Unique determination of the hyperpolarizability tensor ratio through depolarized hyper-Rayleigh scattering under an external electric field

Jisoo Hwang; Byoungchoo Park; H. J. Chang; J. W. Wu

doi:10.1364/JOSAB.19.001620

Journal of the Optical Society of America B
Vol. 19,
Issue 7,
pp. 1620-1626
(2002)
•https://doi.org/10.1364/JOSAB.19.001620

Unique determination of the hyperpolarizability tensor ratio through depolarized hyper-Rayleigh scattering under an external electric field

Jisoo Hwang, Byoungchoo Park, H. J. Chang, and J. W. Wu

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Jisoo Hwang, Byoungchoo Park, H. J. Chang, and J. W. Wu, "Unique determination of the hyperpolarizability tensor ratio through depolarized hyper-Rayleigh scattering under an external electric field," J. Opt. Soc. Am. B 19, 1620-1626 (2002)

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Abstract

We have analyzed the depolarized hyper-Rayleigh scattering from a molecular system with partial macroscopic polar ordering. It is shown that the hyperpolarizability tensor ratio and the molecular dipole moment can be determined simultaneously by the measurement of depolarization as a function of external field strength. As an experimental example, we performed a quantitative analysis of the electric-field-dependent depolarized hyper-Rayleigh signal from a poly-γ-benzyl-l-glutamate solution, obtaining the dipole moment and the ratio of hyperpolarizability components as 4.0 D and $β_{311} / β_{333} = - 0.81,$ respectively.

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$〈 ϕ_{I_{p}}^{6} 〉_{f (θ)}$	$\frac{1}{7} i_{0} (x) + \frac{10}{21} i_{2} (x) + \frac{24}{77} i_{4} (x) + \frac{16}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{4} ϕ_{I_{q}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{21} i_{2} (x) - \frac{16}{385} i_{4} (x) - \frac{8}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{I_{q}}^{2} ϕ_{I_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) - \frac{1}{55} i_{4} (x) + \frac{2}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{4} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) - \frac{3}{55} i_{4} (x) + \frac{2}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{4} 〉_{f (θ)}$	$\frac{3}{35} i_{0} (x) + \frac{3}{14} i_{2} (x) + \frac{201}{3080} i_{4} (x) + \frac{3}{308} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{J_{q}}^{2} 〉_{f (θ)}$	$\frac{2}{105} i_{0} (x) + \frac{1}{42} i_{2} (x) - \frac{23}{770} i_{4} (x) - \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{2} ϕ_{J_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{14} i_{2} (x) + \frac{67}{3080} i_{4} (x) + \frac{1}{308} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}}^{3} ϕ_{J_{q}} 〉_{f (θ)}$	$- \frac{1}{70} i_{0} (x) + \frac{3}{770} i_{4} (x) - \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}} ϕ_{J_{q}} ϕ_{J_{r}}^{2} 〉_{f (θ)}$	$- \frac{1}{210} i_{0} (x) - \frac{1}{168} i_{2} (x) + \frac{23}{3080} i_{4} (x) + \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{K_{p}}^{2} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) - \frac{1}{385} i_{4} (x) + \frac{2}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{K_{q}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{42} i_{2} (x) - \frac{37}{770} i_{4} (x) - \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{2} ϕ_{K_{r}}^{2} 〉_{f (θ)}$	$\frac{8}{105} i_{0} (x) + \frac{9}{42} i_{2} (x) + \frac{529}{6160} i_{4} (x) + \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}} ϕ_{J_{q}} ϕ_{K_{p}} ϕ_{K_{q}} 〉_{f (θ)}$	$- \frac{1}{210} i_{0} (x) + \frac{1}{55} i_{4} (x) - \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}} ϕ_{J_{r}} ϕ_{K_{q}} ϕ_{K_{r}} 〉_{f (θ)}$	$- \frac{1}{42} i_{0} (x) - \frac{1}{14} i_{2} (x) - \frac{19}{616} i_{4} (x) + \frac{4}{231} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}} ϕ_{J_{r}} ϕ_{K_{q}} ϕ_{K_{r}} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) + \frac{1}{168} i_{2} (x) - \frac{51}{3080} i_{4} (x) + \frac{1}{231} i_{6} (x)$

