Abstract

A general class of multipolar molecules is introduced in the context of quadratic nonlinear optics by way of extension of the more specific cases of dipolar and octupolar molecules. An adequate irreducible tensor formalism permits us to define rotationally invariant molecular features that couple to corresponding field tensor components, thereby enabling us to account for a variety of coherent and noncoherent processes such as harmonic light (hyper-Rayleigh) scattering, coherent second-harmonic generation in electrically poled media, and the recently proposed optical poling scheme. Experiments in both harmonic light scattering in solution (for some multipolar molecules) and optical poling (in Disperse Red 1–methyl methacrylate thin films) are analyzed in light of this model. A general tensorial permutation lemma of broad validity allows nonlinear light–matter interactions to be condensed in a statistical medium in compact rotationally invariant expressions: The main tensorial symmetry features for both molecular susceptibility and read–write field polarization tensors that jointly drive these interactions are clearly revealed.

© 1998 Optical Society of America

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1996 (4)

I. D. Morrison, R. G. Denning, W. M. Laidlaw, and M. A. Stammers, Rev. Sci. Instrum. 67, 1445 (1996).
[CrossRef]

P. Kaatz and D. P. Sheldon, Rev. Sci. Instrum. 67, 1439 (1996).
[CrossRef]

M. Kauranen and B. Persoons, J. Chem. Phys. 104, 3445 (1996).
[CrossRef]

B. Kippelen and N. Peyghambarian, Photon. Sci. News 2(2), 18 (1996).

1995 (4)

I. Ledoux, C. Dhenaut, I. D. W. Samuel, and J. Zyss, Nonlinear Opt. 14, 23 (1995).

H. S. Nalwa, T. Watanabe, and S. Miyata, Adv. Mater. 7, 754 (1995).
[CrossRef]

J. Zyss, T. Chauvan, C. Dhenaut, and I. Ledoux, Chem. Phys. 177, 281 (1995).
[CrossRef]

C. Dhenaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgault, and H. Le Bozec, Nature 374, 339 (1995).
[CrossRef]

1994 (3)

J. Zyss and I. Ledoux, Chem. Rev. 94, 77 (1994).
[CrossRef]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
[CrossRef]

C. Fiorini, F. Charra, and J. M. Nunzi, J. Opt. Soc. Am. B 11, 2347 (1994).
[CrossRef]

1993 (3)

H. S. Nalwa, T. Watanabe, and S. Miyata, Opt. Mater. 2, 73 (1993).
[CrossRef]

J. Zyss, J. Chem. Phys. 98, 6583 (1993).
[CrossRef]

G. H. T. Heesink, A. G. T. Ruiter, N. F. van Hulst, and B. Bölger, Phys. Rev. Lett. 71, 999 (1993).
[CrossRef] [PubMed]

1992 (2)

R. Bonneville, Mol. Cryst. Liq. Cryst. Sci. Technol. Sec. B 2, 159 (1992).

K. Clays and A. Persoons, Rev. Sci. Instrum. 63, 2385 (1992).
[CrossRef]

1991 (3)

K. Clays and A. Persoons, Phys. Rev. Lett. 66, 2980 (1991).
[CrossRef] [PubMed]

J. Zyss, Nonlin. Opt. 1, 3 (1991).

B. Boulanger and G. Marnier, J. Phys. Condens. Matter 3, 8327 (1991).
[CrossRef]

1990 (1)

I. Ledoux, J. Zyss, J. Siegel, J. Brienne, and J. M. Lehn, Chem. Phys. Lett. 172, 440 (1990).
[CrossRef]

1987 (2)

1984 (1)

J. Zyss, J. F. Nicoud, and M. Coquillay, J. Chem. Phys. 81, 4160 (1984).
[CrossRef]

1982 (3)

J. L. Oudar and J. Zyss, Phys. Rev. A 26, 2016 (1982).
[CrossRef]

J. Zyss and J. L. Oudar, Phys. Rev. A 26, 2028 (1982).
[CrossRef]

I. Ledoux and J. Zyss, Chem. Phys. 73, 203 (1982).
[CrossRef]

1981 (2)

K. D. Singer and A. F. Garito, J. Chem. Phys. 75, 3572 (1981).
[CrossRef]

J. Zyss, D. S. Chemla, and J. F. Nicoud, J. Chem. Phys. 74, 4800 (1981).
[CrossRef]

1979 (1)

J. Zyss, J. Chem. Phys. 70, 3333, 3341 (1979).

1978 (3)

J. Jerphagnon, D. S. Chemla, and R. Bonneville, Adv. Phys. 27, 609 (1978).
[CrossRef]

D. S. Chemla and R. Bonneville, J. Chem. Phys. 68, 5 (1978).
[CrossRef]

R. Bonneville and D. S. Chemla, Phys. Rev. A 17, 2046 (1978).
[CrossRef]

1977 (2)

J. L. Oudar, J. Chem. Phys. 67, 1626 (1977).
[CrossRef]

J. L. Oudar and D. S. Chemla, J. Chem. Phys. 66, 2664 (1977).
[CrossRef]

1975 (2)

B. F. Levine and C. G. Bethea, J. Chem. Phys. 63, 2666 (1975).
[CrossRef]

D. S. Chemla, J.-L. Oudar, and J. Jerphagnon, Phys. Rev. B 12, 4534 (1975).
[CrossRef]

1970 (1)

D. Maker, Phys. Rev. A 1, 923 (1970).
[CrossRef]

1966 (1)

R. Bersohn, Y. H. Pao, and H. L. Frisch, J. Chem. Phys. 45, 3184 (1966).
[CrossRef]

1965 (2)

S. G. Cyvin, J. E. Rauch, and J. C. Decius, J. Chem. Phys. 43, 4083 (1965).
[CrossRef]

R. W. Terhune, P. D. Maker, and C. M. Savage, Phys. Rev. Lett. 14, 681 (1965).
[CrossRef]

1963 (1)

A. D. Buckingham and R. L. Disch, Proc. R. Soc. London Ser. A 273, 275 (1963).
[CrossRef]

1962 (1)

D. A. Kleinman, Phys. Rev. 126, 1977 (1962).
[CrossRef]

Barzoukas, M.

