Measurements of the hyperpolarizability tensor by means of hyper-Rayleigh scattering

V. Ostroverkhov; R. G. Petschek; K. D. Singer; L. Sukhomlinova; R. J. Twieg; S.-X. Wang; L. C. Chien

doi:10.1364/JOSAB.17.001531

Journal of the Optical Society of America B
Vol. 17,
Issue 9,
pp. 1531-1542
(2000)
•https://doi.org/10.1364/JOSAB.17.001531

Measurements of the hyperpolarizability tensor by means of hyper-Rayleigh scattering

V. Ostroverkhov, R. G. Petschek, K. D. Singer, L. Sukhomlinova, R. J. Twieg, S.-X. Wang, and L. C. Chien

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V. Ostroverkhov, R. G. Petschek, K. D. Singer, L. Sukhomlinova, R. J. Twieg, S.-X. Wang, and L. C. Chien, "Measurements of the hyperpolarizability tensor by means of hyper-Rayleigh scattering," J. Opt. Soc. Am. B 17, 1531-1542 (2000)

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Abstract

We describe improvements to a previously reported hyper-Rayleigh scattering technique [J. Opt. Soc. Am. B 15, 289 (1998) ], for measuring all rotational invariants of the first hyperpolarizability tensor. The full hyper-Rayleigh scattering tensor is expressed in terms of its rotationally invariant components leading to a figures of merit corresponding to each of the rotationally invariant tensors. With elliptically polarized incident light, the polarization state and the intensity of the harmonic light are measured at a scattering angle of 45°. A new analytical fitting method is applied to the signal for two polarization measurements to yield the invariants. We have measured several chiral and nonchiral chromophores to determine the hyperpolarizability tensor with which to analyze the measurement accuracy, to reveal structure property relationships, and to assess the applicability of the chromophores as nonlinear optical materials. Results for a series of chiral chromophores and for Crystal Violet are presented.

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Δ(ℒ)	Explicit Form	c	$c^{'}$
$Δ (0 aa)$	$c ∊_{ijk} ∊_{lmn}$	1/6	–
$Δ (1 ss)$	$c P_{s} ‖ \bar{} P_{s}$	$\sqrt{3} / 5$	–
$Δ (1 mm)$	$c P_{m} \bar{‖} P_{m}$	$\sqrt{3} / 4$	–
$Δ (1 m^{'} m^{'})$	$c P_{m^{'}} \underset{̲}{‖} P_{m^{'}}$	$1 / \sqrt{3}$	–
$Δ (1 sm +)$	$c (P_{m} \bar{‖} P_{s} + P_{s} \bar{‖} P_{m})$	$\sqrt{3 / 40}$	–
$Δ (1 sm -)$	$c (P_{m} \bar{‖} P_{s} - P_{s} \bar{‖} P_{m})$	$\sqrt{3 / 40}$	–
$Δ (1 {sm}^{'} +)$	$c (P_{s} \underset{̲}{‖} P_{m^{'}} + P_{m^{'}} \underset{̲}{‖} P_{s})$	$1 / \sqrt{10}$	–
$Δ (1 {sm}^{'} -)$	$c (P_{s} \underset{̲}{‖} P_{m^{'}} - P_{m^{'}} \underset{̲}{‖} P_{s})$	$1 / \sqrt{10}$	–
$Δ (1 {mm}^{'} +)$	$c (P_{m^{'}} \underset{̲}{‖} P_{m} + P_{m} \underset{̲}{‖} P_{m^{'}})$	$1 / \sqrt{2}$	–
$Δ (1 {mm}^{'} -)$	$c (P_{m^{'}} \underset{̲}{‖} P_{m} - P_{m} \underset{̲}{‖} P_{m^{'}})$	$1 / \sqrt{2}$	–
$Δ (2 mm)$	$c (P_{m} \bar{‖} P_{m}) + c^{'} (P_{m} \bar{\times} P_{m})$	$- 3 / (4 \sqrt{5})$	$1 / \sqrt{5}$
$Δ (2 m^{'} m^{'})$	$c (P_{m^{'}} \underset{̲}{‖} P_{m^{'}}) + c^{'} (P_{m^{'}} \underline{\times} P_{m^{'}})$	$- 1 / \sqrt{5}$	$2 / \sqrt{5}$
$Δ (2 {mm}^{'} +)$	$c (P_{m} \underset{̲}{‖} P_{m^{'}} + P_{m^{'}} \underset{̲}{‖} P_{m}) + c^{'} (P_{m} \underline{\times} P_{m^{'}} + P_{m^{'}} \underline{\times} P_{m})$	$\sqrt{3 / 10}$	$- \sqrt{2 / 15}$
$Δ (2 {mm}^{'} -)$	$c (P_{m} \underset{̲}{‖} P_{m^{'}} - P_{m^{'}} \underset{̲}{‖} P_{m}) + c^{'} (P_{m} \underline{\times} P_{m^{'}} - P_{m^{'}} \underline{\times} P_{m})$	$\sqrt{3 / 10}$	$- \sqrt{2 / 15}$
$Δ (3 ss)$	$c (P_{s} \equiv) + c^{'} P_{s} \bar{‖} P_{s}$	$1 / \sqrt{7}$	$- 3 / (5 \sqrt{7})$

