wn)Tr(adjW·WE),
W=3Wϵ3+3d2Edϵ2(2WEϵ+dEdϵ2WE2)+3dEdϵ(3WEϵ2+dEdϵ3WE2ϵ)+(dEdϵ)33WE3
β=12.d3Edϵ3|E=E0,ϵ=0,
βint=β/βmax
Hij=2PiPj(βint)
x¯=(xx)/(E1E0)1/2,V¯(x¯,{P})=(V(x¯,{P})E0)/(E1E0).
ΔVj(x)=V(x,{Pi+αvij})α|α=0,
V1V22=|P1P2|2,
V1sV2sk2=dxρk(x)|V1sV2s|2,
Hkivij=hjMkivij.
h2j=h1j(1+v1jδM¯v1j)+O(δM¯2)
v2j=v1j+jkv1kh1i(v1kδM¯1v1j)h1jh1k+O(δM¯2).

Metrics