Abstract

We have designed a hyper-Rayleigh scattering scheme to measure six scalar invariants of the squared hyperpolarizability tensor β2. Our theoretical approach expresses the rotational invariants of the irreducible β components as scalars, which eliminates the need for difficult frame transformations. We applied our scheme to several conjugated chiral molecules and found that there are significant Kleinman-disallowed pseudotensor contributions to their hyperpolarizability. These components, along with a large optical rotation and the results of quantum-chemical calculations, indicate a handed nonplanar delocalization of the charge-transfer system in such molecules as predicted by quantum-chemical calculations and are expected to lead to macroscopic second-harmonic generation in axially aligned polymer materials. Pseudotensor contributions to the hyperpolarizability in chiral molecules were found to be as large as the vector contribution in p-nitroaniline. We qualitatively investigated the dispersion in the Kleinman-disallowed components and confirmed that these components are smaller at longer wavelengths.

© 1998 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1997 (1)

S. W. Wong and K. Y. Wong, “Studies of depolarized hyper-Rayleigh scattering of organic molecule,” Opt. Commun. 133, 268–272 (1997).
[CrossRef]

1996 (5)

S. F. Hubbard, R. G. Petschek, and K. D. Singer, “Spectral content and dispersion of hyper-Rayleigh scattering,” Opt. Lett. 21, 1774–1776 (1996).
[CrossRef] [PubMed]

P. Kaatz and D. P. Shelton, “Polarized hyper-Rayleigh light scattering measurements of nonlinear optical chromophores,” J. Chem. Phys. 105, 3918–3929 (1996).
[CrossRef]

I. D. Morrison, R. G. Denning, W. M. Laidlaw, and M. A. Stammers, “Measurement of first hyperpolarizabilities by hyper-Rayleigh scattering,” Rev. Sci. Instrum. 67, 1445–1453 (1996).
[CrossRef]

O. J. F. Noordman and N. F. van Hulst, “Time-resolved hyper-Rayleigh scattering: measuring first hyperpolarizabilities β of fluorescent molecules,” Chem. Phys. Lett. 253, 145–150 (1996).
[CrossRef]

Y. Luo, A. Cesar, and H. Ågren, “The hyperpolarizability of the tricyanomethanide molecular ion in solution,” Chem. Phys. Lett. 252, 389–397 (1996).
[CrossRef]

1995 (6)

C. Denhaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgauld, and H. Le Bozec, “Chiral metal complexes with large octupolar optical nonlinearities,” Nature (London) 374, 339–342 (1995).
[CrossRef]

S. Stadler, R. Deitrich, G. Bourhill, and C. Bräuchle, “Long-wavelength first hyperpolarizability measurements by hyper-Rayleigh scattering,” Opt. Lett. 21, 251–253 (1995).
[CrossRef]

H. S. Nalwa, T. Watanabe, and S. Miyata, “2-d charge-transfer molecules for second order nlo: off-diagonal orientation,” Adv. Mater. 7, 754–758 (1995).
[CrossRef]

M. Kauranen and A. Persoons, “Theory of polarization measurements of second-order nonlinear light scattering,” J. Chem. Phys. 104, 3445–3456 (1995).
[CrossRef]

J. J. Maki, M. Kauranen, and A. Persoons, “Surface second-harmonic generation from chiral materials,” Phys. Rev. B 51, 1425–1433 (1995).
[CrossRef]

J. D. V. Khaydarov, J. H. Andrews, and K. D. Singer, “Pulse compression mechanism in a synchronously pumped optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2199–2208 (1995).
[CrossRef]

1994 (6)

M. Kauranen, T. Verbiest, J. J. Maki, and A. Persoons, “Second-harmonic generation from chiral surfaces,” J. Chem. Phys. 101, 8193–8199 (1994).
[CrossRef]

E. Hendrickx, K. Clays, A. Persoons, C. Dehu, and J. L. Bredàs, “The bacteriorhodopsin chromophore retinal and derivatives: an experimental and theoretical investigation of the second-order optical properties,” J. Am. Chem. Soc. 117, 3547–3555 (1994).
[CrossRef]

A. Persoons, K. Clays, M. Kauranen, E. Hendrickx, E. Put, and W. Bijens, “Characterization of nonlinear optical properties by hyper-scattering techniques,” Synth. Met. 67, 31–38 (1994).
[CrossRef]

P. K. Schmidt and G. W. Rayfield, “Hyper-Rayleigh light scattering from an aqueous suspension of purple membrane,” Appl. Opt. 33, 4286–4292 (1994).
[CrossRef] [PubMed]

T. Verbiest, K. Clays, C. Samyn, J. Wolff, D. Reinhoudt, and A. Persoons, “Investigations of the hyperpolarizability in organic molecules from dipolar to octopolar systems,” J. Am. Chem. Soc. 116, 9320–9323 (1994).
[CrossRef]

J. Zyss and I. Ledoux, “Nonlinear optics in multipolar media: theory and experiments,” Chem. Rev. 94, 77–105 (1994).
[CrossRef]

