Angular dependence of second-harmonic generation in a Langmuir–Blodgett monolayer under a full 360° rotation

Liying Liu; Gongming Wang; Xingze Lu; Jiabiao Zheng; Wencheng Wang

doi:10.1364/JOSAB.14.000382

Journal of the Optical Society of America B
Vol. 14,
Issue 2,
pp. 382-390
(1997)
•https://doi.org/10.1364/JOSAB.14.000382

Angular dependence of second-harmonic generation in a Langmuir–Blodgett monolayer under a full 360° rotation

Liying Liu, Gongming Wang, Xingze Lu, Jiabiao Zheng, and Wencheng Wang

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Liying Liu, Gongming Wang, Xingze Lu, Jiabiao Zheng, and Wencheng Wang, "Angular dependence of second-harmonic generation in a Langmuir–Blodgett monolayer under a full 360° rotation," J. Opt. Soc. Am. B 14, 382-390 (1997)

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Abstract

A theoretical expression is summarized for second-harmonic generation in both reflection and transmission with different polarization combinations. By measuring the angular dependence of a transmitted second-harmonic generation signal of a stilbazium salt Langmuir–Blodgett monolayer, we found that the molecular tilt angle could be conveniently deduced by a single polarization combination $(p \to p)$ in the case that the sample was rotated under a full 360° rotation. This gives us a simple and accurate method to deduce the molecular second-order polarizability and molecular tilt angle. We also show that the multiple reflection in the substrate gives an important contribution for transmitted second-harmonic generation with a large angle of incidence. Consequently, the multiple reflection must be considered if a transmitted second-harmonic generation study is made with a variable angle of incidence, especially with s polarization and fixed detector position.

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Coefficients	Reflection	Transmission
A	$\frac{4 ∊_{g} sin θ {cos}^{2} θ}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ}) (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2}}$	$\frac{8 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{1 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ}) (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{3}}$
B	$\frac{4 ∊_{g} sin θ {cos}^{2} θ}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ}) (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2}}$	$\frac{8 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{1 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2} (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2}}$
C	$\frac{4 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{3}}$	$\frac{8 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$
D	$- \frac{8 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{3}}$	$\frac{16 ∊_{g}^{2} sin θ {cos}^{3} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$
F	$\frac{4 ∊_{g}^{3} {sin}^{3} θ {cos}^{2} θ}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{3}}$	$\frac{8 ∊_{g}^{3} {sin}^{3} θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{1 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$

Coefficients	Reflection	Transmission
A	$\frac{32 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2} (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$	$\frac{16 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2} (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{3}}$
B	$\frac{32 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2} (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$	$\frac{16 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ}) (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$
C	$\frac{32 ∊_{g}^{3} sin θ {cos}^{4} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{6}}$	$\frac{16 ∊_{g}^{3} sin θ {cos}^{4} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{5}}$
D	$- \frac{64 ∊_{g}^{3} sin θ {cos}^{4} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{6}}$	$\frac{32 ∊_{g}^{2} sin θ {cos}^{3} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{5}}$
F	$\frac{32 ∊_{g}^{3} {sin}^{3} θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{6}}$	$\frac{16 ∊_{g}^{3} {sin}^{3} θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{5}}$

Coefficients	Reflection	Transmission
A	$\frac{4 ∊_{g} sin θ {cos}^{2} θ}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ}) (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2}}$	$\frac{8 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{1 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ}) (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{3}}$
B	$\frac{4 ∊_{g} sin θ {cos}^{2} θ}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ}) (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2}}$	$\frac{8 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{1 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2} (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2}}$
C	$\frac{4 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{3}}$	$\frac{8 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$
D	$- \frac{8 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{3}}$	$\frac{16 ∊_{g}^{2} sin θ {cos}^{3} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$
F	$\frac{4 ∊_{g}^{3} {sin}^{3} θ {cos}^{2} θ}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{3}}$	$\frac{8 ∊_{g}^{3} {sin}^{3} θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{1 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$

Coefficients	Reflection	Transmission
A	$\frac{32 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2} (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$	$\frac{16 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2} (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{3}}$
B	$\frac{32 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{2} (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$	$\frac{16 ∊_{g} sin θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ}) (cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{4}}$
C	$\frac{32 ∊_{g}^{3} sin θ {cos}^{4} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{6}}$	$\frac{16 ∊_{g}^{3} sin θ {cos}^{4} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{5}}$
D	$- \frac{64 ∊_{g}^{3} sin θ {cos}^{4} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{6}}$	$\frac{32 ∊_{g}^{2} sin θ {cos}^{3} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{5}}$
F	$\frac{32 ∊_{g}^{3} {sin}^{3} θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)^{3 / 2}}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{6}}$	$\frac{16 ∊_{g}^{3} {sin}^{3} θ {cos}^{2} θ (∊_{g} - {sin}^{2} θ)}{(∊_{g} cos θ + \sqrt{∊_{g} - {sin}^{2} θ})^{5}}$

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Equations (28)

Journal of the Optical Society of America B