Abstract

We study for the first time the effect of the geometry of quantum wire networks on their nonlinear optical properties and show that for some geometries, the first hyperpolarizability is largely enhanced and the second hyperpolarizability is always negative or zero. We use a one-electron model with tight transverse confinement. In the limit of infinite transverse confinement, the transverse wavefunctions drop out of the hyperpolarizabilities, but their residual effects are essential to include in the sum rules. The effects of geometry are manifested in the projections of the transition moments of each wire segment onto the 2D lab frame. Numerical optimization of the geometry of a loop leads to hyperpolarizabilities that rival the best chromophores. We suggest that a combination of geometry and quantum-confinement effects can lead to systems with ultralarge nonlinear response.

© 2012 Optical Society of America

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2012 (1)

2011 (5)

S. Shafei and M. G. Kuzyk, “The effect of extreme confinement on the nonlinear-optical response of quantum wires,” J. Nonlinear Opt. Phys. Mater. 20, 427–441 (2011).
[CrossRef]

A. Feizpour, X. Xing, and A. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (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]

H. Yan, H. Choe, S. Nam, Y. Hu, S. Das, J. Klemic, J. Ellenbogen, and C. Lieber, “Programmable nanowire circuits for nanoprocessors,” Nature 470, 240–244 (2011).
[CrossRef]

X. Chu, M. Yang, and K. Jackson, “The effect of geometry on cluster polarizability: Studies of sodium, copper, and silicon clusters at shape-transition sizes,” J. Chem. Phys. 134, 234505 (2011).
[CrossRef]

2010 (3)

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

M. G. Kuzyk, “A birds-eye view of nonlinear-optical processes: unification through scale invariance,” Nonlinear Opt. Quantum Opt. 40, 1–13 (2010).

2009 (3)

M. G. Kuzyk, “Using fundamental principles to understand and optimize nonlinear-optical materials,” J. Mater. Chem. 19, 7444–7465 (2009).
[CrossRef]

Z. Pastore and R. Blümel, “An exact periodic-orbit formula for the energy levels of the three-pronged star graph,” J. Phys. A 42, 135102 (2009).
[CrossRef]

R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009).
[CrossRef]

2008 (4)

M. C. Kuzyk and M. G. Kuzyk, “Monte Carlo studies of the fundamental limits of the intrinsic hyperpolarizability,” J. Opt. Soc. Am. B 25, 103–110 (2008).
[CrossRef]

S. Keinan, M. J. Therien, D. N. Beratan, and W. T. Yang, “Molecular design of porphyrin-based nonlinear optical materials,” J. Phys. Chem. A 112, 12203–12207 (2008).
[CrossRef]

M. Belloni and R. W. Robinett, “Quantum mechanical sum rules for two model systems,” Am. J. Phys. 76, 798–806 (2008).
[CrossRef]

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

2007 (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]

C. Lieber and Z. Wang, “Functional nanowires,” MRS Bull. 32, 99–108 (2007).
[CrossRef]

2006 (2)

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

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

2005 (1)

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

2004 (1)

K. Tripathy, J. Pérez 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 (1)

I. Roy, T. Y. Ohulchanskyy, 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]

2002 (2)

J. Johnson, H. Yan, R. Schaller, P. Petersen, P. Yang, and R. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2, 279–283 (2002).
[CrossRef]

F. Fernández, “The Thomas-Reiche-Kuhn sum rule for the rigid rotator,” Int. J. Math. Ed. Sci. Technol. 33, 636–640 (2002).

