Effect of applied electric fields on the writing and the readout of photorefractive gratings

Raymond De Vré; Muthu Jeganathan; Jeffrey P. Wilde; Lambertus Hesselink

doi:10.1364/JOSAB.12.000600

Journal of the Optical Society of America B
Vol. 12,
Issue 4,
pp. 600-614
(1995)
•https://doi.org/10.1364/JOSAB.12.000600

Effect of applied electric fields on the writing and the readout of photorefractive gratings

Raymond De Vré, Muthu Jeganathan, Jeffrey P. Wilde, and Lambertus Hesselink

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Raymond De Vré, Muthu Jeganathan, Jeffrey P. Wilde, and Lambertus Hesselink, "Effect of applied electric fields on the writing and the readout of photorefractive gratings," J. Opt. Soc. Am. B 12, 600-614 (1995)

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Abstract

We explore theoretically and experimentally how applying an electric field to a photorefractive crystal affects the recording and the readout of a hologram. We present a unified theory that describes several electric-field-related mechanisms relevant to photorefractive crystals, including amplitude coupling, phase coupling, linear and nonlinear electro-optic effects, and the piezoelectric effect. We analyze the influence of these different effects on the Bragg selectivity and the diffraction efficiency and derive general analytical expressions describing the Bragg detuning effects. Finally, we compare the theory with experiments that we have performed in a strontium barium niobate crystal by recording and retrieving holograms in the presence of an applied electric field.

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Parameter	Value
k_B	1.38 × 102⁻²³ J/K
T	300 K
e	1.602 × 10⁻¹⁹ C
∊₀	8.85 × 10⁻¹² F/m
λ₀	514 nm
∊_r	880
N_D	10²⁵ m⁻³
N_A	1.5 × 10²³ m⁻³
n	2.3
r_effζ(K)	125 pm/V
${\hat{r}}_{33}$	200 pm/V
${\hat{r}}_{13}$	20 pm/V
d₃₃	130 pm/V
l	2.67 mm
Λ_g	5.0 μm (transmission geometry)
Λ_g	0.2 μm (reflection geometry)

Effect	Transmission	Reflection	Remarks
Bragg condition
Electro-optic	$\begin{array}{l} {ξ_{eo}}^{T E} & = \frac{2 π l}{λ_{0}} \frac{n_{e}}{8} {({sin}^{2} {\hat{θ}}_{ref} - {sin}^{2} {\hat{θ}}_{sig}) \\ \times [{\hat{r}}_{33} (E_{0 R}) E_{0 R} - {\hat{r}}_{33} (E_{0 W}) E_{0 W}]} \end{array}$	$\begin{array}{l} {ξ_{eo}}^{R E} & = \frac{2 π l}{λ_{0}} \frac{n_{o}}{8} {2 ({sin}^{2} {\hat{θ}}_{ref} + {sin}^{2} {\hat{θ}}_{sig}) \\ \times [{\hat{r}}_{33} (E_{0 R}) E_{0 R} - {\hat{r}}_{33} (E_{0 W}) E_{0 W}] \\ + 4 {n_{o}}^{2} [{\hat{r}}_{13} (E_{0 R}) E_{0 R} - {\hat{r}}_{13} (E_{0 W}) E_{0 W}]} \end{array}$	Diffraction efficiency enhancement by the nonlinear effects (electrically controlled diffraction)
Phase coupling	${ξ_{PC}}^{T} (z) = - l β tanh \frac{γ z - ln r_{PP}}{2}$	ξ_PC^R = −lβ	r_pp ≪ 1, change in the Bragg condition ξ_PC^T = −lβ
	(Fringe bending)		r_pp ≫ 1, loss of Bragg condition because ξ_PC^T = ξ_PC^T(z)
Piezoelectric	$ξ_{piezo} = \frac{l}{2} d_{33} (E_{0 W} - E_{0 R}) \frac{2 π}{Λ_{g}}$
Diffraction efficiency
Amplitude coupling	$η^{max} = {sin}^{2} [\frac{π n_{1}}{2 λ_{0} cos θ} \int_{0}^{l} m (z) d z]$	$η^{max} = {tanh}^{2} [\frac{π n_{1}}{2 λ_{0} cos θ} \int_{0}^{l} m (z) d z]$	Apodization because of nonuniform grating amplitude
			Decrease in Bragg selectivity because of decrease in effective grating thickness

Parameter	Value
k_B	1.38 × 102⁻²³ J/K
T	300 K
e	1.602 × 10⁻¹⁹ C
∊₀	8.85 × 10⁻¹² F/m
λ₀	514 nm
∊_r	880
N_D	10²⁵ m⁻³
N_A	1.5 × 10²³ m⁻³
n	2.3
r_effζ(K)	125 pm/V
${\hat{r}}_{33}$	200 pm/V
${\hat{r}}_{13}$	20 pm/V
d₃₃	130 pm/V
l	2.67 mm
Λ_g	5.0 μm (transmission geometry)
Λ_g	0.2 μm (reflection geometry)

Effect	Transmission	Reflection	Remarks
Bragg condition
Electro-optic	$\begin{array}{l} {ξ_{eo}}^{T E} & = \frac{2 π l}{λ_{0}} \frac{n_{e}}{8} {({sin}^{2} {\hat{θ}}_{ref} - {sin}^{2} {\hat{θ}}_{sig}) \\ \times [{\hat{r}}_{33} (E_{0 R}) E_{0 R} - {\hat{r}}_{33} (E_{0 W}) E_{0 W}]} \end{array}$	$\begin{array}{l} {ξ_{eo}}^{R E} & = \frac{2 π l}{λ_{0}} \frac{n_{o}}{8} {2 ({sin}^{2} {\hat{θ}}_{ref} + {sin}^{2} {\hat{θ}}_{sig}) \\ \times [{\hat{r}}_{33} (E_{0 R}) E_{0 R} - {\hat{r}}_{33} (E_{0 W}) E_{0 W}] \\ + 4 {n_{o}}^{2} [{\hat{r}}_{13} (E_{0 R}) E_{0 R} - {\hat{r}}_{13} (E_{0 W}) E_{0 W}]} \end{array}$	Diffraction efficiency enhancement by the nonlinear effects (electrically controlled diffraction)
Phase coupling	${ξ_{PC}}^{T} (z) = - l β tanh \frac{γ z - ln r_{PP}}{2}$	ξ_PC^R = −lβ	r_pp ≪ 1, change in the Bragg condition ξ_PC^T = −lβ
	(Fringe bending)		r_pp ≫ 1, loss of Bragg condition because ξ_PC^T = ξ_PC^T(z)
Piezoelectric	$ξ_{piezo} = \frac{l}{2} d_{33} (E_{0 W} - E_{0 R}) \frac{2 π}{Λ_{g}}$
Diffraction efficiency
Amplitude coupling	$η^{max} = {sin}^{2} [\frac{π n_{1}}{2 λ_{0} cos θ} \int_{0}^{l} m (z) d z]$	$η^{max} = {tanh}^{2} [\frac{π n_{1}}{2 λ_{0} cos θ} \int_{0}^{l} m (z) d z]$	Apodization because of nonuniform grating amplitude
			Decrease in Bragg selectivity because of decrease in effective grating thickness

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