1993 (5)

J. Zyss, T. Chauvan, C. Dhenaut, and I. Ledoux, “Harmonic Rayleigh scattering from nonlinear octupolar molecular media: the case of Crystal Violet,” Chem. Phys. 177, 281–296 (1993).
[CrossRef]

J. Zyss, C. Dhenaut, T. Chauvan, and I. Ledoux, “Quadratic nonlinear susceptibility of octupolar chiral ions,” Chem. Phys. Lett. 206, 409–414 (1993).
[CrossRef]

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

G. J. T. Heesink, A. G. T. Ruiter, N. F. Van Hulst, and B. Bölger, “Determination of hyperpolarizability tensor components by depolarized hyper-Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

1992 (2)

R. Wortmann, P. Krämer, C. Glania, S. Lebus, and N. Detzer, “Deviations from Kleinman symmetry of the second-order polarizability tensor in molecules with low-lying perpendicular electronic bands,” Chem. Phys. 173, 99–108 (1992).
[CrossRef]

M. Joffre, D. Yaron, R. Silbey, and J. Zyss, “Second order optical nonlinearity in octopolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

1991 (1)

J. Zyss, “Octupolar organic systems in quadratic nonlinear optics: molecules and materials,” Nonlinear Opt. 1, 3–18 (1991).

1989 (1)

1986 (2)

K. D. Singer, J. E. Sohn, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
[CrossRef]

G. Wagniere, “Theoretical investigation of Kleinman symmetry in molecules,” Appl. Phys. B 41, 169–172 (1986).
[CrossRef]

1985 (1)

M. Kozierowski, “Electric-dipole differential hyper-Rayleigh and hyper-Raman scattering of elliptically polarized light,” Phys. Rev. A 31, 509–510 (1985).
[CrossRef] [PubMed]

1982 (1)

G. Wagniere, “The evaluation of three-dimensional rotational averages,” J. Chem. Phys. 76, 473–480 (1982).
[CrossRef]

1978 (1)

J. Jerphagnon, D. Chemla, and R. Bonneville, “The description of the physical properties of condensed matter using irreducible tensors,” Adv. Phys. 27, 609–650 (1978).
[CrossRef]

1974 (1)

S. Kielich and M. Kozierowski, “Symmetric and antisymmetric second-harmonic elastic light scattering and its angular dependences,” Acta Phys. Pol. A45, 231–251 (1974).

1972 (1)

W. M. McClain, “Polarization dependence of three-photon phenomena for randomly oriented molecules,” J. Chem. Phys. 67, 2264–2274 (1972).
[CrossRef]

1970 (2)

P. D. Maker, “Spectral broadening of elastic second-harmonic light scattering in liquids,” Phys. Rev. A 1, 923–951 (1970).
[CrossRef]

J. Jerphagnon, “Invariants of the third-rank Cartesian tensor: optical nonlinear susceptibilities,” Phys. Rev. B 2, 1091–1098 (1970).
[CrossRef]

1966 (2)

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966).
[CrossRef]

R. Bersohn, Y. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
[CrossRef]

1965 (3)

J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. 138, 1599–1606 (1965).
[CrossRef]

S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[CrossRef]

R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
[CrossRef]

1962 (1)

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Ågren, H.

Y. Luo, A. Cesar, and H. Ågren, “The hyperpolarizability of the tricyanomethanide molecular ion in solution,” Chem. Phys. Lett. 252, 389–397 (1996).
[CrossRef]

Andrews, J. H.

Baldridge, K. K.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Bersohn, R.

R. Bersohn, Y. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
[CrossRef]

Bijens, W.

A. Persoons, K. Clays, M. Kauranen, E. Hendrickx, E. Put, and W. Bijens, “Characterization of nonlinear optical properties by hyper-scattering techniques,” Synth. Met. 67, 31–38 (1994).
[CrossRef]

Boatz, J. A.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Bölger, B.

G. J. T. Heesink, A. G. T. Ruiter, N. F. Van Hulst, and B. Bölger, “Determination of hyperpolarizability tensor components by depolarized hyper-Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

Bonneville, R.

J. Jerphagnon, D. Chemla, and R. Bonneville, “The description of the physical properties of condensed matter using irreducible tensors,” Adv. Phys. 27, 609–650 (1978).
[CrossRef]

Bourgauld, M.

C. Denhaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgauld, and H. Le Bozec, “Chiral metal complexes with large octupolar optical nonlinearities,” Nature (London) 374, 339–342 (1995).
[CrossRef]

Bourhill, G.

Bräuchle, C.

Bredàs, J. L.

E. Hendrickx, K. Clays, A. Persoons, C. Dehu, and J. L. Bredàs, “The bacteriorhodopsin chromophore retinal and derivatives: an experimental and theoretical investigation of the second-order optical properties,” J. Am. Chem. Soc. 117, 3547–3555 (1994).
[CrossRef]

Cesar, A.

Y. Luo, A. Cesar, and H. Ågren, “The hyperpolarizability of the tricyanomethanide molecular ion in solution,” Chem. Phys. Lett. 252, 389–397 (1996).
[CrossRef]

Chauvan, T.

