Calculation of nonlinear-susceptibility tensor components in ferroelectrics: cubic, tetragonal, and rhombohedral symmetries

Rajan Murgan; David R. Tilley; Yoshihiro Ishibashi; Jeff F. Webb; Junaidah Osman

doi:10.1364/JOSAB.19.002007

Journal of the Optical Society of America B
Vol. 19,
Issue 9,
pp. 2007-2021
(2002)
•https://doi.org/10.1364/JOSAB.19.002007

Calculation of nonlinear-susceptibility tensor components in ferroelectrics: cubic, tetragonal, and rhombohedral symmetries

Rajan Murgan, David R. Tilley, Yoshihiro Ishibashi, Jeff F. Webb, and Junaidah Osman

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Rajan Murgan, David R. Tilley, Yoshihiro Ishibashi, Jeff F. Webb, and Junaidah Osman, "Calculation of nonlinear-susceptibility tensor components in ferroelectrics: cubic, tetragonal, and rhombohedral symmetries," J. Opt. Soc. Am. B 19, 2007-2021 (2002)

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Abstract

We present the formalism for the calculation of all second- and third-order nonlinear susceptibility coefficients based on the Landau–Devonshire free-energy expansion for cubic symmetry in the high-temperature paraelectric phase and the Landau–Khalatnikov dynamical equations. Second-order phase transition and single-frequency input waves are considered. Detailed results are given for all nonvanishing tensor elements of the second- and third-order nonlinear optical effects in the paraelectric and the tetragonal and rhombohedral ferroelectric phases.

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Process, K	Susceptibility $χ_{ilmn}^{(3)}$
Third-harmonic generation	$χ_{iijj}^{(3)} = χ_{ijji}^{(3)} = - \frac{β_{2}}{3 ∊_{0}^{3}} r_{3} r_{1}^{3}$
	$(i \neq j)$
$K = 1 / 4$
$χ_{ilmn}^{(3)} (- 3 ω; ω, ω, ω)$
Symmetric on permutation of (lmn)	$χ_{iiii}^{(3)} = - \frac{β_{1}}{∊_{0}^{3}} r_{3} r_{1}^{3}$
Intensity-dependent refractive index	$χ_{iijj}^{(3)} = χ_{ijji}^{(3)} = - \frac{β_{2}}{3 ∊_{0}^{3}} \| r_{1} \|^{2} r_{1}^{2}$
	$(i \neq j)$
$K = 3 / 4$
$χ_{ilmn}^{(3)} (- ω; ω, ω, - ω)$
Symmetric on interchange of (lm)	$χ_{iiii}^{(3)} = - \frac{β_{1}}{∊_{0}^{3}} \| r_{1} \|^{2} r_{1}^{2}$
dc Kerr effect	$χ_{iijj}^{(3)} = χ_{ijji}^{(3)} = - \frac{8 β_{2}}{3 ∊_{0}^{3}} r_{0}^{2} r_{1}^{2}$
	$(i \neq j)$
$K = 3$
$χ_{ilmn}^{(3)} (- ω; 0, 0, ω)$
Symmetric on interchange of (lm)	$χ_{iiii}^{(3)} = - \frac{8 β_{1}}{3 ∊_{0}^{3}} r_{0}^{2} r_{1}^{2}$

Process, K	Susceptibility $χ_{ilm}^{(2)}$
Second-harmonic generation $K = 1 / 2$	$χ_{zzz}^{(2)} = - \frac{3}{∊_{0}^{3}} β_{1} P_{0} s (2 ω) s^{2} (ω)$
$χ_{ilm}^{(2)} (- 2 ω; ω, ω)$	$χ_{zyy}^{(2)} = χ_{zxx}^{(2)} = - \frac{β_{2}}{∊_{0}^{3}} P_{0} s (2 ω) σ^{2} (ω)$
Symmetric on exchange of $(l, m)$	$χ_{xzx}^{(2)} = χ_{yzy}^{(2)} = χ_{xxz}^{(2)} = χ_{yyz}^{(2)} = - \frac{β_{2}}{∊_{0}^{3}} P_{0} σ (2 ω) σ (ω) s (ω)$
Optical rectification $K = 1 / 2$	$χ_{zzz}^{(2)} = - \frac{3}{∊_{0}^{3}} β_{1} P_{0} s (0) s^{*} (ω) s (ω)$
$χ_{ilm}^{(2)} (0; - ω, ω)$	$χ_{zyy}^{(2)} = χ_{zxx}^{(2)} = - \frac{β_{2}}{∊_{0}^{3}} P_{0} s (0) σ^{*} (ω) σ (ω)$
	$χ_{xzx}^{(2)} = χ_{yzy}^{(2)} = - \frac{β_{2}}{∊_{0}^{3}} P_{0} σ (0) σ (ω) s^{*} (ω)$
	$χ_{xxz}^{(2)} = χ_{yyz}^{(2)} = - \frac{β_{2}}{∊_{0}^{3}} P_{0} σ (0) σ^{*} (ω) s (ω)$

