Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques

Lionel Duvillaret; Stéphane Rialland; Jean-Louis Coutaz

doi:10.1364/JOSAB.19.002692

Journal of the Optical Society of America B
Vol. 19,
Issue 11,
pp. 2692-2703
(2002)
•https://doi.org/10.1364/JOSAB.19.002692

Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques

Lionel Duvillaret, Stéphane Rialland, and Jean-Louis Coutaz

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Lionel Duvillaret, Stéphane Rialland, and Jean-Louis Coutaz, "Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques," J. Opt. Soc. Am. B 19, 2692-2703 (2002)

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Abstract

We present a complete analysis of electro-optic sensors for electric field measurement based on three different modulation techniques: amplitude, phase, and polarization state modulation. We treat the most general case, considering both isotropic and anisotropic crystals and taking into account the absorption of the crystal. We derive the optimal configuration of experimental setups for the three studied modulation techniques, and we give the values of the variable physical parameters required to yield the best performance. Finally we compare the three modulation techniques and show that phase or polarization state modulations result in exactly the same performance while amplitude modulation gives slightly enhanced performance.

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Noise Type	Signal-Independent $(σ = 0)$	Signal-Proportional $(σ = 1 / 2)$	Proportional to Square Root of Signal $(σ = 1 / 3)$
Figure of merit	$0.221 \sqrt{η} κ$	0.443	$\frac{0.886 \sqrt{η κ}}{(1 + η^{1 / 3})^{3 / 2}}$
${\tilde{P}}_{1} (ϕ_{E} ≪ 1)$	$\frac{κ}{4} (1 + η \pm 2 \sqrt{η} ϕ_{E})$	$\frac{2 η κ (1 \pm ϕ_{E})}{(1 + \sqrt{η})^{2}}$	$\frac{η^{2 / 3} + η \pm 2 η^{5 / 6} ϕ_{E}}{(1 + η^{1 / 3})^{2}} κ$
χ (rad)	$\pm \arctan (η^{- σ / 2})$
Ψ (rad)	$\frac{1}{2} \arccos [\frac{\pm (1 - η^{σ})}{\sqrt{1 + η^{2 σ} - 2 η^{σ} cos 2 ϕ_{0}}}]$
Θ (rad)	$\frac{1}{4} \arccos [- \frac{1 - η^{σ}}{1 + η^{σ}} \pm \frac{2 η^{σ} sin 2 ϕ_{0}}{(1 + η^{σ}) \sqrt{1 + η^{2 σ} - 2 η^{σ} 2 ϕ_{0}}}]$

Noise Type	Signal-Independent	Signal-Proportional	Proportional to Square Root of Signal
Figure of merit	See Fig. 7	See Fig. 9	See Fig. 11
Figure of merit for $α L = 0$	$\frac{3 \sqrt{3}}{8 π}$	$\frac{\sqrt{3}}{2 π}$	$\frac{3}{4 π}$
${\tilde{P}}_{t} (ϕ_{E} ≪ 1)$	$\frac{3}{4} {(\frac{TF}{π \sqrt{R}})}^{2} (1 + \frac{\sqrt{3} F}{π} ϕ_{E})$
λ	$\frac{2 π nL}{m π - φ_{r} \pm \arcsin [π / (2 \sqrt{3} F)]} m \in Z$
$R_{1} = R_{2} = R$	See Fig. 6	See Fig. 8	See Fig. 10

Noise Type	Signal-Independent	Signal-Proportional	Proportional to Square Root of Signal
Figure of merit	See Fig. 7	See Fig. 9	See Fig. 11
Figure of merit for $α L = 0$	$\frac{3 \sqrt{3}}{8 π}$	$\frac{3 \sqrt{3}}{2 π}$	$\frac{3 \sqrt{3}}{4 π}$
${\tilde{P}}_{t} (ϕ_{E} ≪ 1)$	$\frac{3}{4} {(\frac{TF}{π \sqrt{R}})}^{2} (1 + \frac{\sqrt{3} F}{π} ϕ_{E})$
λ	$\frac{2 π nL}{m π - φ_{r} \pm \arcsin [π / (2 \sqrt{3} F)]} m \in Z$
$R_{1}$	R	$R (1 - \frac{1 - R}{1 + R} α L)$	$R (1 - \frac{4}{5} \frac{1 - R}{1 + R} α L)$
$R_{2}$	R	$R (1 + \frac{1 - R}{1 + R} α L)$	$R (1 + \frac{4}{5} \frac{1 - R}{1 + R} α L)$
$R = (R_{1} R_{2})^{1 / 2}$	See Fig. 6	See Fig. 8	See Fig. 10

Noise Type	Signal-Independent $(σ = 0)$	Signal-Proportional $(σ = 1 / 2)$	Proportional to Square Root of Signal $(σ = 1 / 3)$
Figure of merit	$0.221 \sqrt{κ}$	0.443	$\frac{0.886 \sqrt{κ}}{(1 + κ^{1 / 3})^{3 / 2}}$
${\tilde{P}}_{p} (ϕ_{E} ≪ 1)$	$\frac{1 + κ \pm 2 \sqrt{κ} ϕ_{E}}{4}$	$\frac{κ (1 \pm ϕ_{E})}{1 + κ}$	$\frac{κ^{2 / 3} + κ \pm 2 κ^{5 / 6} ϕ_{E}}{(1 + κ^{1 / 3})^{2}}$
$ϕ_{0}$ (rad)	$\frac{2 π [n (E_{Ω} = 0) - 1] L}{λ} + (2 m + 1) \frac{π}{2} m \in Z$
$R_{1}$	$\frac{κ^{σ}}{1 + κ^{σ}}$
$R_{2}$	$\frac{1}{1 + κ^{σ}}$

Noise Type	Signal-Independent	Signal-Proportional	Proportional to Square Root of Signal
PSM	0.221	0.443	0.313
AM in transmission	0.207	0.276	0.239
AM in reflection	0.207	0.827	0.413
PM	0.221	0.443	0.313

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Equations (54)

Journal of the Optical Society of America B