J. Zyss, T. Chauvan, C. Dhenaut, and I. Ledoux, “Harmonic Rayleigh scattering from nonlinear octupolar molecular media: the case of Crystal Violet,” Chem. Phys. 177, 281–296 (1993).
[CrossRef]

J. Zyss, C. Dhenaut, T. Chauvan, and I. Ledoux, “Quadratic nonlinear susceptibility of octupolar chiral ions,” Chem. Phys. Lett. 206, 409–414 (1993).
[CrossRef]

Chemla, D.

J. Jerphagnon, D. Chemla, and R. Bonneville, “The description of the physical properties of condensed matter using irreducible tensors,” Adv. Phys. 27, 609–650 (1978).
[CrossRef]

Clays, K.

A. Persoons, K. Clays, M. Kauranen, E. Hendrickx, E. Put, and W. Bijens, “Characterization of nonlinear optical properties by hyper-scattering techniques,” Synth. Met. 67, 31–38 (1994).
[CrossRef]

E. Hendrickx, K. Clays, A. Persoons, C. Dehu, and J. L. Bredàs, “The bacteriorhodopsin chromophore retinal and derivatives: an experimental and theoretical investigation of the second-order optical properties,” J. Am. Chem. Soc. 117, 3547–3555 (1994).
[CrossRef]

T. Verbiest, K. Clays, C. Samyn, J. Wolff, D. Reinhoudt, and A. Persoons, “Investigations of the hyperpolarizability in organic molecules from dipolar to octopolar systems,” J. Am. Chem. Soc. 116, 9320–9323 (1994).
[CrossRef]

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

Cyvin, S. J.

S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[CrossRef]

Decius, J. C.

S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[CrossRef]

Dehu, C.

E. Hendrickx, K. Clays, A. Persoons, C. Dehu, and J. L. Bredàs, “The bacteriorhodopsin chromophore retinal and derivatives: an experimental and theoretical investigation of the second-order optical properties,” J. Am. Chem. Soc. 117, 3547–3555 (1994).
[CrossRef]

Deitrich, R.

Denhaut, C.

C. Denhaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgauld, and H. Le Bozec, “Chiral metal complexes with large octupolar optical nonlinearities,” Nature (London) 374, 339–342 (1995).
[CrossRef]

Denning, R. G.

I. D. Morrison, R. G. Denning, W. M. Laidlaw, and M. A. Stammers, “Measurement of first hyperpolarizabilities by hyper-Rayleigh scattering,” Rev. Sci. Instrum. 67, 1445–1453 (1996).
[CrossRef]

Detzer, N.

R. Wortmann, P. Krämer, C. Glania, S. Lebus, and N. Detzer, “Deviations from Kleinman symmetry of the second-order polarizability tensor in molecules with low-lying perpendicular electronic bands,” Chem. Phys. 173, 99–108 (1992).
[CrossRef]

Dhenaut, C.

J. Zyss, T. Chauvan, C. Dhenaut, and I. Ledoux, “Harmonic Rayleigh scattering from nonlinear octupolar molecular media: the case of Crystal Violet,” Chem. Phys. 177, 281–296 (1993).
[CrossRef]

J. Zyss, C. Dhenaut, T. Chauvan, and I. Ledoux, “Quadratic nonlinear susceptibility of octupolar chiral ions,” Chem. Phys. Lett. 206, 409–414 (1993).
[CrossRef]

Dirk, C. W.

Dupuis, M.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Elbert, S. T.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Engersen, J. F. J.

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

Frisch, H. L.

R. Bersohn, Y. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
[CrossRef]

Giordmaine, J. A.

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966).
[CrossRef]

J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. 138, 1599–1606 (1965).
[CrossRef]

Glania, C.

R. Wortmann, P. Krämer, C. Glania, S. Lebus, and N. Detzer, “Deviations from Kleinman symmetry of the second-order polarizability tensor in molecules with low-lying perpendicular electronic bands,” Chem. Phys. 173, 99–108 (1992).
[CrossRef]

Gordon, H. M.

Gordon, M. S.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Heesink, G. J. T.

G. J. T. Heesink, A. G. T. Ruiter, N. F. Van Hulst, and B. Bölger, “Determination of hyperpolarizability tensor components by depolarized hyper-Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

Hendrickx, E.

E. Hendrickx, K. Clays, A. Persoons, C. Dehu, and J. L. Bredàs, “The bacteriorhodopsin chromophore retinal and derivatives: an experimental and theoretical investigation of the second-order optical properties,” J. Am. Chem. Soc. 117, 3547–3555 (1994).
[CrossRef]

A. Persoons, K. Clays, M. Kauranen, E. Hendrickx, E. Put, and W. Bijens, “Characterization of nonlinear optical properties by hyper-scattering techniques,” Synth. Met. 67, 31–38 (1994).
[CrossRef]

Hubbard, S. F.

Jensen, J. H.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Jerphagnon, J.