Bersohn, R.

R. Bersohn, Y. H. Pao, and H. L. Frisch, J. Chem. Phys. 45, 3184 (1966).
[CrossRef]

Bethea, C. G.

B. F. Levine and C. G. Bethea, J. Chem. Phys. 63, 2666 (1975).
[CrossRef]

Bölger, B.

G. H. T. Heesink, A. G. T. Ruiter, N. F. van Hulst, and B. Bölger, Phys. Rev. Lett. 71, 999 (1993).
[CrossRef] [PubMed]

Bonneville, R.

R. Bonneville, Mol. Cryst. Liq. Cryst. Sci. Technol. Sec. B 2, 159 (1992).

R. Bonneville and D. S. Chemla, Phys. Rev. A 17, 2046 (1978).
[CrossRef]

J. Jerphagnon, D. S. Chemla, and R. Bonneville, Adv. Phys. 27, 609 (1978).
[CrossRef]

D. S. Chemla and R. Bonneville, J. Chem. Phys. 68, 5 (1978).
[CrossRef]

Boulanger, B.

B. Boulanger and G. Marnier, J. Phys. Condens. Matter 3, 8327 (1991).
[CrossRef]

Bourgault, M.

C. Dhenaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgault, and H. Le Bozec, Nature 374, 339 (1995).
[CrossRef]

Brienne, J.

I. Ledoux, J. Zyss, J. Siegel, J. Brienne, and J. M. Lehn, Chem. Phys. Lett. 172, 440 (1990).
[CrossRef]

Buckingham, A. D.

A. D. Buckingham and R. L. Disch, Proc. R. Soc. London Ser. A 273, 275 (1963).
[CrossRef]

Charra, F.

C. Fiorini, F. Charra, and J. M. Nunzi, J. Opt. Soc. Am. B 11, 2347 (1994).
[CrossRef]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
[CrossRef]

Chauvan, T.

J. Zyss, T. Chauvan, C. Dhenaut, and I. Ledoux, Chem. Phys. 177, 281 (1995).
[CrossRef]

Chemla, D. S.

J. Zyss, D. S. Chemla, and J. F. Nicoud, J. Chem. Phys. 74, 4800 (1981).
[CrossRef]

R. Bonneville and D. S. Chemla, Phys. Rev. A 17, 2046 (1978).
[CrossRef]

D. S. Chemla and R. Bonneville, J. Chem. Phys. 68, 5 (1978).
[CrossRef]

J. Jerphagnon, D. S. Chemla, and R. Bonneville, Adv. Phys. 27, 609 (1978).
[CrossRef]

J. L. Oudar and D. S. Chemla, J. Chem. Phys. 66, 2664 (1977).
[CrossRef]

D. S. Chemla, J.-L. Oudar, and J. Jerphagnon, Phys. Rev. B 12, 4534 (1975).
[CrossRef]

Clays, K.

K. Clays and A. Persoons, Rev. Sci. Instrum. 63, 2385 (1992).
[CrossRef]

K. Clays and A. Persoons, Phys. Rev. Lett. 66, 2980 (1991).
[CrossRef] [PubMed]

Coquillay, M.

J. Zyss, J. F. Nicoud, and M. Coquillay, J. Chem. Phys. 81, 4160 (1984).
[CrossRef]

Cyvin, S. G.

S. G. Cyvin, J. E. Rauch, and J. C. Decius, J. Chem. Phys. 43, 4083 (1965).
[CrossRef]

Decius, J. C.

S. G. Cyvin, J. E. Rauch, and J. C. Decius, J. Chem. Phys. 43, 4083 (1965).
[CrossRef]

Denning, R. G.

I. D. Morrison, R. G. Denning, W. M. Laidlaw, and M. A. Stammers, Rev. Sci. Instrum. 67, 1445 (1996).
[CrossRef]

Dhenaut, C.

I. Ledoux, C. Dhenaut, I. D. W. Samuel, and J. Zyss, Nonlinear Opt. 14, 23 (1995).

C. Dhenaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgault, and H. Le Bozec, Nature 374, 339 (1995).
[CrossRef]

J. Zyss, T. Chauvan, C. Dhenaut, and I. Ledoux, Chem. Phys. 177, 281 (1995).
[CrossRef]

Disch, R. L.

A. D. Buckingham and R. L. Disch, Proc. R. Soc. London Ser. A 273, 275 (1963).
[CrossRef]

Fiorini, C.

C. Fiorini, F. Charra, and J. M. Nunzi, J. Opt. Soc. Am. B 11, 2347 (1994).
[CrossRef]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
[CrossRef]

Frémaux, P.

Frisch, H. L.

R. Bersohn, Y. H. Pao, and H. L. Frisch, J. Chem. Phys. 45, 3184 (1966).
[CrossRef]

Garito, A. F.

K. D. Singer and A. F. Garito, J. Chem. Phys. 75, 3572 (1981).
[CrossRef]

Heesink, G. H. T.

G. H. T. Heesink, A. G. T. Ruiter, N. F. van Hulst, and B. Bölger, Phys. Rev. Lett. 71, 999 (1993).
[CrossRef] [PubMed]

Jerphagnon, J.

J. Jerphagnon, D. S. Chemla, and R. Bonneville, Adv. Phys. 27, 609 (1978).
[CrossRef]

D. S. Chemla, J.-L. Oudar, and J. Jerphagnon, Phys. Rev. B 12, 4534 (1975).
[CrossRef]

Josse, D.

Kaatz, P.

P. Kaatz and D. P. Sheldon, Rev. Sci. Instrum. 67, 1439 (1996).
[CrossRef]

Kauranen, M.