$〈 ϕ_{I_{p}}^{6} 〉_{f (θ)}$	$\frac{1}{7} i_{0} (x) + \frac{10}{21} i_{2} (x) + \frac{24}{77} i_{4} (x) + \frac{16}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{4} ϕ_{I_{q}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{21} i_{2} (x) - \frac{16}{385} i_{4} (x) - \frac{8}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{I_{q}}^{2} ϕ_{I_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) - \frac{1}{55} i_{4} (x) + \frac{2}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{4} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) - \frac{3}{55} i_{4} (x) + \frac{2}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{4} 〉_{f (θ)}$	$\frac{3}{35} i_{0} (x) + \frac{3}{14} i_{2} (x) + \frac{201}{3080} i_{4} (x) + \frac{3}{308} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{J_{q}}^{2} 〉_{f (θ)}$	$\frac{2}{105} i_{0} (x) + \frac{1}{42} i_{2} (x) - \frac{23}{770} i_{4} (x) - \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{2} ϕ_{J_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{14} i_{2} (x) + \frac{67}{3080} i_{4} (x) + \frac{1}{308} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}}^{3} ϕ_{J_{q}} 〉_{f (θ)}$	$- \frac{1}{70} i_{0} (x) + \frac{3}{770} i_{4} (x) - \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}} ϕ_{J_{q}} ϕ_{J_{r}}^{2} 〉_{f (θ)}$	$- \frac{1}{210} i_{0} (x) - \frac{1}{168} i_{2} (x) + \frac{23}{3080} i_{4} (x) + \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{K_{p}}^{2} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) - \frac{1}{385} i_{4} (x) + \frac{2}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{K_{q}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{42} i_{2} (x) - \frac{37}{770} i_{4} (x) - \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{2} ϕ_{K_{r}}^{2} 〉_{f (θ)}$	$\frac{8}{105} i_{0} (x) + \frac{9}{42} i_{2} (x) + \frac{529}{6160} i_{4} (x) + \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}} ϕ_{J_{q}} ϕ_{K_{p}} ϕ_{K_{q}} 〉_{f (θ)}$	$- \frac{1}{210} i_{0} (x) + \frac{1}{55} i_{4} (x) - \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}} ϕ_{J_{r}} ϕ_{K_{q}} ϕ_{K_{r}} 〉_{f (θ)}$	$- \frac{1}{42} i_{0} (x) - \frac{1}{14} i_{2} (x) - \frac{19}{616} i_{4} (x) + \frac{4}{231} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}} ϕ_{J_{r}} ϕ_{K_{q}} ϕ_{K_{r}} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) + \frac{1}{168} i_{2} (x) - \frac{51}{3080} i_{4} (x) + \frac{1}{231} i_{6} (x)$

	$α = \sum_{p} β_{ppp}^{2}$	$β_{1} = \sum_{p, q} β_{ppp} β_{pqq}$	$β_{2} = \sum_{p, q} β_{ppp} (β_{qpq} + β_{qqp})$	$β_{3} = \sum_{p, q} β_{pqq}^{2}$
$A 1$	$〈 Φ_{I_{p}}^{6} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{4} Φ_{I_{q}}^{2} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{4} Φ_{I_{q}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{4} Φ_{I_{q}}^{2} 〉_{f (θ)}$
$A 2$	$〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{4} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{4} 〉_{f (θ)}$
$A 3$	$〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{K_{p}}^{2} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{K_{q}}^{2} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 4$	$8 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{4} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)} + 〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{4} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)} + 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$
$A 5$	$4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{4} 〉_{f (θ)}$	$8 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)}$	$4 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)} + 4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$	$4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$
$A 6$	$4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{K_{p}}^{2} 〉_{f (θ)}$	$8 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$8 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{K_{q}}^{2} 〉_{f (θ)}$
	$β_{4} = \sum_{p, q} β_{pqq} (β_{qpq} + β_{qqp})$	$β_{5} = \sum_{p, q} (β_{qpq} + β_{qqp})^{2}$	$γ_{1} = \sum_{p, q, r} β_{pqq} β_{prr}$	$γ_{2} = \sum_{p, q, r} (β_{qpq} + β_{qqp}) (β_{rpr} + β_{rrp})$

$A 1$	$2 〈 Φ_{I_{p}}^{4} Φ_{I_{q}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{4} Φ_{I_{q}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{I_{q}}^{2} Φ_{I_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{I_{q}}^{2} Φ_{I_{r}}^{2} 〉_{f (θ)}$
$A 2$	$2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 3$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{K_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$
$A 4$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 5$	$4 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)} + 4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)} + 〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{4} 〉_{f (θ)} + 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$	$4 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)} + 3 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 6$	$8 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{K_{q}}^{2} 〉_{f (θ)} + 2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$4 〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)} + 2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$
	$γ_{3} = \sum_{p, q, r} β_{pqq} (β_{rpr} + β_{rrp})$	$δ_{1} = \sum_{p, q, r} (β_{pqr} + β_{prq})^{2}$		$δ_{2} = \sum_{p, q, r} (β_{qpq} + β_{prp}) (β_{qpr} + β_{qrp})$
$A 1$	$2 〈 Φ_{I_{p}}^{2} Φ_{I_{q}}^{2} Φ_{I_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{2} 〈 Φ_{I_{p}}^{2} Φ_{I_{q}}^{2} Φ_{I_{r}}^{2} 〉_{f (θ)}$		$〈 Φ_{I_{p}}^{2} Φ_{I_{q}}^{2} Φ_{I_{r}}^{2} 〉_{f (θ)}$
$A 2$	$2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{2} 〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)}$		$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 3$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$	$\frac{1}{2} 〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$		$〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$
$A 4$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)} + 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{2} 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$		$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 5$	$8 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)} + 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$		$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)} + 3 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 6$	$8 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{K_{r}}^{2} 〉_{f (θ)}$		$2 〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)} + 2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$

Abstract

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Equations (7)

Journal of the Optical Society of America B