Biases induced by retardance and diattenuation in the measurements of long-baseline interferometers

Open Access

Table 2

Astrometric bias.

Diatt.	Partial polarization	x (ɛ =+1)	y (ɛ = −1)	Total (ɛ = 0)
	P = 0	$\frac{1}{2} \cos 2 θ \vec{α_{0}}$ ${1 \over 2}\cos 2\theta \overrightarrow {{\alpha _0}}$	$- \frac{1}{2} \cos 2 θ \vec{α_{0}}$ $- {1 \over 2}\cos 2\theta \overrightarrow {{\alpha _0}}$	0
δ_τ = 0	Circular	$\frac{1}{4} [2 \cos 2 θ + P \sin 4 θ \cos γ] \vec{α_{0}}$ ${1 \over 4}[2\cos 2\theta + P\sin 4\theta \cos \gamma ]\overrightarrow {{\alpha _0}}$	$\frac{1}{4} [- 2 \cos 2 θ + P \sin 4 θ \cos γ] \vec{α_{0}}$ ${1 \over 4}[ - 2\cos 2\theta + P\sin 4\theta \cos \gamma ]\overrightarrow {{\alpha _0}}$	0
	Linear	$\frac{1}{4} [2 \cos 2 θ + 2 P \cos 2 α \sin^{2} 2 θ + P \sin 4 θ \sin 2 α \cos γ] \vec{α_{0}}$ ${1 \over 4}\left[ {2\cos 2\theta + 2P\cos 2\alpha {{\sin }^2}2\theta + P\sin 4\theta \sin 2\alpha \cos \gamma } \right]\overrightarrow {{\alpha _0}}$	$\frac{1}{4} [- 2 \cos 2 θ + 2 P \cos 2 α \sin^{2} 2 θ + P \sin 4 θ \sin 2 α \cos γ] \vec{α_{0}}$ ${1 \over 4}\left[ { - 2\cos 2\theta + 2P\cos 2\alpha {{\sin }^2}2\theta + P\sin 4\theta \sin 2\alpha \cos \gamma } \right]\overrightarrow {{\alpha _0}}$	$\frac{1}{2} P \cos 2 α \vec{α_{0}}$ ${1 \over 2}P\cos 2\alpha \overrightarrow {{\alpha _0}}$
	P = 0	$\frac{1}{2} [\cos 2 θ + \frac{δ τ}{τ} \sin^{2} 2 θ] \vec{α_{0}}$ ${1 \over 2}\left[ {\cos 2\theta + {{\delta \tau } \over \tau }{{\sin }^2}2\theta } \right]\overrightarrow {{\alpha _0}}$	$- \frac{1}{2} [\cos 2 θ + \frac{δ τ}{τ} \sin^{2} 2 θ] \vec{α_{0}}$ $- {1 \over 2}\left[ {\cos 2\theta + {{\delta \tau } \over \tau }{{\sin }^2}2\theta } \right]\overrightarrow {{\alpha _0}}$	$\frac{1}{2} \frac{δ τ}{τ} \vec{α_{0}}$ ${1 \over 2}{{\delta \tau } \over \tau }\overrightarrow {{\alpha _0}}$
δ_τ ≠ 0	Circular	$\frac{1}{4} [2 \cos 2 θ + 2 \frac{δ τ}{τ} \sin^{2} 2 θ + P \sin 4 θ \cos γ] \vec{α_{0}}$ ${1 \over 4}\left[ {2\cos 2\theta + 2{{\delta \tau } \over \tau }{{\sin }^2}2\theta + P\sin 4\theta \cos \gamma } \right]\overrightarrow {{\alpha _0}}$	$\frac{1}{4} [- 2 \cos 2 θ + 2 \frac{δ τ}{τ} \sin^{2} 2 θ + P \sin 4 θ \cos γ] \vec{α_{0}}$ ${1 \over 4}\left[ { - 2\cos 2\theta + 2{{\delta \tau } \over \tau }{{\sin }^2}2\theta + P\sin 4\theta \cos \gamma } \right]\overrightarrow {{\alpha _0}}$	$\frac{1}{2} \frac{δ τ}{τ} \vec{α_{0}}$ ${1 \over 2}{{\delta \tau } \over \tau }\overrightarrow {{\alpha _0}}$
	Linear	$\frac{1}{4} [2 \cos 2 θ + 2 (P \cos 2 α + \frac{δ τ}{τ}) \sin^{2} 2 θ + P \sin 4 θ \sin 2 α \cos γ] \vec{α_{0}}$ ${1 \over 4}\left[ {2\cos 2\theta + 2\left( {P\cos 2\alpha + {{\delta \tau } \over \tau }} \right){{\sin }^2}2\theta + P\sin 4\theta \sin 2\alpha \cos \gamma } \right]\overrightarrow {{\alpha _0}}$	$\frac{1}{4} [- 2 \cos 2 θ + 2 (P \cos 2 α + \frac{δ τ}{τ}) \sin^{2} 2 θ + P \sin 4 θ \sin 2 α \cos γ] \vec{α_{0}}$ ${1 \over 4}\left[ { - 2\cos 2\theta + 2\left( {P\cos 2\alpha + {{\delta \tau } \over \tau }} \right){{\sin }^2}2\theta + P\sin 4\theta \sin 2\alpha \cos \gamma } \right]\overrightarrow {{\alpha _0}}$	$\frac{1}{2} (P \cos 2 α + \frac{δ τ}{τ}) \vec{α_{0}}$ ${1 \over 2}\left( {P\cos 2\alpha + {{\delta \tau } \over \tau }} \right)\overrightarrow {{\alpha _0}}$

Notes. Astrometric bias as a function of the diattenuation properties of the interferometer, of the polarization of the source and of the output polarization axis (either split or combined polarizations). The expressions are the direct application of Eq. (36). Differential retardance, degree of polarization and diattenuation, and supposed to be small.

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