J. Jerphagnon, D. Chemla, and R. Bonneville, “The description of the physical properties of condensed matter using irreducible tensors,” Adv. Phys. 27, 609–650 (1978).
[CrossRef]

J. Jerphagnon, “Invariants of the third-rank Cartesian tensor: optical nonlinear susceptibilities,” Phys. Rev. B 2, 1091–1098 (1970).
[CrossRef]

Joffre, M.

M. Joffre, D. Yaron, R. Silbey, and J. Zyss, “Second order optical nonlinearity in octopolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

Kaatz, P.

P. Kaatz and D. P. Shelton, “Polarized hyper-Rayleigh light scattering measurements of nonlinear optical chromophores,” J. Chem. Phys. 105, 3918–3929 (1996).
[CrossRef]

Katz, H. E.

Kauranen, M.

J. J. Maki, M. Kauranen, and A. Persoons, “Surface second-harmonic generation from chiral materials,” Phys. Rev. B 51, 1425–1433 (1995).
[CrossRef]

M. Kauranen and A. Persoons, “Theory of polarization measurements of second-order nonlinear light scattering,” J. Chem. Phys. 104, 3445–3456 (1995).
[CrossRef]

M. Kauranen, T. Verbiest, J. J. Maki, and A. Persoons, “Second-harmonic generation from chiral surfaces,” J. Chem. Phys. 101, 8193–8199 (1994).
[CrossRef]

A. Persoons, K. Clays, M. Kauranen, E. Hendrickx, E. Put, and W. Bijens, “Characterization of nonlinear optical properties by hyper-scattering techniques,” Synth. Met. 67, 31–38 (1994).
[CrossRef]

Kelderman, E.

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

Khaydarov, J. D. V.

Kielich, S.

S. Kielich and M. Kozierowski, “Symmetric and antisymmetric second-harmonic elastic light scattering and its angular dependences,” Acta Phys. Pol. A45, 231–251 (1974).

King, L. A.

Kleinman, D. A.

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Koseki, S.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Kozierowski, M.

M. Kozierowski, “Electric-dipole differential hyper-Rayleigh and hyper-Raman scattering of elliptically polarized light,” Phys. Rev. A 31, 509–510 (1985).
[CrossRef] [PubMed]

S. Kielich and M. Kozierowski, “Symmetric and antisymmetric second-harmonic elastic light scattering and its angular dependences,” Acta Phys. Pol. A45, 231–251 (1974).

Krämer, P.

R. Wortmann, P. Krämer, C. Glania, S. Lebus, and N. Detzer, “Deviations from Kleinman symmetry of the second-order polarizability tensor in molecules with low-lying perpendicular electronic bands,” Chem. Phys. 173, 99–108 (1992).
[CrossRef]

Laidlaw, W. M.

I. D. Morrison, R. G. Denning, W. M. Laidlaw, and M. A. Stammers, “Measurement of first hyperpolarizabilities by hyper-Rayleigh scattering,” Rev. Sci. Instrum. 67, 1445–1453 (1996).
[CrossRef]

Lalama, S. J.

K. D. Singer, J. E. Sohn, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
[CrossRef]

Le Bozec, H.

C. Denhaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgauld, and H. Le Bozec, “Chiral metal complexes with large octupolar optical nonlinearities,” Nature (London) 374, 339–342 (1995).
[CrossRef]

Lebus, S.

R. Wortmann, P. Krämer, C. Glania, S. Lebus, and N. Detzer, “Deviations from Kleinman symmetry of the second-order polarizability tensor in molecules with low-lying perpendicular electronic bands,” Chem. Phys. 173, 99–108 (1992).
[CrossRef]

Ledoux, I.

C. Denhaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgauld, and H. Le Bozec, “Chiral metal complexes with large octupolar optical nonlinearities,” Nature (London) 374, 339–342 (1995).
[CrossRef]

J. Zyss and I. Ledoux, “Nonlinear optics in multipolar media: theory and experiments,” Chem. Rev. 94, 77–105 (1994).
[CrossRef]

J. Zyss, T. Chauvan, C. Dhenaut, and I. Ledoux, “Harmonic Rayleigh scattering from nonlinear octupolar molecular media: the case of Crystal Violet,” Chem. Phys. 177, 281–296 (1993).
[CrossRef]

J. Zyss, C. Dhenaut, T. Chauvan, and I. Ledoux, “Quadratic nonlinear susceptibility of octupolar chiral ions,” Chem. Phys. Lett. 206, 409–414 (1993).
[CrossRef]

Luo, Y.

Y. Luo, A. Cesar, and H. Ågren, “The hyperpolarizability of the tricyanomethanide molecular ion in solution,” Chem. Phys. Lett. 252, 389–397 (1996).
[CrossRef]

Maker, P. D.

P. D. Maker, “Spectral broadening of elastic second-harmonic light scattering in liquids,” Phys. Rev. A 1, 923–951 (1970).
[CrossRef]

R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
[CrossRef]

Maki, J. J.