Process, K	Susceptibility, $χ_{ilmn}^{(3)}$
Third-harmonic generation $K = 1 / 4$	$χ_{xxxx} = χ_{yyyy} = \frac{σ (3 ω) σ^{3} (ω)}{∊_{0}^{3}} [\frac{2 β_{2}^{2} P_{0}^{2} s (2 ω)}{∊_{0}^{2}} - β_{1}]$
$χ_{ilmn}^{(3)} (- 3 ω; ω, ω, ω)$ Symmetric on permutation of $(lmn)$	$χ_{zzzz} = \frac{β_{1} s (3 ω) s^{3} (ω)}{∊_{0}^{3}} [\frac{18 P_{0}^{2} β_{1} s (2 ω)}{∊_{0}^{2}} - 1]$
	$χ_{yxxy} = χ_{xxyy} = \frac{β_{2}}{3 ∊_{0}^{3}} σ (3 ω) σ^{3} (ω) [\frac{2 β_{2} P_{0}^{2} s (2 ω)}{∊_{0}^{2}} - 1]$
	$χ_{xxzz} = χ_{yyzz} = \frac{β_{2} σ (3 ω) σ (ω) s^{2} (ω)}{3 ∊_{0}^{3}} [\frac{4 β_{2} P_{0}^{2} σ (2 ω) + 6 P_{0}^{2} β_{1} s (2 ω)}{∊_{0}^{2}} - 1]$
	$χ_{zyyz} = χ_{zxxz} = \frac{β_{2} s (3 ω) σ^{2} (ω) s (ω)}{3 ∊_{0}^{3}} [\frac{4 β_{2} P_{0}^{2} σ (2 ω) + 6 P_{0}^{2} β_{1} s (2 ω)}{∊_{0}^{2}} - 1]$
dc Kerr effect $K = 3$	$χ_{xxxx} = χ_{yyyy} = \frac{4 σ^{2} (ω) σ (0)}{∊_{0}^{3}} [\frac{4 β_{2}^{2} P_{0}^{2} s (0) σ (ω)}{∊_{0}^{2}} - β_{1} σ (0)]$
$χ_{ilmn}^{(3)} (- ω; 0, 0, ω)$ Symmetric on interchange of $(lm)$	$χ_{zzzz} = \frac{β_{1} s^{2} (ω) s^{2} (0)}{∊_{0}^{3}} [\frac{144 β_{1} P_{0}^{2} s (ω)}{∊_{0}^{2}} - 4]$
	$χ_{yxxy} = χ_{xyyx} = - \frac{2 β_{2} σ^{2} (ω) σ^{2} (0)}{3 ∊_{0}^{3}}$
	$χ_{yxyx} = χ_{xxyy} = \frac{σ^{2} (ω) σ^{2} (0) β_{2}}{∊_{0}^{3}} [\frac{8 β_{2} P_{0}^{2} s (ω)}{∊_{0}^{2}} - \frac{14}{9}]$
	$χ_{yzzy} = χ_{xzzx} = \frac{4 σ^{2} (ω) s^{2} (0) β_{2}}{∊_{0}^{3}} [\frac{4 β_{2} P_{0}^{2} σ (ω)}{∊_{0}^{2}} - \frac{1}{3}]$
	$χ_{yyzz} = χ_{xxzz} = χ_{zzyy} = χ_{zzxx} = \frac{4 β_{2} σ (ω) s (ω) s (0) σ (0)}{∊_{0}^{3}} \{[\frac{6 β_{1} P_{0}^{2} s (ω)}{∊_{0}^{2}} + \frac{2 P_{0}^{2} β_{2} σ (ω)}{∊_{0}^{2}}] - 1\}$
	$χ_{zyyz} = χ_{zxxz} = \frac{4 β_{2} σ^{2} (0) s^{2} (ω)}{∊_{0}^{3}} [\frac{4 β_{2} P_{0}^{2} σ (ω)}{∊_{0}^{2}} - \frac{1}{3}]$
Intensity-dependent refractive index $K = 3 / 4$	$χ_{xxxx} = χ_{yyyy} = \frac{σ^{3} (ω) σ^{*} (ω)}{∊_{0}^{3}} [\frac{2 β_{2}^{2} P_{0}^{2} s (0)}{∊_{0}^{2}} - β_{1}]$
$χ_{ilmn}^{(3)} (- ω; - ω, ω, ω)$ Symmetric on interchange of $(lm)$	$χ_{zzzz} = \frac{β_{1} s^{3} (ω) s^{*} (ω)}{∊_{0}^{3}} [\frac{18 β_{1} P_{0}^{2} s (0)}{∊_{0}^{2}} - 1]$
	$χ_{yxxy} = χ_{xyyx} = - \frac{β_{2} σ^{3} (ω) σ^{*} (ω)}{3 ∊_{0}^{3}}$
	$χ_{yxyx} = χ_{xxyy} = \frac{σ^{3} (ω) σ^{*} (ω) β_{2}}{2 ∊_{0}^{3}} [\frac{2 β_{2} P_{0}^{2} s (ω)}{∊_{0}^{2}} - \frac{2}{3}]$
	$χ_{yzzy} = χ_{xzzx} = \frac{β_{2} σ^{2} (ω) s (ω) s^{*} (ω)}{∊_{0}^{3}} [\frac{3 β_{2} P_{0}^{2} σ (0)}{∊_{0}^{2}} - \frac{1}{3}]$
	$χ_{yyzz} = χ_{xxzz} = \frac{β_{2} σ (ω) s (ω)}{2 ∊_{0}^{3}} \{[\frac{6 β_{1} P_{0}^{2} σ (ω) s^{} (ω) s (0) + 2 P_{0}^{2} β_{2} s (ω) σ^{} (ω) σ (0)}{∊_{0}^{2}}] - \frac{1}{3} [σ (ω) s^{} (ω) + s (ω) σ^{} (ω)]\}$
	$χ_{zyyz} = χ_{zxxz} = \frac{β_{2} s^{2} (ω) σ (ω) σ^{*} (ω)}{∊_{0}^{3}} [\frac{2 β_{2} P_{0}^{2} σ (0)}{∊_{0}^{2}} - \frac{1}{3}]$
	$χ_{zzyy} = χ_{zzxx} = \frac{β_{2} σ (ω) s (ω)}{2 ∊_{0}^{3}} \{[\frac{6 β_{1} P_{0}^{2} σ^{} (ω) s (ω) s (0) + 2 P_{0}^{2} β_{2} s^{} (ω) σ (ω) σ (0)}{∊_{0}^{2}}] - \frac{1}{3} [σ^{} (ω) s (ω) + s^{} (ω) σ (ω)]\}$