M. Kauranen and B. Persoons, J. Chem. Phys. 104, 3445 (1996).
[CrossRef]

Kippelen, B.

B. Kippelen and N. Peyghambarian, Photon. Sci. News 2(2), 18 (1996).

Kleinman, D. A.

D. A. Kleinman, Phys. Rev. 126, 1977 (1962).
[CrossRef]

Kuzyk, M. G.

Laidlaw, W. M.

I. D. Morrison, R. G. Denning, W. M. Laidlaw, and M. A. Stammers, Rev. Sci. Instrum. 67, 1445 (1996).
[CrossRef]

Le Bozec, H.

C. Dhenaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgault, and H. Le Bozec, Nature 374, 339 (1995).
[CrossRef]

Ledoux, I.

C. Dhenaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgault, and H. Le Bozec, Nature 374, 339 (1995).
[CrossRef]

J. Zyss, T. Chauvan, C. Dhenaut, and I. Ledoux, Chem. Phys. 177, 281 (1995).
[CrossRef]

I. Ledoux, C. Dhenaut, I. D. W. Samuel, and J. Zyss, Nonlinear Opt. 14, 23 (1995).

J. Zyss and I. Ledoux, Chem. Rev. 94, 77 (1994).
[CrossRef]

I. Ledoux, J. Zyss, J. Siegel, J. Brienne, and J. M. Lehn, Chem. Phys. Lett. 172, 440 (1990).
[CrossRef]

I. Ledoux and J. Zyss, Chem. Phys. 73, 203 (1982).
[CrossRef]

Lehn, J. M.

I. Ledoux, J. Zyss, J. Siegel, J. Brienne, and J. M. Lehn, Chem. Phys. Lett. 172, 440 (1990).
[CrossRef]

Levine, B. F.

B. F. Levine and C. G. Bethea, J. Chem. Phys. 63, 2666 (1975).
[CrossRef]

Maker, D.

D. Maker, Phys. Rev. A 1, 923 (1970).
[CrossRef]

Maker, P. D.

R. W. Terhune, P. D. Maker, and C. M. Savage, Phys. Rev. Lett. 14, 681 (1965).
[CrossRef]

Marnier, G.

B. Boulanger and G. Marnier, J. Phys. Condens. Matter 3, 8327 (1991).
[CrossRef]

Miyata, S.

H. S. Nalwa, T. Watanabe, and S. Miyata, Adv. Mater. 7, 754 (1995).
[CrossRef]

H. S. Nalwa, T. Watanabe, and S. Miyata, Opt. Mater. 2, 73 (1993).
[CrossRef]

Morley, J. O.

Morrison, I. D.

I. D. Morrison, R. G. Denning, W. M. Laidlaw, and M. A. Stammers, Rev. Sci. Instrum. 67, 1445 (1996).
[CrossRef]

Nalwa, H. S.

H. S. Nalwa, T. Watanabe, and S. Miyata, Adv. Mater. 7, 754 (1995).
[CrossRef]

H. S. Nalwa, T. Watanabe, and S. Miyata, Opt. Mater. 2, 73 (1993).
[CrossRef]

Nicoud, J. F.

J. Zyss, J. F. Nicoud, and M. Coquillay, J. Chem. Phys. 81, 4160 (1984).
[CrossRef]

J. Zyss, D. S. Chemla, and J. F. Nicoud, J. Chem. Phys. 74, 4800 (1981).
[CrossRef]

Nicoud, J.-F.

Nunzi, J. M.

C. Fiorini, F. Charra, and J. M. Nunzi, J. Opt. Soc. Am. B 11, 2347 (1994).
[CrossRef]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
[CrossRef]

Oudar, J. L.

J. L. Oudar and J. Zyss, Phys. Rev. A 26, 2016 (1982).
[CrossRef]

J. Zyss and J. L. Oudar, Phys. Rev. A 26, 2028 (1982).
[CrossRef]

J. L. Oudar, J. Chem. Phys. 67, 1626 (1977).
[CrossRef]

J. L. Oudar and D. S. Chemla, J. Chem. Phys. 66, 2664 (1977).
[CrossRef]

Oudar, J.-L.

D. S. Chemla, J.-L. Oudar, and J. Jerphagnon, Phys. Rev. B 12, 4534 (1975).
[CrossRef]

Pao, Y. H.

R. Bersohn, Y. H. Pao, and H. L. Frisch, J. Chem. Phys. 45, 3184 (1966).
[CrossRef]

Persoons, A.

K. Clays and A. Persoons, Rev. Sci. Instrum. 63, 2385 (1992).
[CrossRef]

K. Clays and A. Persoons, Phys. Rev. Lett. 66, 2980 (1991).
[CrossRef] [PubMed]

Persoons, B.

M. Kauranen and B. Persoons, J. Chem. Phys. 104, 3445 (1996).
[CrossRef]

Peyghambarian, N.

B. Kippelen and N. Peyghambarian, Photon. Sci. News 2(2), 18 (1996).

Rauch, J. E.

S. G. Cyvin, J. E. Rauch, and J. C. Decius, J. Chem. Phys. 43, 4083 (1965).
[CrossRef]

Ruiter, A. G. T.

G. H. T. Heesink, A. G. T. Ruiter, N. F. van Hulst, and B. Bölger, Phys. Rev. Lett. 71, 999 (1993).
[CrossRef] [PubMed]

Samuel, I. D. W.

I. Ledoux, C. Dhenaut, I. D. W. Samuel, and J. Zyss, Nonlinear Opt. 14, 23 (1995).

C. Dhenaut, I. Ledoux, I. D. W. Samuel, J. Zyss, M. Bourgault, and H. Le Bozec, Nature 374, 339 (1995).
[CrossRef]

Savage, C. M.

R. W. Terhune, P. D. Maker, and C. M. Savage, Phys. Rev. Lett. 14, 681 (1965).
[CrossRef]

Sheldon, D. P.