2001 (3)

M. G. Kuzyk, “Quantum limits of the hyper-Rayleigh scattering susceptibility,” IEEE J. Sel. Topics Quantum Electron. 7, 774–780 (2001).
[CrossRef]

M. Huang, S. Mao, H. Feick, H. Yan, Y. Wu, H. Kind, E. Weber, R. Russo, and P. Yang, “Room-temperature ultraviolet nanowire nanolasers,” Science 292, 1897–1899 (2001).
[CrossRef]

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

2000 (3)

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]

V. Ostroverkhov, O. Ostroverkhova, R. Petschek, K. Singer, L. Sukhomlinova, R. Twieg, S. Wang, and L. Chien, “Optimization of the molecular hyperpolarizability for second harmonic generation in chiral media,” Chem. Phys. 257, 263–274 (2000).
[CrossRef]

1999 (2)

J. Ballesteros, J. Solis, R. Serna, and C. Afonso, “Nanocrystal size dependence of the third-order nonlinear optical response of cu: Alo thin films,” Appl. Phys. Lett. 74, 2791 (1999).
[CrossRef]

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]

1998 (1)

G. Banfi, V. Degiorgio, and D. Ricard, “Nonlinear optical properties of semiconductor nanocrystals,” Adv. Phys. 47, 447–510 (1998).
[CrossRef]

1997 (2)

J. Xia and K. Cheah, “Quantum confinement effect in thin quantum wires,” Phys. Rev. B 55, 15688–15693 (1997).
[CrossRef]

E. Hadjimichael, W. Currie, and S. Fallieros, “The Thomas–Reiche–Kuhn sum rule and the rigid rotator,” Am. J. Phys. 65, 335–341 (1997).
[CrossRef]

1996 (1)

R. Ashoori, “Electrons in artificial atoms,” Nature 379, 413–419 (1996).
[CrossRef]

1993 (1)

R. Chen, D. Lin, and B. Mendoza, “Enhancement of the third-order nonlinear optical susceptibility in si quantum wires,” Phys. Rev. B 48, 11879–11882 (1993).
[CrossRef]

1992 (2)

G. Sanders and Y. Chang, “Theory of optical properties of quantum wires in porous silicon,” Phys. Rev. B 45, 9202–9213 (1992).
[CrossRef]

F. Buda, J. Kohanoff, and M. Parrinello, “Optical properties of porous silicon: a first-principles study,” Phys. Rev. Lett. 69, 1272–1275 (1992).
[CrossRef]

1986 (1)

1984 (1)

L. Brus, “Electron–electron and electron-hole interactions in small semiconductor crystallites: the size dependence of the lowest excited electronic state,” J. Chem. Phys. 80, 4403 (1984).
[CrossRef]

1971 (2)

R. C. Jaklevic, J. Lambe, M. Mikkor, and W. C. Vassell, “Observation of electron standing waves in a crystalline box,” Phys. Rev. Lett. 26, 88–92 (1971).
[CrossRef]

B. J. Orr and J. F. Ward, “Perturbation theory of the non-linear optical polarization of an isolated system,” Mol. Phys. 20, 513–526 (1971).
[CrossRef]

1962 (1)

R. Kubo, “Electronic properties of metallic fine particles. i,” J. Phys. Soc. Jpn. 17, 975–986 (1962).
[CrossRef]

1937 (1)

H. Frohlich, “Die spezifische wärme der elektronen kleiner metallteilchen bei tiefen temperaturen,” Physica 4, 406–412 (1937).

Afonso, C.

J. Ballesteros, J. Solis, R. Serna, and C. Afonso, “Nanocrystal size dependence of the third-order nonlinear optical response of cu: Alo thin films,” Appl. Phys. Lett. 74, 2791 (1999).
[CrossRef]

Ananthavel, S. P.

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]

Ashoori, R.

R. Ashoori, “Electrons in artificial atoms,” Nature 379, 413–419 (1996).
[CrossRef]

Atherton, T.

Ballesteros, J.

J. Ballesteros, J. Solis, R. Serna, and C. Afonso, “Nanocrystal size dependence of the third-order nonlinear optical response of cu: Alo thin films,” Appl. Phys. Lett. 74, 2791 (1999).
[CrossRef]

Banfi, G.