Material	$β_{1 mm}^{2} / β_{1 ss}^{2}$	$β_{1 sm +}^{2} / β_{1 ss}^{2}$	$β_{1 sm -}^{2} / β_{1 ss}^{2}$	$β_{2 mm}^{2} / β_{1 ss}^{2}$	$β_{3 ss}^{2} / β_{1 ss}^{2}$	$β_{1 ss}^{2} / (β_{1 ss}^{2})_{pNA}$
1064 nm
1, pNa	$0.16 \pm 0.08$	$- 0.02 \pm 0.03$	$0.17 \pm 0.02$	$0.04 \pm 0.04$	$0.54 \pm 0.04$	1
2	$0.22 \pm 0.04$	$0.11 \pm 0.02$	$- 0.05 \pm 0.01$	$0.11 \pm 0.05$	$0.52 \pm 0.03$	∼8.5
5	$0.6 \pm 0.2$	$0.05 \pm 0.01$	$0.25 \pm 0.01$	$0.02 \pm 0.02$	$0.41 \pm 0.02$	∼82
6, Q-BNH	$0.15 \pm 0.10$	$- 0.06 \pm 0.07$	$0.13 \pm 0.03$	$0.19 \pm 0.05$	$0.47 \pm 0.04$	∼11.5
7	$0.13 \pm 0.03$	$0.10 \pm 0.02$	$0.15 \pm 0.02$	$0.08 \pm 0.02$	$0.51 \pm 0.02$	∼8.7
8, Crystal Violet	$1.2 \pm 0.2$	$- 1.3 \pm 0.1$	$0.08 \pm 0.03$	$0.8 \pm 0.2$	$1.7 \pm 0.2$	∼712
1340 nm
1, pNA	$0.18 \pm 0.05$	$- 0.03 \pm 0.02$	$0.057 \pm 0.002$	$0.00 \pm 0.05$	$0.53 \pm 0.02$	1
3	$0.05 \pm 0.01$	$0.04 \pm 0.01$	$0.04 \pm 0.01$	$0.11 \pm 0.06$	$0.47 \pm 0.01$	∼31.4
4	$0.11 \pm 0.02$	$0.13 \pm 0.01$	$0.01 \pm 0.01$	$0.00 \pm 0.01$	$0.40 \pm 0.01$	∼93.6

Material	$‖ β_{1 ss} ‖$ $(\times 10^{- 30} esu)$	$‖ β_{1 mm} ‖$ $(\times 10^{- 30} esu)$	$‖ β_{2 mm} ‖$ $(\times 10^{- 30} esu)$	$‖ β_{3 ss} ‖$ $(\times 10^{- 30} esu)$
1064 pm
1, pNA	$11.2 \pm 1.6$	$4.6 \pm 1.2$	$2.1 \pm 2.6$	$8.2 \pm 1.2$
2	$33.2 \pm 4.8$	$15.5 \pm 2.6$	$10.8 \pm 2.9$	$23.9 \pm 3.5$
5	$110 \pm 16$	$26.8 \pm 6.0$	$15.7 \pm 7.6$	$70 \pm 10$
6, Q-BNH	$38.3 \pm 5.9$	$14.7 \pm 5.7$	$16.9 \pm 3.3$	$26.3 \pm 4.3$
7	$33.6 \pm 4.7$	$12.1 \pm 2.2$	$9.6 \pm 1.6$	$23.9 \pm 3.4$
8, Crystal Violet	$305 \pm 58$	$341 \pm 72$	$276 \pm 52$	$398 \pm 81$
1340 nm
pNA	$8.1 \pm 1.5$	$3.4 \pm 0.8$	$0.0 \pm 1.9$	$5.9 \pm 1.1$
3	$44.9 \pm 7.7$	$10.2 \pm 1.9$	$14.7 \pm 4.8$	$30.9 \pm 5.3$
4	$78 \pm 13$	$25.7 \pm 4.7$	$0.0 \pm 1.0$	$49.1 \pm 8.4$