P. Kaatz and D. P. Sheldon, Rev. Sci. Instrum. 67, 1439 (1996).
[CrossRef]

Siegel, J.

I. Ledoux, J. Zyss, J. Siegel, J. Brienne, and J. M. Lehn, Chem. Phys. Lett. 172, 440 (1990).
[CrossRef]

Singer, K. D.

Sohn, J. E.

Stammers, M. A.

I. D. Morrison, R. G. Denning, W. M. Laidlaw, and M. A. Stammers, Rev. Sci. Instrum. 67, 1445 (1996).
[CrossRef]

Terhune, R. W.

R. W. Terhune, P. D. Maker, and C. M. Savage, Phys. Rev. Lett. 14, 681 (1965).
[CrossRef]

van Hulst, N. F.

G. H. T. Heesink, A. G. T. Ruiter, N. F. van Hulst, and B. Bölger, Phys. Rev. Lett. 71, 999 (1993).
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Other (13)

J. Zyss, I. Ledoux, and J.-F. Nicoud, in Molecular Nonlinear Optics: Materials, Physics and Devices, J. Zyss, ed. (Academic, Boston, Mass., 1993), pp. 130–201.

C. Fiorini, F. Charra, J. M. Nunzi, I. D. W. Samuel, and J. Zyss, Opt. Lett. 20, 2469 (1995); J. Zyss, I. D. W. Samuel, C. Fiorini, F. Charra, and J. M. Nunzi, in Organic Thin Films for Photonic Applications, Vol. 21 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 264–267.
[CrossRef]

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[CrossRef]

J. Zyss, in Nonlinear Optics: Fundamentals, Materials and Devices, S. Miyata, ed. (North-Holland, Amsterdam, 1992), pp. 33–478.

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D. S. Chemla and J. Zyss, eds., Molecular Nonlinear Optics: Materials, Physics and Devices (Academic, Boston, Mass. 1993).

N. P. Prasad and D. J. Williams, Introduction to Nonlinear Optical Effects in Molecules and Polymers (Wiley, New York, 1991).

J. Zyss ed., “Molecular Nonlinear Optics: Materials, Physics and Devices” (Academic, New York, 1994).

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992).

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E. P. Wigner, Group Theory (Academic, New York, 1959).

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Figures (14)

Fig. 1
Fig. 1

Schematic representation of prototypical noncentrosymmetric planar C2v symmetry aromatic charge-transfer molecules: (a) dipolar system for which βJ=10 and |βxxx||βxyy|; (b) multipolar system for which βJ=10, βJ=30, and |βxxx| <|βxyy|; (c) octupolar system for which βJ=1=0, βJ=30, and βxxx=-βxyy.

Fig. 2
Fig. 2

Laboratory reference frame {X, Y, Z} with the fundamental incoming beam propagating along Z and polarized in the (X, Y) plane. The observation direction is along Y, with polarization analysis along X or Z. Orthogonal (e.g., ϕ-independent configuration permitting retrieval of 2βZXX2=2βXYY2) and coplanar (e.g., ϕ-dependent configuration permitting retrieval of βXXX2 and βZXX2) polarization configurations correspond to analyzed polarization directions that are, respectively, parallel and perpendicular to the reflection plane.

Fig. 3
Fig. 3

Experimental harmonic scattered intensities I2ω [curve (a), with no analyzer] and IZ2ω [curve (b), with an analyzer along the Z direction] for a NPP solution in chloroform, plotted as functions of the incoming polarization angle ϕ as defined in Fig. 2. IZ2ω is ϕ independent.

Fig. 4
Fig. 4

Dependence of the macroscopic depolarization D and the squared nonlinear anisotropy ρ2 (2D normalization) as functions of the Cartesian nonlinear anisotropy u=βxyy/βxxx related to the in-plane anisotropy of the β tensor in the case of C2v symmetry. Experimental D(u) values are plotted for various molecules, namely, NPP, NPAN, POM, DNDAB, DADC, CRVI, and DR1. Note that ρ goes to zero for a pure dipole in the case of 2D tensorial normalization of the molecular β (see Fig. 7 below for comparison with a more general 3D model). Abbreviations of the molecules are defined in text.

Fig. 5
Fig. 5

Experimental setup for the HLS experiment: λ/2’s, half-wave plates; P, Glan polarizer; L’s, converging lenses; F1, F2, filters selecting fundamental and harmonic light; PMT’s, photomultiplier tubes.

Fig. 6
Fig. 6

Molecular structure of various types of molecule investigated by HLS: DR1 (Cv symmetry), DNDAB and DADC (C2v symmetries), NPP, POM, and NPAN (quasi-1D symmetries); CRVI (D3h symmetry). Abbreviations defined in text.

Fig. 7
Fig. 7

Dependence of the squared nonlinear anisotropy ρ2 in the spherical formalism as a function of its Cartesian counterpart u=βxyy/βxxx for planar C2v molecules in 2D (dotted curve) and 3D (solid curve) formalisms. In both cases ρ2 is diverging for octupolar molecular symmetry (e.g., u=-1) and is minimal for u=1/3. In the 2D (3D) formalism the minimal value of ρ2 is 0 (1/4).

Fig. 8
Fig. 8

Schematic representation of collinear forward optical poling configurations at the write and read steps. In the {X, Y, Z} laboratory reference frame the fundamental incoming beams are propagating along Z and are polarized in the transverse (X, Y) plane. The observation direction is also along Z, with eventual polarization analysis along X or Y.

Fig. 9
Fig. 9

Harmonic intensity analyzed along the vertical Y direction as a function of the read polarization angle ϕ in the case of linearly polarized writing fields Eω//X and E2ω//Y for dipolar (dotted curve) and octupolar (solid curve) molecular symmetries. The sidelobes in the latter case designate the presence of βJ=3 octupolar molecular components.

