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

Open Access

Table 1

Visibility phase shifts.

Diatt.	Partial polarization	x (ε = +1)	y (ε = −1)	Total (ε = 0)
τ_x = τ_y	P = 0	$atan (\cos 2 θ \tan (\frac{1}{2} ΔΔ ψ))$ ${\mathop{\rm atan}\nolimits} \left( {\cos 2\theta \tan \left( {{1 \over 2}\Delta \Delta \psi } \right)} \right)$	$- atan (\cos 2 θ \tan (\frac{1}{2} ΔΔ ψ))$ $- {\mathop{\rm atan}\nolimits} \left( {\cos 2\theta \tan \left( {{1 \over 2}\Delta \Delta \psi } \right)} \right)$	0
	Circular	$atan (\frac{\cos 2 θ \sin (\frac{1}{2} ΔΔ ψ)}{\cos (\frac{1}{2} ΔΔ ψ) - P \sin 2 θ \cos γ})$ ${\mathop{\rm atan}\nolimits} \left( {{{\cos 2\theta \sin \left( {{1 \over 2}\Delta \Delta \psi } \right)} \over {\cos \left( {{1 \over 2}\Delta \Delta \psi } \right) - P\sin 2\theta \cos \gamma }}} \right)$	$- atan (\frac{\cos 2 θ \sin (\frac{1}{2} ΔΔ ψ)}{\cos (\frac{1}{2} ΔΔ ψ) - P \sin 2 θ \cos γ})$ $-{\mathop{\rm atan}\nolimits} \left( {{{\cos 2\theta \sin \left( {{1 \over 2}\Delta \Delta \psi } \right)} \over {\cos \left( {{1 \over 2}\Delta \Delta \psi } \right) - P\sin 2\theta \cos \gamma }}} \right)$	0
	Linear	$atan (\frac{[P \cos 2 α + \cos 2 θ] \sin (\frac{1}{2} ΔΔ ψ)}{[1 + P \cos 2 θ \cos 2 α] \cos (\frac{1}{2} ΔΔ ψ) - P \sin 2 θ \sin 2 α \cos γ})$ ${\mathop{\rm atan}\nolimits} \left( {{{[P\cos 2\alpha + \cos 2\theta ]\sin \left( {{1 \over 2}\Delta \Delta \psi } \right)} \over {[1 + P\cos 2\theta \cos 2\alpha ]\cos \left( {{1 \over 2}\Delta \Delta \psi } \right) - P\sin 2\theta \sin 2\alpha \cos \gamma }}} \right)$	$atan (\frac{[P \cos 2 α - \cos 2 θ] \sin (\frac{1}{2} ΔΔ ψ)}{[1 - P \cos 2 θ \cos 2 α] \cos (\frac{1}{2} ΔΔ ψ) + P \sin 2 θ \sin 2 α \cos γ})$ ${\mathop{\rm atan}\nolimits} \left( {{{[P\cos 2\alpha - \cos 2\theta ]\sin \left( {{1 \over 2}\Delta \Delta \psi } \right)} \over {[1 - P\cos 2\theta \cos 2\alpha ]\cos \left( {{1 \over 2}\Delta \Delta \psi } \right) + P\sin 2\theta \sin 2\alpha \cos \gamma }}} \right)$	$atan (P \cos 2 α \tan (\frac{1}{2} ΔΔ ψ))$ ${\mathop{\rm atan}\nolimits} \left( {P\cos 2\alpha \tan \left( {{1 \over 2}\Delta \Delta \psi } \right)} \right)$