Process, K	Susceptibility, $χ_{ilm}^{(2)}$
Second-harmonic generation $K = 1 / 2$	$χ_{zzz}^{(2)} = χ_{xxx}^{(2)} = χ_{yyy}^{(2)} = χ_{1}^{SHG} = \frac{∊_{0} [({\hat{O}}_{2} - q) ({vs}^{2} + 2 {uh}_{1}) + 2 q (v σ^{2} + {uh}_{2})]}{8 ({\hat{O}}_{2} + q) ({\hat{O}}_{2} - 2 q)}$
$χ_{ilm}^{(2)} (- 2 ω; ω, ω)$ Symmetric on exchange of $(l, m)$	$χ_{xxy}^{(2)} = χ_{xxz}^{(2)} = χ_{yyx}^{(2)} = χ_{yyz}^{(2)} = χ_{zzx}^{(2)} = χ_{zzy}^{(2)} = χ_{3}^{SHG} = \frac{∊_{0} [{\hat{O}}_{2} (2 vs σ + 2 {uh}_{2}) + q (2 v σ^{2} + 2 {uh}_{3})]}{8 ({\hat{O}}_{2} + q) ({\hat{O}}_{2} - 2 q)}$
	$χ_{xyy}^{(2)} = χ_{xzz}^{(2)} = χ_{yxx}^{(2)} = χ_{yzz}^{(2)} = χ_{zxx}^{(2)} = χ_{zyy}^{(2)} = χ_{2}^{SHG} = \frac{∊_{0} [{\hat{O}}_{2} (v σ^{2} + {uh}_{2}) + q ({vs}^{2} + 2 {uh}_{1})]}{8 ({\hat{O}}_{2} + q) ({\hat{O}}_{2} - 2 q)}$
Optical rectification $K = 1 / 2$ $χ_{ilm}^{(2)} (0; - ω, ω)$	$χ_{zzz}^{(2)} = χ_{xxx}^{(2)} = χ_{yyy}^{(2)} = χ_{1}^{OR} = \frac{∊_{0} [(p + q) (v \| s \|^{2} - 2 {ug}_{1}) - 2 q (v \| σ \|^{2} + {ug}_{2})]}{8 (q - p) (p + 2 q)}$
	$χ_{xxy}^{(2)} = χ_{xxz}^{(2)} = χ_{yyx}^{(2)} = χ_{yyz}^{(2)} = χ_{zzx}^{(2)} = χ_{zzy}^{(2)} = χ_{3}^{OR} = \frac{∊_{0} [(p + q) (vs σ^{} + {ug}_{2}) - q ({vs}^{} σ + {ug}_{2} + v \| σ \|^{2} + 2 {ug}_{1})]}{8 (q - p) (p + 2 q)}$
	$χ_{xyx}^{(2)} = χ_{xzx}^{(2)} = χ_{yxy}^{(2)} = χ_{yzy}^{(2)} = χ_{zxz}^{(2)} = χ_{zyz}^{(2)} = χ_{4}^{OR} = \frac{∊_{0} [(p + q) ({vs}^{} σ + {ug}_{2}) - q (vs σ^{} + {ug}_{2} + v \| σ \|^{2} + 2 {ug}_{1})]}{8 (q - p) (p + 2 q)}$
	$χ_{xyy}^{(2)} = χ_{xzz}^{(2)} = χ_{yxx}^{(2)} = χ_{yzz}^{(2)} = χ_{zxx}^{(2)} = χ_{zyy}^{(2)} = χ_{2}^{OR} = \frac{∊_{0} [p (v \| σ \|^{2} + {ug}_{2}) - q (v \| s \|^{2} + 2 {ug}_{1})]}{8 (q - p) (p + 2 q)}$