Fig. 10
Fig. 10

Polar plots I2ω(ϕ) of the second-harmonic signal read out by a linearly polarized fundamental beam (with the polarization angle ϕ defined as in Fig. 8) in the case of grafted DR1–MMA spin-coated films. Squares correspond to experimental values; solid curves, to theoretical fits according to Eqs. (72) in the four following configurations: top left (top right), analyzer at 2ω along X(Y), E2ω and Eω writing fields linearly polarized along X; bottom left (bottom right), analyzer at 2ω along X(Y), E2ω and Eω writing fields linearly polarized along Y(X).

Fig. 11
Fig. 11

Harmonic intensity analyzed along the vertical X direction as a function of the read polarization angle ϕ in the case of dipolar (dotted curve) and octupolar (solid curve) writing fields for an arbitrary molecular anisotropy (taken as ρ=1 in the present case). The distinctive presence of four lobes is a characteristic feature of the interaction with an octupolar writing field EJ=3 component.

Fig. 12
Fig. 12

Variations of the spherical macroscopic anisotropy ρχ(2)=χJ=3(2)/χJ=1(2) as a function of the ellipticity parameter δ of the harmonic component E2ω of the writing field tensor for the cases of optical poling of a dipolar molecule (dotted curve), an octupolar molecule (solid curve), and an intermediate multipolar molecule (dashed curve). For all the molecular configurations, variation of δ permits continuous tuning of the macroscopic anisotropy from a dipolar [ρχ(2)=1/4] to an octupolar [ρχ(2)=] configuration.

Fig. 13
Fig. 13

Polar plot of the squared normalized amplitude of the macroscopic nonlinearity per molecule χ(2)2/β4 as a function of the angle δ (tan δ is the ellipticity of the polarization of the second-harmonic writing beam; the fundamental one is circular) for a dipolar molecule [dotted curve; the macroscopic χ(2) magnitude is maximized for δ=π/4, e.g., a dipolar writing field tensor] and an octupolar molecule (dashed curve; the macroscopic magnitude is maximized for δ=-π/4, e.g., an octupolar writing field) and an intermediate multipolar molecule with ρ2=14/3 (solid curve; the macroscopic magnitude is then constant for any writing field anisotropy).  

Fig. 14
Fig. 14

Polar plot of the normalized harmonic intensity per molecule as a function of the read polarization angle ϕ defined in Fig. 8 for an octupolar writing field tensor and an arbitrary molecular anisotropy (set to ρ=1 in the present case). The analyzed signal along the X (dotted–dashed curve) and Y (dotted curve) directions add to a ϕ-independent signal in the absence of an analyzer (solid curve).

Tables (8)

Tables Icon

Table 1 Experimental Results for HLS Depolarization Ratio D, Molecular Averaged Hyperpolarizability β2, and Cartesian Molecular Anisotropy u for Various Types of Molecule a

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Table 2 Cartesian ijk Cubic Tensorial Basis Elements in Terms of the CmJ Reduced Spherical Tensorial Basis Elements

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Table 3 Cartesian ijk Cubic Tensorial Basis Elements in Terms of the CmJ Reduced Spherical Tensorial Basis Elements

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Table 4 Cartesian ij Quadratic Tensorial Basis Elements in Terms of the CmJ Reduced Spherical Tensorial Basis Elements

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Table 5 Experimental Results for Molecular Anisotropy ρ Inferred from Macroscopic Depolarization Ratio D for the Same Molecules as in Table 1 a

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Table 6 Spherical Components EmJ and Norms EJ2 of the Writing Field Tensors for Various Polarization Configurations and Linear Polarizations

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Table 7 Spherical Components EmJ and Norms EJ2 of the Writing Field Tensors for Various Polarization Configurations for Eω Circular and E2ω Elliptical a

Tables Icon

Table 8 Spherical Components EmJ and Norms EJ2 of the Writing Field Tensors for Elliptical Polarizations for Eω and E2ω a

Equations (216)