J. J. Maki, M. Kauranen, and A. Persoons, “Surface second-harmonic generation from chiral materials,” Phys. Rev. B 51, 1425–1433 (1995).
[CrossRef]

M. Kauranen, T. Verbiest, J. J. Maki, and A. Persoons, “Second-harmonic generation from chiral surfaces,” J. Chem. Phys. 101, 8193–8199 (1994).
[CrossRef]

Matsunaga, N.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

McClain, W. M.

W. M. McClain, “Polarization dependence of three-photon phenomena for randomly oriented molecules,” J. Chem. Phys. 67, 2264–2274 (1972).
[CrossRef]

Miyata, S.

H. S. Nalwa, T. Watanabe, and S. Miyata, “2-d charge-transfer molecules for second order nlo: off-diagonal orientation,” Adv. Mater. 7, 754–758 (1995).
[CrossRef]

Montgomery, J. A.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Morrison, I. D.

I. D. Morrison, R. G. Denning, W. M. Laidlaw, and M. A. Stammers, “Measurement of first hyperpolarizabilities by hyper-Rayleigh scattering,” Rev. Sci. Instrum. 67, 1445–1453 (1996).
[CrossRef]

Nalwa, H. S.

H. S. Nalwa, T. Watanabe, and S. Miyata, “2-d charge-transfer molecules for second order nlo: off-diagonal orientation,” Adv. Mater. 7, 754–758 (1995).
[CrossRef]

Nguyen, K. A.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Noordman, O. J. F.

O. J. F. Noordman and N. F. van Hulst, “Time-resolved hyper-Rayleigh scattering: measuring first hyperpolarizabilities β of fluorescent molecules,” Chem. Phys. Lett. 253, 145–150 (1996).
[CrossRef]

Pao, Y.

R. Bersohn, Y. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
[CrossRef]

Persoons, A.

M. Kauranen and A. Persoons, “Theory of polarization measurements of second-order nonlinear light scattering,” J. Chem. Phys. 104, 3445–3456 (1995).
[CrossRef]

J. J. Maki, M. Kauranen, and A. Persoons, “Surface second-harmonic generation from chiral materials,” Phys. Rev. B 51, 1425–1433 (1995).
[CrossRef]

M. Kauranen, T. Verbiest, J. J. Maki, and A. Persoons, “Second-harmonic generation from chiral surfaces,” J. Chem. Phys. 101, 8193–8199 (1994).
[CrossRef]

A. Persoons, K. Clays, M. Kauranen, E. Hendrickx, E. Put, and W. Bijens, “Characterization of nonlinear optical properties by hyper-scattering techniques,” Synth. Met. 67, 31–38 (1994).
[CrossRef]

T. Verbiest, K. Clays, C. Samyn, J. Wolff, D. Reinhoudt, and A. Persoons, “Investigations of the hyperpolarizability in organic molecules from dipolar to octopolar systems,” J. Am. Chem. Soc. 116, 9320–9323 (1994).
[CrossRef]

E. Hendrickx, K. Clays, A. Persoons, C. Dehu, and J. L. Bredàs, “The bacteriorhodopsin chromophore retinal and derivatives: an experimental and theoretical investigation of the second-order optical properties,” J. Am. Chem. Soc. 117, 3547–3555 (1994).
[CrossRef]

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

Petschek, R. G.

Put, E.

A. Persoons, K. Clays, M. Kauranen, E. Hendrickx, E. Put, and W. Bijens, “Characterization of nonlinear optical properties by hyper-scattering techniques,” Synth. Met. 67, 31–38 (1994).
[CrossRef]

Rauch, J. E.

S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[CrossRef]

Rayfield, G. W.

Reinhoudt, D.

T. Verbiest, K. Clays, C. Samyn, J. Wolff, D. Reinhoudt, and A. Persoons, “Investigations of the hyperpolarizability in organic molecules from dipolar to octopolar systems,” J. Am. Chem. Soc. 116, 9320–9323 (1994).
[CrossRef]

Reinhoudt, D. N.

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

Rentzepis, P. M.

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966).
[CrossRef]

Ruiter, A. G. T.

G. J. T. Heesink, A. G. T. Ruiter, N. F. Van Hulst, and B. Bölger, “Determination of hyperpolarizability tensor components by depolarized hyper-Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

Samuel, I. D. W.

C. Denhaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgauld, and H. Le Bozec, “Chiral metal complexes with large octupolar optical nonlinearities,” Nature (London) 374, 339–342 (1995).
[CrossRef]

Samyn, C.

T. Verbiest, K. Clays, C. Samyn, J. Wolff, D. Reinhoudt, and A. Persoons, “Investigations of the hyperpolarizability in organic molecules from dipolar to octopolar systems,” J. Am. Chem. Soc. 116, 9320–9323 (1994).
[CrossRef]

Savage, C. M.

R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
[CrossRef]

Schmidt, M. W.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Schmidt, P. K.

Shelton, D. P.

P. Kaatz and D. P. Shelton, “Polarized hyper-Rayleigh light scattering measurements of nonlinear optical chromophores,” J. Chem. Phys. 105, 3918–3929 (1996).
[CrossRef]

Silbey, R.