Process, K	Susceptibility, $χ_{ilmn}^{(3)}$
Third-harmonic generation $K = 1 / 4$	$χ_{xxxx} = χ_{yyyy} = χ_{zzzz} = \frac{({\hat{O}}_{3} - q) τ_{1} + 2 q τ_{2}}{∊_{0} ({\hat{O}}_{3} + q) ({\hat{O}}_{3} - 2 q)}$
$χ_{ilmn}^{(3)} (- 3 ω; ω, ω, ω)$ Symmetric on permutation of (lmn)	$χ_{yxxx} = χ_{zxxx} = χ_{xyyy} = χ_{zyyy} = χ_{xzzz} = χ_{yzzz} = \frac{({\hat{O}}_{3} - q) τ_{2} + q (τ_{1} + τ_{2})}{∊_{0} ({\hat{O}}_{3} + q) ({\hat{O}}_{3} - 2 q)}$
	$χ_{xxyy} = χ_{yyxx} = χ_{yyzz} = χ_{zzyy} = χ_{zzxx} = χ_{xxzz} = \frac{({\hat{O}}_{3} - q) τ_{4} + q τ_{3}}{∊_{0} ({\hat{O}}_{3} + q) ({\hat{O}}_{3} - 2 q)}$
	$χ_{xxxy} = χ_{xxxz} = χ_{yyyx} = χ_{yyyz} = χ_{zzzx} = χ_{zzzy} = \frac{({\hat{O}}_{3} - q) τ_{3} + q τ_{4}}{∊_{0} ({\hat{O}}_{3} + q) ({\hat{O}}_{3} - 2 q)}$
	$χ_{xyzz} = χ_{zxyy} = χ_{yzxx} = χ_{xzyy} = χ_{yxzz} = χ_{zyxx} = \frac{q (τ_{3} + τ_{4})}{∊_{0} ({\hat{O}}_{3} + q) ({\hat{O}}_{3} - 2 q)}$
Intensity-dependent refractive index $K = 3 / 4$	$χ_{xxxx} = χ_{yyyy} = χ_{zzzz} = \frac{({\hat{O}}_{1} - q) ρ_{1} + 2 q ρ_{2}}{∊_{0} ({\hat{O}}_{1} + q) ({\hat{O}}_{1} - 2 q)}$
$χ_{ilmn}^{(3)} (- ω; - ω, ω, ω)$ Symmetric on interchange of (lm)	$χ_{xxxy} = χ_{xxxz} = χ_{yyyx} = χ_{yyyz} = χ_{zzzx} = χ_{zzzy} = \frac{({\hat{O}}_{1} - q) ρ_{3} + q (ρ_{7} + ρ_{8})}{∊_{0} ({\hat{O}}_{1} + q) ({\hat{O}}_{1} - 2 q)}$
	$χ_{yxxx} = χ_{zxxx} = χ_{xyyy} = χ_{zyyy} = χ_{xzzz} = χ_{yzzz} = \frac{({\hat{O}}_{1} - q) ρ_{2} + q (ρ_{1} + ρ_{2})}{∊_{0} ({\hat{O}}_{1} + q) ({\hat{O}}_{1} - 2 q)}$
	$χ_{xyyx} = χ_{xzzx} = χ_{yxxy} = χ_{yzzy} = χ_{zxxz} = χ_{zyyz} = \frac{({\hat{O}}_{1} - q) ρ_{7} + q (ρ_{3} + ρ_{8})}{∊_{0} ({\hat{O}}_{1} + q) ({\hat{O}}_{1} - 2 q)}$
	$χ_{xzyy} = χ_{xyzz} = χ_{yxzz} = χ_{yzxx} = χ_{zyxx} = χ_{zxyy} = \frac{({\hat{O}}_{1} - q) ρ_{9} + q (ρ_{6} + ρ_{4})}{∊_{0} ({\hat{O}}_{1} + q) ({\hat{O}}_{1} - 2 q)}$