G. Banfi, V. Degiorgio, and D. Ricard, “Nonlinear optical properties of semiconductor nanocrystals,” Adv. Phys. 47, 447–510 (1998).
[CrossRef]

Barlow, S.

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

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]

Belloni, M.

M. Belloni and R. W. Robinett, “Quantum mechanical sum rules for two model systems,” Am. J. Phys. 76, 798–806 (2008).
[CrossRef]

Beratan, D. N.

S. Keinan, M. J. Therien, D. N. Beratan, and W. T. Yang, “Molecular design of porphyrin-based nonlinear optical materials,” J. Phys. Chem. A 112, 12203–12207 (2008).
[CrossRef]

Bergey, E. J.

I. Roy, T. Y. Ohulchanskyy, 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]

Bethe, H. A.

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Plenum, 1977).

Blümel, R.

Z. Pastore and R. Blümel, “An exact periodic-orbit formula for the energy levels of the three-pronged star graph,” J. Phys. A 42, 135102 (2009).
[CrossRef]

Brédas, J.

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

Brus, L.

L. Brus, “Electron–electron and electron-hole interactions in small semiconductor crystallites: the size dependence of the lowest excited electronic state,” J. Chem. Phys. 80, 4403 (1984).
[CrossRef]

Buda, F.

F. Buda, J. Kohanoff, and M. Parrinello, “Optical properties of porous silicon: a first-principles study,” Phys. Rev. Lett. 69, 1272–1275 (1992).
[CrossRef]

Chang, Y.

G. Sanders and Y. Chang, “Theory of optical properties of quantum wires in porous silicon,” Phys. Rev. B 45, 9202–9213 (1992).
[CrossRef]

Cheah, K.

J. Xia and K. Cheah, “Quantum confinement effect in thin quantum wires,” Phys. Rev. B 55, 15688–15693 (1997).
[CrossRef]

Chen, R.

R. Chen, D. Lin, and B. Mendoza, “Enhancement of the third-order nonlinear optical susceptibility in si quantum wires,” Phys. Rev. B 48, 11879–11882 (1993).
[CrossRef]

Chien, L.

V. Ostroverkhov, O. Ostroverkhova, R. Petschek, K. Singer, L. Sukhomlinova, R. Twieg, S. Wang, and L. Chien, “Optimization of the molecular hyperpolarizability for second harmonic generation in chiral media,” Chem. Phys. 257, 263–274 (2000).
[CrossRef]

Choe, H.

H. Yan, H. Choe, S. Nam, Y. Hu, S. Das, J. Klemic, J. Ellenbogen, and C. Lieber, “Programmable nanowire circuits for nanoprocessors,” Nature 470, 240–244 (2011).
[CrossRef]

Chu, X.

X. Chu, M. Yang, and K. Jackson, “The effect of geometry on cluster polarizability: Studies of sodium, copper, and silicon clusters at shape-transition sizes,” J. Chem. Phys. 134, 234505 (2011).
[CrossRef]

Clays, K.

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

K. Tripathy, J. Pérez 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]

Coe, B. J.

K. Tripathy, J. Pérez 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]

Cumpston, B. H.

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]

Currie, W.

E. Hadjimichael, W. Currie, and S. Fallieros, “The Thomas–Reiche–Kuhn sum rule and the rigid rotator,” Am. J. Phys. 65, 335–341 (1997).
[CrossRef]

Das, S.

H. Yan, H. Choe, S. Nam, Y. Hu, S. Das, J. Klemic, J. Ellenbogen, and C. Lieber, “Programmable nanowire circuits for nanoprocessors,” Nature 470, 240–244 (2011).
[CrossRef]

Degiorgio, V.

G. Banfi, V. Degiorgio, and D. Ricard, “Nonlinear optical properties of semiconductor nanocrystals,” Adv. Phys. 47, 447–510 (1998).
[CrossRef]

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[CrossRef]

Sun, C.