M. Joffre, D. Yaron, R. Silbey, and J. Zyss, “Second order optical nonlinearity in octopolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

Singer, K. D.

Sohn, J. E.

Stadler, S.

Stammers, M. A.

I. D. Morrison, R. G. Denning, W. M. Laidlaw, and M. A. Stammers, “Measurement of first hyperpolarizabilities by hyper-Rayleigh scattering,” Rev. Sci. Instrum. 67, 1445–1453 (1996).
[CrossRef]

Starmans, W. A. J.

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

Su, S. J.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Terhune, R. W.

R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
[CrossRef]

Van Duynhoven, J. P. M.

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

van Hulst, N. F.

O. J. F. Noordman and N. F. van Hulst, “Time-resolved hyper-Rayleigh scattering: measuring first hyperpolarizabilities β of fluorescent molecules,” Chem. Phys. Lett. 253, 145–150 (1996).
[CrossRef]

G. J. T. Heesink, A. G. T. Ruiter, N. F. Van Hulst, and B. Bölger, “Determination of hyperpolarizability tensor components by depolarized hyper-Rayleigh scattering,” Phys. Rev. Lett. 71, 999–1002 (1993).
[CrossRef] [PubMed]

Verbiest, T.

T. Verbiest, K. Clays, C. Samyn, J. Wolff, D. Reinhoudt, and A. Persoons, “Investigations of the hyperpolarizability in organic molecules from dipolar to octopolar systems,” J. Am. Chem. Soc. 116, 9320–9323 (1994).
[CrossRef]

M. Kauranen, T. Verbiest, J. J. Maki, and A. Persoons, “Second-harmonic generation from chiral surfaces,” J. Chem. Phys. 101, 8193–8199 (1994).
[CrossRef]

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

Verboom, W.

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

Wagniere, G.

G. Wagniere, “Theoretical investigation of Kleinman symmetry in molecules,” Appl. Phys. B 41, 169–172 (1986).
[CrossRef]

G. Wagniere, “The evaluation of three-dimensional rotational averages,” J. Chem. Phys. 76, 473–480 (1982).
[CrossRef]

Watanabe, T.

H. S. Nalwa, T. Watanabe, and S. Miyata, “2-d charge-transfer molecules for second order nlo: off-diagonal orientation,” Adv. Mater. 7, 754–758 (1995).
[CrossRef]

Wecht, K. W.

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966).
[CrossRef]

Windus, T. L.

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

Wolff, J.

T. Verbiest, K. Clays, C. Samyn, J. Wolff, D. Reinhoudt, and A. Persoons, “Investigations of the hyperpolarizability in organic molecules from dipolar to octopolar systems,” J. Am. Chem. Soc. 116, 9320–9323 (1994).
[CrossRef]

Wong, K. Y.

S. W. Wong and K. Y. Wong, “Studies of depolarized hyper-Rayleigh scattering of organic molecule,” Opt. Commun. 133, 268–272 (1997).
[CrossRef]

Wong, S. W.

S. W. Wong and K. Y. Wong, “Studies of depolarized hyper-Rayleigh scattering of organic molecule,” Opt. Commun. 133, 268–272 (1997).
[CrossRef]

Wortmann, R.

R. Wortmann, P. Krämer, C. Glania, S. Lebus, and N. Detzer, “Deviations from Kleinman symmetry of the second-order polarizability tensor in molecules with low-lying perpendicular electronic bands,” Chem. Phys. 173, 99–108 (1992).
[CrossRef]

Yaron, D.

M. Joffre, D. Yaron, R. Silbey, and J. Zyss, “Second order optical nonlinearity in octopolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

Zyss, J.

C. Denhaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgauld, and H. Le Bozec, “Chiral metal complexes with large octupolar optical nonlinearities,” Nature (London) 374, 339–342 (1995).
[CrossRef]

J. Zyss and I. Ledoux, “Nonlinear optics in multipolar media: theory and experiments,” Chem. Rev. 94, 77–105 (1994).
[CrossRef]

J. Zyss, T. Chauvan, C. Dhenaut, and I. Ledoux, “Harmonic Rayleigh scattering from nonlinear octupolar molecular media: the case of Crystal Violet,” Chem. Phys. 177, 281–296 (1993).
[CrossRef]

J. Zyss, C. Dhenaut, T. Chauvan, and I. Ledoux, “Quadratic nonlinear susceptibility of octupolar chiral ions,” Chem. Phys. Lett. 206, 409–414 (1993).
[CrossRef]

M. Joffre, D. Yaron, R. Silbey, and J. Zyss, “Second order optical nonlinearity in octopolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

J. Zyss, “Octupolar organic systems in quadratic nonlinear optics: molecules and materials,” Nonlinear Opt. 1, 3–18 (1991).

Acta Phys. Pol. (1)

S. Kielich and M. Kozierowski, “Symmetric and antisymmetric second-harmonic elastic light scattering and its angular dependences,” Acta Phys. Pol. A45, 231–251 (1974).