	$χ_{xxyy} = χ_{yyzz} = χ_{zzxx} = χ_{yyxx} = χ_{zzyy} = χ_{xxzz} = \frac{({\hat{O}}_{1} - q) ρ_{4} + q (ρ_{6} + ρ_{9})}{∊_{0} ({\hat{O}}_{1} + q) ({\hat{O}}_{1} - 2 q)}$
	$χ_{xyxx} = χ_{xzxx} = χ_{yxyy} = χ_{yzyy} = χ_{zxzz} = χ_{zyzz} = \frac{({\hat{O}}_{1} - q) ρ_{6} + q (ρ_{4} + ρ_{9})}{∊_{0} ({\hat{O}}_{1} + q) ({\hat{O}}_{1} - 2 q)}$
	$χ_{xyyz} = χ_{xzzy} = χ_{yxxz} = χ_{yzzx} = χ_{zxxy} = χ_{zyyx} = \frac{({\hat{O}}_{1} - q) ρ_{8} + q (ρ_{3} + ρ_{7})}{∊_{0} ({\hat{O}}_{1} + q) ({\hat{O}}_{1} - 2 q)}$
	$χ_{xxyz} = χ_{yyxz} = χ_{zzyx} = χ_{xzxy} = χ_{yzyx} = χ_{zxzy} = \frac{({\hat{O}}_{1} + q) ρ_{5}}{∊_{0} ({\hat{O}}_{1} + q) ({\hat{O}}_{1} - 2 q)}$
dc Kerr effect $K = 3$	$χ_{xxxx} = χ_{yyyy} = χ_{zzzz} = \frac{(p + q) (c_{1} + d_{1} + d_{2}) + 2 q (c_{2} + d_{3} + d_{4} + d_{5})}{(p + 2 q) (q - p)}$
$χ_{ilmn}^{(3)} (- ω; 0, 0, ω)$ Symmetric on interchange of (lm)	$χ_{xxxy} = χ_{xxxz} = χ_{yyyx} = χ_{yyyz} = χ_{zzzx} = χ_{zzzy} = \frac{(p + q) (c_{3} + c_{4} + d_{6} + d_{7})}{(p + 2 q) (q - p)}$
	$χ_{yxxx} = χ_{zxxx} = χ_{xyyy} = χ_{zyyy} = χ_{xzzz} = χ_{yzzz} = \frac{q (c_{1} + d_{1} + d_{2}) - p (c_{2} + d_{3} + d_{4} + d_{5})}{(p + 2 q) (p - q)}$
	$χ_{xxyz} = χ_{yyzx} = χ_{zzxy} = \frac{(p + q) (c_{6} + c_{8})}{(q - p) (p + 2 q)}$
	$χ_{yzxx} = χ_{zyxx} = χ_{zxyy} = χ_{xzyy} = χ_{xyzz} = χ_{yxzz} = \frac{q (c_{6} + c_{5})}{(p - q) (p + 2 q)}$
	$χ_{xxyy} = χ_{yyzz} = χ_{zzxx} = χ_{yyxx} = χ_{zzyy} = χ_{xxzz} = \frac{(p + q) (c_{6} + c_{5})}{(q - p) (p + 2 q)}$
	$χ_{xyyz} = χ_{xzzy} = χ_{yzzx} = χ_{yxxz} = χ_{zxxy} = χ_{zyyx} = \frac{q (c_{3} + c_{4} + d_{6} + d_{7})}{(p - q) (p + 2 q)}$
	$χ_{xyzx} = χ_{yxzy} = χ_{zxyz} = \frac{q (c_{6} + c_{8})}{(p - q) (p + 2 q)}$

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Journal of the Optical Society of America B