Z. Zhou, Z. Liu, Z. Li, X. Huang, and C. Sun, “Shape effect of graphene quantum dot on enhancing second order nonlinear optical response and spin multiplicity in nh2-gqd-no2 systems,” J. Phys. Chem. C115, 16282–16286 (2011).

Sun, H.-B.

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

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

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S. Keinan, M. J. Therien, D. N. Beratan, and W. T. Yang, “Molecular design of porphyrin-based nonlinear optical materials,” J. Phys. Chem. A 112, 12203–12207 (2008).
[CrossRef]

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K. Tripathy, J. Pérez 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]

Twieg, R.

V. Ostroverkhov, O. Ostroverkhova, R. Petschek, K. Singer, L. Sukhomlinova, R. Twieg, S. Wang, and L. Chien, “Optimization of the molecular hyperpolarizability for second harmonic generation in chiral media,” Chem. Phys. 257, 263–274 (2000).
[CrossRef]

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R. C. Jaklevic, J. Lambe, M. Mikkor, and W. C. Vassell, “Observation of electron standing waves in a crystalline box,” Phys. Rev. Lett. 26, 88–92 (1971).
[CrossRef]

Wang, S.

V. Ostroverkhov, O. Ostroverkhova, R. Petschek, K. Singer, L. Sukhomlinova, R. Twieg, S. Wang, and L. Chien, “Optimization of the molecular hyperpolarizability for second harmonic generation in chiral media,” Chem. Phys. 257, 263–274 (2000).
[CrossRef]

Wang, Z.

C. Lieber and Z. Wang, “Functional nanowires,” MRS Bull. 32, 99–108 (2007).
[CrossRef]

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B. J. Orr and J. F. Ward, “Perturbation theory of the non-linear optical polarization of an isolated system,” Mol. Phys. 20, 513–526 (1971).
[CrossRef]

Watkins, D. S.

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

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

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M. Huang, S. Mao, H. Feick, H. Yan, Y. Wu, H. Kind, E. Weber, R. Russo, and P. Yang, “Room-temperature ultraviolet nanowire nanolasers,” Science 292, 1897–1899 (2001).
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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]

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M. Huang, S. Mao, H. Feick, H. Yan, Y. Wu, H. Kind, E. Weber, R. Russo, and P. Yang, “Room-temperature ultraviolet nanowire nanolasers,” Science 292, 1897–1899 (2001).
[CrossRef]

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J. Xia and K. Cheah, “Quantum confinement effect in thin quantum wires,” Phys. Rev. B 55, 15688–15693 (1997).
[CrossRef]

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A. Feizpour, X. Xing, and A. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).
[CrossRef]

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H. Yan, H. Choe, S. Nam, Y. Hu, S. Das, J. Klemic, J. Ellenbogen, and C. Lieber, “Programmable nanowire circuits for nanoprocessors,” Nature 470, 240–244 (2011).
[CrossRef]

J. Johnson, H. Yan, R. Schaller, P. Petersen, P. Yang, and R. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2, 279–283 (2002).
[CrossRef]

M. Huang, S. Mao, H. Feick, H. Yan, Y. Wu, H. Kind, E. Weber, R. Russo, and P. Yang, “Room-temperature ultraviolet nanowire nanolasers,” Science 292, 1897–1899 (2001).
[CrossRef]

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R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009).
[CrossRef]

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X. Chu, M. Yang, and K. Jackson, “The effect of geometry on cluster polarizability: Studies of sodium, copper, and silicon clusters at shape-transition sizes,” J. Chem. Phys. 134, 234505 (2011).
[CrossRef]

Yang, P.

R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009).
[CrossRef]

J. Johnson, H. Yan, R. Schaller, P. Petersen, P. Yang, and R. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2, 279–283 (2002).
[CrossRef]

M. Huang, S. Mao, H. Feick, H. Yan, Y. Wu, H. Kind, E. Weber, R. Russo, and P. Yang, “Room-temperature ultraviolet nanowire nanolasers,” Science 292, 1897–1899 (2001).
[CrossRef]

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S. Keinan, M. J. Therien, D. N. Beratan, and W. T. Yang, “Molecular design of porphyrin-based nonlinear optical materials,” J. Phys. Chem. A 112, 12203–12207 (2008).
[CrossRef]

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

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

Zhou, Z.