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PI2ω=J,KβIJK EJω EKω,
βIJK(Ω)=RΩ(β)IJK=ijkRIi RJj RKk(Ω)βijk.
II2ωPI2ω PI2ω*Ω,
PI2ω PI2ω*Ω
=NJ,K,M,NβIJK(Ω)βIMN(Ω)*ΩEJω EKω EMω* ENω*,
P2ωP2ω*Ω=NI transvJKMNβIJK βIMNΩ EJω EKω EMω*ENω*,
II2ωPI2ωPI2ω*Ω=NSJKMNI EJω EKω EMω*ENω*,
PX2ΩPY2ΩPZ2Ω=NSXXXXXSYYYYXSXXYYXSXXXXYSYYYYYSXXYYYSXXXXZSYYYYZSXXYYZ×|EXω|4|EYω|4|EXω|2|EYω|2.
S=βXXX2βZXX2βXXX2+βZXX2βZXX2βXXX2βXXX2+βZXX2βZXX2βZXX22βZXX2.
PX2PZ2=NβXXX2cos2 ϕ+βZXX2sin2 ϕβZXX2E4,
IX2ω+IZ2ω(βXXX2+βZXX2)cos2 ϕ+2βZXX2sin2 ϕ.
βIJKβLMNΩ=ijklmnRIiRJjRKkRLlRMmRNn(Ω)Ωβijkβlmn=18π2 ijklmnθ=0πψ=02πφ=02πRIiRNnβijkβlmn×sin θdθdφdψ.
βXXX2=17 iβiii2+635 ijβiiiβijj+935 ijβijj2+635 ijkβijjβikk+1235 ijkβikj2,
βZXX2=135 iβiii2-2105 ijβiiiβijj-11105 ijβijj2-2105 ijkβijjβikk+835 ijkβikj2.
βXXX2=17 βxxx2+635 βxxxβxyy+935 βxyy2,
βZXX2=135 βxxx2-2105 βxxxβxyy+11105 βxyy2.
D(u)=3-2u+11u215+18u+27u2.
β=βJ=1βJ=3.
ρ=βJ=3/βJ=1.
β=3/4[(βxxx+βxyy)x+(βyyy+βyxx)y](xx+yy)+1/4(βxxx-3βxyy)x(xx-3yy)+1/4(βyyy-3βyxx)y(yy-3xx).
βJ=12D2=3/4[(βxxx+βxyy)2+(βyyy+βyxx)2],
βJ=32D2=1/4[(βxxx-3βxyy)2+(βyyy-3βyxx)2].
(ρ2D)2=βJ=32D2βJ=12D2=13 1-3u1+u2,
β=J=1,3-JmJβmJCmJ(θ,φ).
βJ2=-JmJ|βmJ|2,
βJ=12=35 iβiii2+65 ijβiiiβijj+35 ijβijj2+65 ijkβijjβikk,
βJ=32=25 iβiii2-65 ijβiiiβijj+125 ijβijj2-65 ijkβijjβikk+6ijkβikj2.
βJ=12=3/5(βxxx+βxyy)2,
βJ=32=1/20[3(βxxx+βxyy)2+5(βxxx-3βxyy)2].
ρ2=βJ=32βJ=12=112 3(1+u)2+5(1-3u)2(1+u)2.
αJ=02=1/3(αxx+αyy+αzz)2,
αJ=22=1/3[(αxx-αyy)2+(αxx-αzz)2+(αzz-αyy)2],
βXXX2=970 (βxxx+βxyy)2+170 (βxxx-3βxyy)2,
βZXX2=2105 (βxxx+βxyy)2+1105 (βxxx-3βxyy)2.
βXXX2=135 (7βJ=12+2βJ=32),
βZXX2=145 βJ=12+127 βJ=32,
D=βZXX2βXXX2=19 7+12ρ27+2ρ2.
UV=IJKUIJKVIJK=m,J(-1)mUmJV-mJ.
(UV)IJKLMN=UIJKVLMN.
P2ω=βF,
z(Ω)Ω=Ωz(Ω)f(Ω)dΩ.
RΩ(β)=m,JβmJmDmmJ(Ω)CmJ,
RΩ(β)mJ=mβmJDmmJ(Ω).
F=m,JFmJCmJ(θ,φ),
PP*=(βF)(β*F*).
IPP*Q=NJ 12J+1 (βJβJ)(FJFJ)=NJ 12J+1 βJ2FJ2,
(βJβJ)=m=-JJ|βmJ|2=βJ2,
(FJFJ)=(AJ cos2 ϕ+BJ sin2 ϕ)(Eω)4.
I23 13 βJ=12+17 βJ=32cos2 ϕ+215 13 βJ=12+47 βJ=32sin2 ϕ.
IX15 βJ=12+235 βJ=32cos2 ϕ+145 βJ=12+4105 βJ=32sin2 ϕ.
PΩ=χ(2)F,
PΩ=m,J(-1)m[χ(2)]mJF-mJ.
W(Ω)=W0-TE,
f(Ω)exp[-W(Ω)/kT]=1-W(Ω)/kT+.
f(Ω)1-W0kT+1kT m,J(-1)mRΩ(T)mJE-mJ,
ΩRΩ(β)dΩ=0;
χ(2)=NRΩ(β)ΩN 1kT Ωβ(Ω)(TE)dΩ.
χ(2)=NRΩ(β)ΩN 1kT Ωm,J(-1)mE-mJmTmJDmmJ(Ω)×q,q,KβqKDqqK(Ω)CqKdΩ.
ΩDmmJ(Ω)DllK(Ω)dΩ
=8π22J+1 δJKδm,-lδm,-l(-1)m-m.
χ(2)=N 1kT m,Jm12J+1 (-1)mE-mJTmJβ-mJC-mJ.
 