Material	$β_{1 mm}^{2} / β_{1 ss}^{2}$	$β_{1 sm +}^{2} / β_{1 ss}^{2}$	$β_{1 sm -}^{2} / β_{1 ss}^{2}$	$β_{2 mm}^{2} / β_{1 ss}^{2}$	$β_{3 ss}^{2} / β_{1 ss}^{2}$
CV–acetone	$1.2 \pm 0.2$	$- 1.3 \pm 0.1$	$0.08 \pm 0.03$	$0.8 \pm 0.2$	$1.7 \pm 0.2$
CV–methanol	$0.5 \pm 0.2$	$- 0.9 \pm 0.1$	$- 0.22 \pm 0.04$	$1.1 \pm 0.1$	$1.3 \pm 0.2$
CV–dioxane	$0.9 \pm 0.2$	$- 0.8 \pm 0.2$	$0.29 \pm 0.02$	$0.90 \pm 0.03$	$1.9 \pm 0.3$

Δ(ℒ)	Explicit Form	c	$c^{'}$
$Δ (0 aa)$	$c ∊_{ijk} ∊_{lmn}$	1/6	–
$Δ (1 ss)$	$c P_{s} ‖ \bar{} P_{s}$	$\sqrt{3} / 5$	–
$Δ (1 mm)$	$c P_{m} \bar{‖} P_{m}$	$\sqrt{3} / 4$	–
$Δ (1 m^{'} m^{'})$	$c P_{m^{'}} \underset{̲}{‖} P_{m^{'}}$	$1 / \sqrt{3}$	–
$Δ (1 sm +)$	$c (P_{m} \bar{‖} P_{s} + P_{s} \bar{‖} P_{m})$	$\sqrt{3 / 40}$	–
$Δ (1 sm -)$	$c (P_{m} \bar{‖} P_{s} - P_{s} \bar{‖} P_{m})$	$\sqrt{3 / 40}$	–
$Δ (1 {sm}^{'} +)$	$c (P_{s} \underset{̲}{‖} P_{m^{'}} + P_{m^{'}} \underset{̲}{‖} P_{s})$	$1 / \sqrt{10}$	–
$Δ (1 {sm}^{'} -)$	$c (P_{s} \underset{̲}{‖} P_{m^{'}} - P_{m^{'}} \underset{̲}{‖} P_{s})$	$1 / \sqrt{10}$	–
$Δ (1 {mm}^{'} +)$	$c (P_{m^{'}} \underset{̲}{‖} P_{m} + P_{m} \underset{̲}{‖} P_{m^{'}})$	$1 / \sqrt{2}$	–
$Δ (1 {mm}^{'} -)$	$c (P_{m^{'}} \underset{̲}{‖} P_{m} - P_{m} \underset{̲}{‖} P_{m^{'}})$	$1 / \sqrt{2}$	–
$Δ (2 mm)$	$c (P_{m} \bar{‖} P_{m}) + c^{'} (P_{m} \bar{\times} P_{m})$	$- 3 / (4 \sqrt{5})$	$1 / \sqrt{5}$
$Δ (2 m^{'} m^{'})$	$c (P_{m^{'}} \underset{̲}{‖} P_{m^{'}}) + c^{'} (P_{m^{'}} \underline{\times} P_{m^{'}})$	$- 1 / \sqrt{5}$	$2 / \sqrt{5}$
$Δ (2 {mm}^{'} +)$	$c (P_{m} \underset{̲}{‖} P_{m^{'}} + P_{m^{'}} \underset{̲}{‖} P_{m}) + c^{'} (P_{m} \underline{\times} P_{m^{'}} + P_{m^{'}} \underline{\times} P_{m})$	$\sqrt{3 / 10}$	$- \sqrt{2 / 15}$
$Δ (2 {mm}^{'} -)$	$c (P_{m} \underset{̲}{‖} P_{m^{'}} - P_{m^{'}} \underset{̲}{‖} P_{m}) + c^{'} (P_{m} \underline{\times} P_{m^{'}} - P_{m^{'}} \underline{\times} P_{m})$	$\sqrt{3 / 10}$	$- \sqrt{2 / 15}$
$Δ (3 ss)$	$c (P_{s} \equiv) + c^{'} P_{s} \bar{‖} P_{s}$	$1 / \sqrt{7}$	$- 3 / (5 \sqrt{7})$

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Journal of the Optical Society of America B