Z. Zhou, Z. Liu, Z. Li, X. Huang, and C. Sun, “Shape effect of graphene quantum dot on enhancing second order nonlinear optical response and spin multiplicity in nh2-gqd-no2 systems,” J. Phys. Chem. C115, 16282–16286 (2011).

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M. G. Kuzyk, “Quantum limits of the hyper-Rayleigh scattering susceptibility,” IEEE J. Sel. Topics Quantum Electron. 7, 774–780 (2001).
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K. Tripathy, J. Pérez 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).
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M. G. Kuzyk and D. S. Watkins, “The effects of geometry on the hyperpolarizability,” J. Chem. Phys. 124, 244104 (2006).
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J. Pérez-Moreno, K. Clays, and M. G. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128, 084109 (2008).
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[CrossRef]

MRS Bull. (1)

C. Lieber and Z. Wang, “Functional nanowires,” MRS Bull. 32, 99–108 (2007).
[CrossRef]

Nano Lett. (1)

J. Johnson, H. Yan, R. Schaller, P. Petersen, P. Yang, and R. Saykally, “Near-field imaging of nonlinear optical mixing in single zinc oxide nanowires,” Nano Lett. 2, 279–283 (2002).
[CrossRef]

Nat. Photonics (1)

R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009).
[CrossRef]

Nature (4)

H. Yan, H. Choe, S. Nam, Y. Hu, S. Das, J. Klemic, J. Ellenbogen, and C. Lieber, “Programmable nanowire circuits for nanoprocessors,” Nature 470, 240–244 (2011).
[CrossRef]

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[CrossRef]

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[CrossRef]

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).
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[CrossRef]

M. Huang, S. Mao, H. Feick, H. Yan, Y. Wu, H. Kind, E. Weber, R. Russo, and P. Yang, “Room-temperature ultraviolet nanowire nanolasers,” Science 292, 1897–1899 (2001).
[CrossRef]

Other (4)

Z. Zhou, Z. Liu, Z. Li, X. Huang, and C. Sun, “Shape effect of graphene quantum dot on enhancing second order nonlinear optical response and spin multiplicity in nh2-gqd-no2 systems,” J. Phys. Chem. C115, 16282–16286 (2011).

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Figures (5)

Fig. 1.
Fig. 1.

2D closed quantum loop. (0, 0) is the starting point of the loop. (x,y) is the electron’s coordinate at point v with respect to (xn,yn). θ is the angle that this wire segment makes with the x axis. Each wire segment confines the electron in the transverse direction.

Fig. 2.
Fig. 2.

Three triangles with largest βint(0.049).

Fig. 3.
Fig. 3.

Distribution of βint over its full range of triangular configurations as quantified by the ratio of the smallest angle, θmin, and largest angle, θmax. Points in blue represent isosceles triangles.

Fig. 4.
Fig. 4.

βint for isosceles triangles with two points fixed at (0, 10) and (0,10), and the third point varying along the x axis. The triangle in the picture represents the largest value of βint=0.049.

Fig. 5.
Fig. 5.

Distribution of γint over its full range of triangular configurations as quantified by the ratio of the smallest to largest angle.