[χ(2)]mJ=N 1kT m(-1)mTmJβ-mJ 12J+1 EmJ=N 1kT 12J+1 (TJβJ)EmJ.
[χ(2)]J=NkT 12J+1 (TJβJ)EJ.
(TE)β=J 12J+1 (Tβ)EJ.
[χ(2)]J=1=[χ(2)]01=N3kT (μβJ=1)E,
[χ(2)]J=3=0.
PΩ=[χ(2)]01F01=N3kT (μβJ=1)E01F01.
χZZZ(2)=-35 N μ0E03kT β01.
χZZZ(2)=N μ0E05kT (βzzz+βzxx+βzyy).
χZXX(2)=-115 N μ0E03kT β01,
χZXX(2)=N μ0E015kT (βzzz+βzxx+βzyy).
P01(Ω)=183ω (μ01E2ω)*(μ01Eω)(ΔμEω)+c.c.,
P01(Ω)β(E2ω*EωEω)=βE,
f(Ω)18π2-κP01(Ω)=18π2-κβE,
χ(2)=Nβ(Ω)Ω=NΩβ(Ω)f(Ω)dΩ.
χ(2)NΩβ(Ω)18π2-κP01(Ω)dΩΩβ(Ω)P01(Ω)dΩ,
χ(2)Ωβ(Ω)(βE)dΩ.
E=E2ω*EωEω=m,JEmJCmJ(θ, φ).
β(Ω)(βE)Ω=J 12J+1 (βJβJ)EJ.
χ(2)=J[χ(2)]JJ 12J+1 βJ2EJ,
[χ(2)]mJ12J+1 βJ2EmJ.
PΩJ 12J+1 βJ2(EJFJ).
FX,11(ϕ)=-130 (3 cos2 ϕ+sin2 ϕ)+2j30 cos ϕ sin ϕ×(Fω)2,
FX,13(ϕ)=1230 (3 cos2 ϕ+sin2 ϕ)-j30 cos ϕ sin ϕ(Fω)2,
FX,33(ϕ)=122 (sin2 ϕ-cos2 ϕ)+j2 cos ϕ sin ϕ×(Fω)2.
FY,11(ϕ)=-230 cos ϕ sin ϕ+j30 (3 sin2 ϕ+cos2 ϕ)×(Fω)2,
FY,13(ϕ)=130 cos ϕ sin ϕ-j230 (3 sin2 ϕ+cos2 ϕ)×(Fω)2,
FY,33(ϕ)=12 cos ϕ sin ϕ+j22 (cos2 ϕ-sin2 ϕ)×(Fω)2.
I(ϕ)13 βJ=12[E11F11*(ϕ)+c.c.]+17 βJ=32[E13F13*(ϕ)+E33F33*(ϕ)+c.c.]2.
ρχ(2)=37 ρmolec2ρE,
χ(2)2=13 βJ=122EJ=12+17 βJ=322EJ=32.
cos2 Φ=βJ=12βJ=12+βJ=32=11+ρ2,
sin2 Φ=βJ=32βJ=12+βJ=32=ρ21+ρ2.
χ(2)2β4=19 cos4 ΦEJ=1(δ)2+149 sin4 ΦEJ=3(δ)2.
χ(2)2β4 (ρ, δ)=19 11+ρ22EJ=1(δ)2+149 ρ21+ρ22EJ=3(δ)2.
χ(2)2β4 (ρ, δ)=A(ρ2)cos2δ-π4+B(ρ2)sin2δ-π4,
A(ρ2)=115 49 11+ρ22+149 ρ21+ρ22,
B(ρ2)=149 ρ21+ρ22.
PP*ΩHLS=NJ=1,3 12J+1 βJ2FJ2,
PΩSHG=N3kT (μβJ=1)E01F01,
PΩOPJ=1,3 12J+1 βJ2(EJFJ),
PI2ωPI2ω*Ω=NJ,K,L,MSIJKIMNEJω EKω EMω* ENω*
SIJKIMN=βIJK βIMNΩ(+permut.),
βIJKβLMNΩ
=ijklmnRIiRJjRKkRLlRMmRNnΩβijkβlmn,
PX2PY2PZ2=SEX4EY4EX2EY2.
S=βXXX2βXYY2(βXYX+βXXY)2+2βXXXβXYYβYXX2βYYY2(βYYX+βYXY)2+2βYXXβYYYβZXX2βZYY2(βZYX+βZXY)2+2βZXXβZYY.
βXXX2=βYYY2=βZZZ2.
βYXX2=βXYY2=βZXX2=βZYY2.
βZXX2=1/2(βZYX+βZXY)2+2βZXXβZYY.
1/2(βXYX+βXXY)2+2βXXXβXYY
=1/2(βYYX+βYXY)2+2βYYYβYXX.
βXXX2+βZXX2=(βXYX+βXXY)2+2βXXXβXYY.
β2=βXXX2+βZXX2.
S=βXXX2βZXX2βXXX2+βZXX2βZXX2βXXX2βXXX2+βZXX2βZXX2βZXX22βZXX2,
PrPϕ0Pθ0=R(θ0, ϕ0)PXPYPZ,
Pu2Pϕ02Pθ02=R(θ0, ϕ0)PX2PY2PZ2PXPYPXPZPYPZ.
PXPY=N(2βXXXβYYX+βYXX2EX3EY+2βYYYβXXY+βXYY2EY3EX).
βXXXβYYX=βXXX2-3βZXX22.
PXPY=N(βXXX2-βZXX2)(EX3EY+EY3EX).
PX2PY2PZ2PXPY
=βXXX2βZXX2βXXX2+βZXX20βZXX2βXXX2βXXX2+βZXX20βZXX2βZXX22βZXX20000βXXX2-βZXX2×EX4EY4EX2EY2EX3EY+EXEY3.
x=sin θ cos ϕ,y=sin θ sin ϕ,z=cos θ.
αmJ=i,jλijmJαij,
α00=-13 (αxx+αyy+αzz);
α02=16 (2αzz-αxx-αyy),
α±12=(αxz±iαyz),
α±22=12 (αxx-αyy±2iαxy).
βmJ=i,j,kλijkmJβijk
β01=315 (βzzz+βzxx+βzyy),
β±11=330 (βxxx+βxyy+βxzz)+3i30 (βyyy+βyxx+βyzz);
β03=110 [2βzzz-3(βzxx+βzyy)],
β±13=±3230 [(βxxx+βxyy)-4βxzz]-3i230 [(βyyy+βyxx)-4βyzz],
β±23=323 [(βzxx-βzyy)2iβxyz],
β±33=±122 (-βxxx+3βxyy)+i22 (-βyyy+3βyxx).
β01=0,β03=0,β±23=0.
β±11=330 [(βxxx+βxyy)i(βyyy+βyxx)],
β±13=±3230 [(βxxx+βxyy)i(βyyy+βyxx)],
β±33=±122 [(-βxxx+3βxyy)+i(-βyyy+3βyxx)].
β±11=330 (βxxx+βxyy),
β±13=±3230 (βxxx+βxyy),
β±33=±122 (-βxxx+3βxyy).
β±11=0,β±13=0,