τ_x ≠ τ_y	P = 0	$atan (\frac{[τ_{x}^{2} \cos^{2} θ - τ_{y}^{2} \sin^{2} θ]}{[τ_{x}^{2} \cos^{2} θ + τ_{y}^{2} \sin^{2} θ]} \tan (\frac{1}{2} ΔΔ ψ))$ ${\mathop{\rm atan}\nolimits} \left( {{{\left[ {\tau _x^2{{\cos }^2}\theta - \tau _y^2{{\sin }^2}\theta } \right]} \over {\left[ {\tau _x^2{{\cos }^2}\theta + \tau _y^2{{\sin }^2}\theta } \right]}}\tan \left( {{1 \over 2}\Delta \Delta \psi } \right)} \right)$	$atan (\frac{[τ_{x}^{2} \sin^{2} θ - τ_{y}^{2} \cos^{2} θ]}{[τ_{x}^{2} \sin^{2} θ + τ_{y}^{2} \cos^{2} θ]} \tan (\frac{1}{2} ΔΔ ψ))$ ${\mathop{\rm atan}\nolimits} \left( {{{\left[ {\tau _x^2{{\sin }^2}\theta - \tau _y^2{{\cos }^2}\theta } \right]} \over {\left[ {\tau _x^2{{\sin }^2}\theta + \tau _y^2{{\cos }^2}\theta } \right]}}\tan \left( {{1 \over 2}\Delta \Delta \psi } \right)} \right)$	$atan (\frac{(τ_{x}^{2} - τ_{y}^{2})}{(τ_{x}^{2} + τ_{y}^{2})} \tan (\frac{1}{2} ΔΔ ψ))$ ${\mathop{\rm atan}\nolimits} \left( {{{\left( {\tau _x^2 - \tau _y^2} \right)} \over {\left( {\tau _x^2 + \tau _y^2} \right)}}\tan \left( {{1 \over 2}\Delta \Delta \psi } \right)} \right)$
	circ.	$atan (\frac{[τ_{x}^{2} \cos^{2} θ - τ_{y}^{2} \sin^{2} θ] \sin (\frac{1}{2} ΔΔ ψ)}{[τ_{x}^{2} \cos^{2} θ + τ_{y}^{2} \sin^{2} θ] \cos (\frac{1}{2} ΔΔ ψ) - P \sin 2 θ \cos γ})$ ${\mathop{\rm atan}\nolimits} \left( {{{\left[ {\tau _x^2{{\cos }^2}\theta - \tau _y^2{{\sin }^2}\theta } \right]\sin \left( {{1 \over 2}\Delta \Delta \psi } \right)} \over {\left[ {\tau _x^2{{\cos }^2}\theta + \tau _y^2{{\sin }^2}\theta } \right]\cos \left( {{1 \over 2}\Delta \Delta \psi } \right) - P\sin 2\theta \cos \gamma }}} \right)$ `	$atan (\frac{[τ_{x}^{2} \sin^{2} θ - τ_{y}^{2} \cos^{2} θ] \sin (\frac{1}{2} ΔΔ ψ)}{[τ_{x}^{2} \sin^{2} θ + τ_{y}^{2} \cos^{2} θ] \cos (\frac{1}{2} ΔΔ ψ) + P \sin 2 θ \cos γ})$ ${\mathop{\rm atan}\nolimits} \left( {{{\left[ {\tau _x^2{{\sin }^2}\theta - \tau _y^2{{\cos }^2}\theta } \right]\sin \left( {{1 \over 2}\Delta \Delta \psi } \right)} \over {\left[ {\tau _x^2{{\sin }^2}\theta + \tau _y^2{{\cos }^2}\theta } \right]\cos \left( {{1 \over 2}\Delta \Delta \psi } \right) + P\sin 2\theta \cos \gamma }}} \right)$	$atan (\frac{(τ_{x}^{2} - τ_{y}^{2})}{(τ_{x}^{2} + τ_{y}^{2})} \tan (\frac{1}{2} ΔΔ ψ))$ ${\mathop{\rm atan}\nolimits} \left( {{{\left( {\tau _x^2 - \tau _y^2} \right)} \over {\left( {\tau _x^2 + \tau _y^2} \right)}}\tan \left( {{1 \over 2}\Delta \Delta \psi } \right)} \right)$
	lin.	$atan (\frac{[τ_{x}^{2} \cos^{2} θ (1 + P \cos 2 α) - τ_{y}^{2} \sin^{2} θ (1 - P \cos 2 α)] \sin (\frac{1}{2} ΔΔ ψ)}{[τ_{x}^{2} \cos^{2} θ (1 + P \cos 2 α) + τ_{y}^{2} \sin^{2} θ (1 - P \cos 2 α)] \cos (\frac{1}{2} ΔΔ ψ) - P \sin 2 θ \sin 2 α \cos γ})$ ${\mathop{\rm atan}\nolimits} \left( {{{\left[ {\tau _x^2{{\cos }^2}\theta (1 + P\cos 2\alpha ) - \tau _y^2{{\sin }^2}\theta (1 - P\cos 2\alpha )} \right]\sin \left( {{1 \over 2}\Delta \Delta \psi } \right)} \over {\left[ {\tau _x^2{{\cos }^2}\theta (1 + P\cos 2\alpha ) + \tau _y^2{{\sin }^2}\theta (1 - P\cos 2\alpha )} \right]\cos \left( {{1 \over 2}\Delta \Delta \psi } \right) - P\sin 2\theta \sin 2\alpha \cos \gamma }}} \right)$	$atan (\frac{[τ_{x}^{2} \sin^{2} θ (1 + P \cos 2 α) - τ_{y}^{2} \cos^{2} θ (1 - P \cos 2 α)] \sin (\frac{1}{2} ΔΔ ψ)}{[τ_{x}^{2} \sin^{2} θ (1 + P \cos 2 α) + τ_{y}^{2} \cos^{2} θ (1 - P \cos 2 α)] \cos (\frac{1}{2} ΔΔ ψ) + P \sin 2 θ \sin 2 α \cos γ})$ ${\mathop{\rm atan}\nolimits} \left( {{{\left[ {\tau _x^2{{\sin }^2}\theta (1 + P\cos 2\alpha ) - \tau _y^2{{\cos }^2}\theta (1 - P\cos 2\alpha )} \right]\sin \left( {{1 \over 2}\Delta \Delta \psi } \right)} \over {\left[ {\tau _x^2{{\sin }^2}\theta (1 + P\cos 2\alpha ) + \tau _y^2{{\cos }^2}\theta (1 - P\cos 2\alpha )} \right]\cos \left( {{1 \over 2}\Delta \Delta \psi } \right) + P\sin 2\theta \sin 2\alpha \cos \gamma }}} \right)$	$atan (\frac{(τ_{x}^{2} - τ_{y}^{2}) + (τ_{x}^{2} + τ_{y}^{2}) P \cos 2 α}{(τ_{x}^{2} + τ_{y}^{2}) + (τ_{x}^{2} - τ_{y}^{2}) P \cos 2 α} \tan (\frac{1}{2} ΔΔ ψ))$ ${\mathop{\rm atan}\nolimits} \left( {{{\left( {\tau _x^2 - \tau _y^2} \right) + \left( {\tau _x^2 + \tau _y^2} \right)P\cos 2\alpha } \over {\left( {\tau _x^2 + \tau _y^2} \right) + \left( {\tau _x^2 - \tau _y^2} \right)P\cos 2\alpha }}\tan \left( {{1 \over 2}\Delta \Delta \psi } \right)} \right)$

Notes. Visibility phase shift 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. (27).

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