Adv. Mater. (1)

H. S. Nalwa, T. Watanabe, and S. Miyata, “2-d charge-transfer molecules for second order nlo: off-diagonal orientation,” Adv. Mater. 7, 754–758 (1995).
[CrossRef]

Adv. Phys. (1)

J. Jerphagnon, D. Chemla, and R. Bonneville, “The description of the physical properties of condensed matter using irreducible tensors,” Adv. Phys. 27, 609–650 (1978).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

G. Wagniere, “Theoretical investigation of Kleinman symmetry in molecules,” Appl. Phys. B 41, 169–172 (1986).
[CrossRef]

Appl. Phys. Lett. (1)

K. D. Singer, J. E. Sohn, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
[CrossRef]

Chem. Mater. (1)

E. Kelderman, W. A. J. Starmans, J. P. M. Van Duynhoven, W. Verboom, J. F. J. Engersen, D. N. Reinhoudt, T. Verbiest, K. Clays, and A. Persoons, “Triphenylcarbinol derivatives as molecules for second-order nonlinear optics,” Chem. Mater. 6, 412–417 (1993).
[CrossRef]

Chem. Phys. (2)

J. Zyss, T. Chauvan, C. Dhenaut, and I. Ledoux, “Harmonic Rayleigh scattering from nonlinear octupolar molecular media: the case of Crystal Violet,” Chem. Phys. 177, 281–296 (1993).
[CrossRef]

R. Wortmann, P. Krämer, C. Glania, S. Lebus, and N. Detzer, “Deviations from Kleinman symmetry of the second-order polarizability tensor in molecules with low-lying perpendicular electronic bands,” Chem. Phys. 173, 99–108 (1992).
[CrossRef]

Chem. Phys. Lett. (3)

J. Zyss, C. Dhenaut, T. Chauvan, and I. Ledoux, “Quadratic nonlinear susceptibility of octupolar chiral ions,” Chem. Phys. Lett. 206, 409–414 (1993).
[CrossRef]

Y. Luo, A. Cesar, and H. Ågren, “The hyperpolarizability of the tricyanomethanide molecular ion in solution,” Chem. Phys. Lett. 252, 389–397 (1996).
[CrossRef]

O. J. F. Noordman and N. F. van Hulst, “Time-resolved hyper-Rayleigh scattering: measuring first hyperpolarizabilities β of fluorescent molecules,” Chem. Phys. Lett. 253, 145–150 (1996).
[CrossRef]

Chem. Rev. (1)

J. Zyss and I. Ledoux, “Nonlinear optics in multipolar media: theory and experiments,” Chem. Rev. 94, 77–105 (1994).
[CrossRef]

J. Am. Chem. Soc. (2)

T. Verbiest, K. Clays, C. Samyn, J. Wolff, D. Reinhoudt, and A. Persoons, “Investigations of the hyperpolarizability in organic molecules from dipolar to octopolar systems,” J. Am. Chem. Soc. 116, 9320–9323 (1994).
[CrossRef]

E. Hendrickx, K. Clays, A. Persoons, C. Dehu, and J. L. Bredàs, “The bacteriorhodopsin chromophore retinal and derivatives: an experimental and theoretical investigation of the second-order optical properties,” J. Am. Chem. Soc. 117, 3547–3555 (1994).
[CrossRef]

J. Chem. Phys. (8)

M. Joffre, D. Yaron, R. Silbey, and J. Zyss, “Second order optical nonlinearity in octopolar aromatic systems,” J. Chem. Phys. 97, 5607–5615 (1992).
[CrossRef]

P. Kaatz and D. P. Shelton, “Polarized hyper-Rayleigh light scattering measurements of nonlinear optical chromophores,” J. Chem. Phys. 105, 3918–3929 (1996).
[CrossRef]

W. M. McClain, “Polarization dependence of three-photon phenomena for randomly oriented molecules,” J. Chem. Phys. 67, 2264–2274 (1972).
[CrossRef]

S. J. Cyvin, J. E. Rauch, and J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering): selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Example of the graphic notation used to represent (a) rank-6 intensity coefficients and (b) the effects of the permutation operations.

Fig. 2
Fig. 2

Reduction of rank-6 coefficients to polarization-dependent scalars.

Fig. 3
Fig. 3

Polar plots of the tensor coefficients for the six allowed values of β2 in HRS.

Fig. 4
Fig. 4

Chiral molecules used in this study with λmax values determined with a UV–visible spectrophotometer.

Fig. 5
Fig. 5

Wave functions of a chiral camphorquinone molecule in the AM1 optimized geometry. The contours of the wave function are shown.

Fig. 6
Fig. 6

Experimental setup for determining rotational invariants with HRS: OPO, optical parametric amplifier; OPA, optical parametric amplifier; TC, temperature controller; BS’s, beam splitters; M’s, silvered mirrors; F’s, color filters; Pol.’s, linear polarizers; WP, quarter-wave plate; BC, Berek compensator; L’s, lenses; QC, quartz crystal; PMT’s, photomultiplier tubes; HV’s, high-voltage supplies; MC, motion controller; OSC, digital oscilloscope; PD, photodiode; PC, computer.