β±33=±2βxxx.
β±33=0,
β±11=430 βxxx,
β±13=±230 βxxx.
β±11=330 βxxx,
β±13=±3230 βxxx,
β±33=122 βxxx.
β±11=0,β±13=0,β±33=0,β01=0,
β03=0,
β±23=3iβxyz.
PP*=(βF)(β*F*),
P=m,J(-1)mRΩ(β)mJF-mJ,
PP*Ω
=Nm,J,q,K(-1)m+qRΩ(β)mJRΩ(β)qK*ΩF-mJF-qK*,
RΩ(β)mJRΩ(β)qK*Ω=ΩRΩ(β)mJRΩ(β)qK*dΩΩdΩ.
RΩ(β)mJ=mβmJDmmJ(Ω).
ΩRΩ(β)mJRΩ(β)qK*dΩ
=m,qβmJβqK*ΩDmmJ(Ω)DqqK*(Ω)dΩ.
Ω m,lDmmJ(Ω)DllK*(Ω)dΩ=8π22J+1 δJKδm,lδm,l,
ΩRΩ(β)mJRΩ(β)qK*dΩ=m 8π22J+1 βmJβmJ*δJKδmq.
RΩ(β)mJRΩ(β)qK*Ω=12J+1 βJ2δJKδmq,
PP*=Nm,J 12J+1 βJ2(F-mJF-mJ*).
PP*=NJ 12J+1 βJ2(FJFJ),
FX=(cos2 ϕXXX+sin2 ϕXYY+2 cos ϕ sin ϕXXY)E2.
FX=cos2 ϕ-310 (C11-C-11)+12 310 (C13-C-13)-122 (C33-C-33)E2+sin2 ϕ-130 (C11-C-11)+1230 (C13-C-13)+122 (C33-C-33)E2+2 cos ϕ sin ϕi30 (C11+C-11)-i230×(C13+C-13)+i22 (C33+C-33)E2.
(FX)11=-310 cos2 ϕ-130 sin2 ϕ+2i30 cos ϕ sin ϕE2,
(FX)-11=310 cos2 ϕ 130 sin2 ϕ+2i30 cos ϕ sin ϕE2=-[(FX)11]*.
FXJ=1FXJ=1=m=0,±1(FX)mJ=1(FX)-mJ=1(-1)m= |(FX)11|2+|(FX)-11|2,
FXJ=1FXJ=1=35 cos2 ϕ+115 sin2 ϕE4.
FXJ=3FXJ=3=25 cos2 ϕ+415 sin2 ϕE4.
PP*X=N13 βJ=1235 cos2 ϕ+115 sin2 ϕE4+17 βJ=3225 cos2 ϕ+415 sin2 ϕE4.
FZ=(cos2 ϕZXX+sin2 ϕZYY+2 cos ϕ sin ϕZXY)E2.
FZJ=1FZJ=1=115 E4,
FZJ=3FZJ=3=415 E4.
PP*Z=N145 βJ=12+4105 βJ=32E4.
PP*=NJ 12J+1 βJ2(AJ cos2 ϕ+BJ sin2 ϕ)E4,
A1=23,A3=23,B1=215,B3=815.
βXXX2=135 (7βJ=12+2βJ=32).
βZXX2=145 βJ=12+127 βJ=32.
E01=315 EZZZ+EZXX+EXZX+EXXZ3+EZYY+EYZY+EYYZ3,
E±11=330 EXXX+EXYY+EYXY+EYYX3+EXZZ+EZXZ+EZZX3+3i30 EYYY+EYXX+EXYX+EXXY3+EYZZ+EZYZ+EZZY3,
E±11¯=±126 (EYYX+EYXY-2EXYY)-i26 (EXXY+EXYX-2EYXX),
E±12=-126 (EYYX+EYXY-2EXYY)±i26 (EXXY+EXYX-2EYXX),
E03=110 [2EZZZ-(EZXX+EXZX+EXXZ+EZYY+EYZY+EYYZ)],
E±13=±3230 EXXX+EXYY+EYXY+EYYX3-4 EXZZ+EZXZ+EZZX3-3i230 EYYY+EYXX+EXYX+EXXY3-4 EYZZ+EZYZ+EZZY3,
E±23=323 EZXX+EXZX+EXXZ3-EZYY+EYZY+EYYZ3+2i EXYZ+EZXY+EYZX3,
E±33=±122 [-EXXX+(EXYY+EYXY+EYYX)]+i22 [-EYYY+(EYXX+EXYX+EXXY)].
EXXX=a12a2*EXYY=b12a2*EYXY=EYYX=a1b1b2*,
EYYY=b12b2*EYXX=a12b2*EXYX=EXXY=a1b1a2*.
EJ=12=35 cos2 ϕ+115 sin2 ϕE2,
EJ=32=25 cos2 ϕ+415 sin2 ϕE2,
EJ=22=EJ=1¯2=13 sin2 ϕE2,
ρE2(δ)=14+154 tan δ-1tan δ+12,
I=PXΩ2+PYΩ2.
PXΩJ 12J+1 βJ2(EJFXJ),
PYΩJ 12J+1 βJ2(EJFYJ).
[PXΩ]2β4=A2(δ, Φ)cos4 ϕ+B2(δ, Φ)sin4 ϕ+A(δ, Φ)B(δ, Φ)cos2 ϕ sin2 ϕ,
[PYΩ]2β4=C2(δ, Φ)cos2 ϕ sin2 ϕ,
A(δ, Φ)=12 115 cos2 Φ+120 17 sin2 Φcosδ-π4+12 128 sin2 Φsinδ-π4,
B(δ, Φ)=12 145 cos2 Φ+160 17 sin2 Φcosδ-π4-12 128 sin2 Φsinδ-π4,
C(δ, Φ)=12 245 cos2 Φ+130 17 sin2 Φcosδ-π4-12 114 sin2 Φsinδ-π4.
I(δ, Φ, ϕ)β4=A2(δ, Φ)cos4 ϕ+B2(δ, Φ)sin4 ϕ+[C2(δ, Φ)+2A(δ, Φ)B(δ, Φ)]×cos2 ϕ sin2 ϕ.
A2(δ, Φ)=B2(δ, Φ),
C2(δ, Φ)+2A(δ, Φ)B(δ, Φ)=2A2(δ, Φ),
I(δ, Φ, ϕ)=A2(δ, Φ).
A(δ, Φ)=-B(δ, Φ),C(δ, Φ)=±2A(δ, Φ),
A(δ, Φ)=B(δ, Φ),C(δ, Φ)=0.
245 cos2 Φ+8105 sin2 Φcos δ
=115 sin2 Φ-245 cos2 Φsin δ,
245 cos2 Φ-115 sin2 Φcos δ
=-245 cos2 Φ-8105 sin2 Φsin δ.
tan δ=(14+24 tan2 Φ)(-2+3 tan2 Φ)=(2-3 tan2 Φ)(-14-24 tan2 Φ),

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