Tables (1)

Tables Icon

Table 1. Numerical Results for Diagonal Components, i.e., p=q and κ=λ in Eq. (28) for Triangles, 100 Longitudinal and 200 Transverse States

Equations (49)

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

ψ(s)=Aeikss+Beikss,
V(τ)=gδ(τ),
η0(τ)=k0τek0τ|τ|,
k0τ=mg2,
E0τ=2k0τ22m.
ηνeven(τ)=2acos(νπ)sin(2νπa|τ|)
ηνodd(τ)=2acos(νπ)sin(2νπaτ),
Eν=2kντ22m,
kντa2=νπ.
H(s,τ)=22m(s2+τ2)gδ(τ)Hs+Hτ,
Hs=22ms2andHτ=22mτ2gδ(τ).
ψli(s)=1Rexp[±(iklss+ϕli)]Θ[s]Θ[Lis],
s=(xxn)2+(yyn)2.
kls=2πlR,
ϕli(s)=2πlR·s(0,i1)
s(0,i1)=j=1i1Lj=j=1n(xjxj1)2+(yjyj1)2.
Θ[xx0]={1xx00x<x0,
Els=2π22l2mR2l=±1,±2,
ϕ(s,τ)=ψm(s)ημ(τ)=k=1Eϕk(s,τ).
xpκ,qλpκ|x|qλ=dxxϕpκ*(x)ϕqλ(x)=kidxxkϕpkk*(x)ϕqλk(x).
x=scosθi±τsinθi+x1i,
(xsi)pκ,qλ=δκλReikqps(0,i1)×[(cosθikqp2x2ikqpi)eikqpLicosθikqp2+x1ikqpi].
(xsi)pκ,pλ=δκλ(x2i+x1i)2LiR.
(xτi)pκ,qλ=sinθiidsψpi(s)ψqi(s)idηηηκ(τ)ηλ(τ)sinθi×Ipqi×Jκλi,
Ipqi={LiRp=qeikqpss(0,i1)ikqpsR(eikqpsLi1)pq.
Jκλi={82(1)κπκa5/2k0τ3/2(a2k0τ2+4π2κ2)2κ(λ)>0,λ(κ)=02(1+(1)κ+λ+1)a|κλ|π2(κ2λ2)2λκ0a4|κ|=|λ|0.
n(EnEm+Ep2)xmnxnp=22mδmp.
δκλi,n[(EnsEps+Eqs2)xpns,ixnqs,i]+i,ν[(EντEκτ+Eλτ2)Ipqixκντ,ixνλτ,i]=22mδpqδκλ,
βxxx=3e3n,mx0nx¯nmxm0En0Em0,
βmax=31/4(em1/2)3N3/2E107/2.
βintββmax=(34)3/4n,mξ0nξnmξm0enem,
ξij=xijx01max,ei=Ei0E10,
x01max=(22mE10)1/2,
βint=f(E)G(X),
f(E)=(1E)3/2(E2+32E+1),
G(X)=34X32(1X4),
γint=14(n,m,lξ0nξ¯nmξ¯mlξl0enemeln,m|ξ0n|2|ξm0|2en2em),
|N=k=1E|Nk)
M|=(Mj|j=1E,
N|M=(Mj|j=1Ei=1E|Ni)=δNM
N|M=dsψ*(s)ψ(s)=kEdxkϕNk*(xk)ϕMk(xk)=k=1E(Mk|Nk).
|NM|=j=1E|Mj)(Ni|i=1E.
N|O|M=(Nj|j=1EOi=1E|Mi)=i=1E(Ni|O|Mi).
N|AB|M,=PN|A|PP|B|M=P(Nj|j=1EAi=1E|Pi)(Pk|k=1EBl=1E|Ml)=P(j=1E(Nj|A|Pj))(l=1E(Pl|B|Ml)),
N|[[H,x],x]|M=N|Hxx+xxH2xHx|M=22mδNM.
N|Hxx|M=PENN|x|PP|x|M.
i,j=1E(P[EP12(EN+EM)](Ni|x|Pi)(Pj|x|Mj))=22mi=1E(Ni|Mi)=22mδNM.
xk=1x1k+1τscosθ1sτsinθ,yk=1y1k+1τssinθ+1sτcosθ,
(pκ|xk|qλ)=δκ,λ0Lkdsψpk*(s)(x1k+scosθ)ψqk(s)Ipq+dτϕpk*(τ)τsinθϕpk(τ),

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