Fig. 7
Fig. 7

Polar plots of real data with fitted (solid) curves and Kleinman-allowed (dashed) curves for (a) pNA and (b) Q-KCh.

Tables (4)

Tables Icon

Table 1 Allowed Representations According to Tensor Rank

Tables Icon

Table 2 Formal Expressions of the Δ Tensors with Calculated Weight Factors

Tables Icon

Table 3 Normalized β2 Values for the Dyes in This Study

Tables Icon

Table 4 Values of β1,ss and β2,mm (×10-30 esu) at λ=1265 and λ=1350 nm

Equations (45)

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IILω3=GNBIJKLMN2(ω3=-ω2-ω1; ω2, ω1)(E1,Jω1)×(E1,Kω1)*(E2,Mω2)(E2,Nω2)*,
BIJKLMN2(ω3=-ω2-ω1; ω2, ω1)
=βIJK(ω3; ω1, ω2)βLMN*(ω3; ω1, ω2),
βpls1110(111)(22)3,
β1(11)sym1(02)1123,
β13,
βijk=βijk(3)+½[ijkβlk(2)+iklβlj(2)]+βi(1)δjk+½[βj(1)δik+βk(1)δij],
BIJKLMN2=LΔIJK,LMN(L)β2(L),
σ,σβ2(L, σ, σ, +)Re(tσ,σ)
-β2(L, σ, σ, -)Im(tσ,σ)0,
ijk,lmn=02[Δ(L)ijk,lmnΔ(L)ijk,lmn]=δL,L,
Δ(0)ijk,lmn=c(0)ijklmn,
Tr(Δ(L, a/s/m/m, a/s/m/m)ijk,lmn2)=1.
βi(1, 1)=δjkβkji,
βi(1, 2)=δjkβkij,
βi(1, 3)=δjkβijk,
β(1, s)=[β(1, 1)+β(1, 2)+β(1, 3)]
β(1, m)=[2β(1, 1)-β(1, 2)-β(1, 3)],
β(1, m)=[β(1, 2)-β(1, 3)].
βlk(2, 1)½(lijβijk+kijβijl)+δklmjiβmji,
βlk(2, 2)½(ijlβkij+ijkβlij)+δklmjiβmji.
βijk(3)=ijkβijk+nβ(1, n)iujk,
Ps={1+(12)+(13)+(23)+(123)+(132)},
Pm={2+2(23)-(123)-(132)-(12)-(13)},
Pm={2-2(23)-(123)-(132)+(12)+(13)},
I=IILeo,I*eo,L,
I=BIJKLMN2eo,I*ei,Jei,Keo,Lei,M*ei,N*
=LΔ(L, P1, P2)β(L)2.
Δ(1, ss)=c1,ss19[AA*+4BB*+4 Re(BA*C)],
Δ(1, mm)=c1,mm49[AA*+BB*-2 Re(BA*C)],
Δ(1, sm)+=c1,sm+49[AA*-2BB*+Re(BA*C)],
Δ(1, sm)-=c1,sm -4i3 [Im(C*AB*)],
Δ(2, mm)=c2,mm23(1-CC*)+c2,mm49[AA*+BB*-2 Re(BA*C)],
Δ(3, ss)=c3,ss13(1+2CC*)+c3,ss19[AA*+4BB*+4 Re(BA*C)],
e^i=cos(αi-γi)(sin γiyˆ+cos γixˆ)+i sin(γi-αi)×(cos γiyˆ-sin γixˆ).
e^o=cos(αo-γo)[sin γoyˆ+cos γo(zˆ sin θ+xˆ cos θ)]+i sin(γo-αo)[cos γoyˆ-sin γo(zˆ sin θ+xˆ cos θ)].
e^o=azˆ+b[cos κ(xˆ cos ϕ+yˆ sin ϕ)+i sin κ(yˆ cos ϕ-xˆ sin ϕ)],
e^i=eιψ2 [xˆ(cos 2γ-i)+yˆ sin 2γ],
|e^ie^i*|2(e^oe^o*)=|a|2+|b|2=1,
(e^ie^i*)|e^oe^i|2=|b|2[½ cos 2κ sin 2ϕ sin 4γ+2 sin 2κ cos2 ϕ cos 2γ+cos2 2γ cos 2ϕ+1],
(eie^i*)|e^o*e^i|2=|b|2[½ cos 2κ sin 2ϕ sin 4γ-2 sin 2κ cos2 ϕ cos 2γ+cos2 2γ cos 2ϕ+1],
(e^oe^o*)|e^ie^i|2=¼ cos2 2γ,
(e^oe^i)(e^o*e^i)(e^i*)2
=|b|2/32 cos 2γ[½ cos 2κ sin 2ϕ×sin 4γ-2 cos 2κ cos2 ϕ+i sin 2κ sin2 2γ],
σμ22[2(χ2)μ2],

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