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3 A fringe tracking unit with air-filled delay line

3.1 Coherencing operation

A fringe tracking unit is needed in order to reduce the OPD variations, either systematic or random, to a fraction of the coherence length. The delay line is permanently adjusted for this purpose, and its motion is controlled by the error signal of the tracking unit. Simultaneous observations in at least two different wavebands are needed. Let $\sigma_{1}$ and $\sigma_{2}$ the mean wavenumbers of the two bands, that we call guiding wavenumbers . The phase difference is, with (17) and :
\Delta \phi_{12} \simeq 2\pi \Delta \sigma_{12} [3\sigma_{0}...
\sigma_{1}\sigma_{2} + \sigma_{2}^{2})] \beta L_\mathrm{del} .\end{displaymath} (19)

Maintaining zero phase difference whatever the delay length gives the condition for zero group delay tracking at $\sigma_{0}$:
\sigma_{0}^{2} = \frac{1}{3} (\sigma_{1}^{2} + \sigma_{1}\sigma_{2} +
\sigma_{2}^{2}).\end{displaymath} (20)
The wavenumber with zero group delay has to be precisely known. As it stands at the residual phase maximum, $\sigma_{0}$ is further called leading wavenumber . Indeed, laser metrology of the delay line has to be corrected for air longitudinal dispersion, either at $\sigma_{0}$ for group delay metrology, with two colors, or at $\sigma'_{0}$ for optical length or phase measurements, with a single color. These two wavenumbers are approximately given by (20) and (16), to a few parts in 103. More accurate expressions would be obtained with the next term in the power expansion of the refractive index.

3.2 Choice of the guiding wavenumbers

The two observed bands should be large enough for sensitivity purposes, and with a convenient separation: not too close for a good sensitivity to delay variations, and not too far away in order to easily solve for the ambiguity in phase difference. The pair H and K bands seems to be a good compromise, or more precisely the $K_\mathrm{s}$ band for reduced thermal contribution.

With $\sigma_{1} = \sigma_{K_\mathrm{s}}$, and $\sigma_{2} = \sigma_{H}$, respectively 0.463 and 0.606 $\mu$m-1, the leading wavenumber will be:

\sigma_{0} = 0.536~\mu{\rm m}^{-1},~{\rm or}~\lambda_{0}=1.865~\mu{\rm m}.\end{displaymath}

The ambiguity in the phase difference and the sensitivity to delay variations are both given by the relation between phase difference and additional path length $\delta L$:
\delta\phi_{12} = 2\pi\Delta\sigma_{12}\delta L.\end{displaymath} (21)
With $\Delta\sigma_{12}=0.143$, we obtain a $2\pi$ variation for $\delta L \simeq$$\mu$m, and a phase sensitivity of about 1 degree per 20 nm.

\includegraphics [width=8cm,clip]{ds1718f2.eps}\end{figure} Figure 2: Sine fringe patterns in the H and $K_\mathrm{s}$bands (full and dashed line respectively) for different air delay length L, from 0 to 30 m. The relative bandwidth is 0.1, with square-shaped band filter. The phase difference ($\phi_{H}-\phi_{K_\mathrm{s}}$) also shown keeps to be the same whatever the delay length

Fringe patterns are shown in Fig. 2, for four different values of the delay length L, ranging from 0 to 30 m. The amplitude or pattern envelope is here solely determined by the coherence due to the width and shape of the bandpath filters in each observed bands, and the source emission spectrum is supposed to be flat. With a square-shaped bandpath filter of width $\Delta \lambda$, centered at $\lambda =
1/\sigma$, and with resolution $R = \lambda/\Delta \lambda$, the coherence is $\gamma_{\rm c}(l) = {{\rm sinc}}(\pi\sigma_{i}l/R_{i})$with i=H or $K_\mathrm{s}$, l being the additional delay relative to zero group delay, in each band. The relative bandwidth at the two observed wavebands is supposed to be 0.1, or R = 10. The effect of longitudinal dispersion is clearly seen on the position of maximum coherence, shifted on either side of zero delay.

Another choice could be the J and $K_\mathrm{s}$ bands. With $\sigma_{1} = \sigma_{K_\mathrm{s}}$, and $\sigma_{2} = \sigma_{J}$, respectively 0.463 and 0.80 $\mu$m-1, the leading wavenumber being: $\sigma_{0}=0.638~\mu$m-1, or $\lambda_{0}=1.565~\mu$m, that is within the H band. With $\Delta\sigma_{12}=0.337$, a $2\pi$ variation in phase difference is obtained for $\delta L \simeq 3~\mu$m, and a sensitivity of about 1 phase degree per 8 nm. The phase ambiguity is here more critical, but this could be solved with a third band observation, for example in the H band.

A third choice would be the two guiding wavenumbers in the same near-IR band, say the $K_\mathrm{s}$ band. The difference between the two wavenumbers cannot exceed about 0.05 $\mu$m-1, so that a $2\pi$ variation is reached for $\delta L \simeq
20~\mu$m, and the sensitivity is about 1 phase degree per 62 nm. This last choice can be thought of as a preliminary or complementary setup for finding the fringe pattern, the $(K_\mathrm{s},H)$ pair being favored for high precision measurements.

3.3 Coherence and delay range

Table 1: Residual group delay $\Delta L_\mathrm{g}$ in 4 IR bands per meter of delay length, and air delay length L95 responsible for 5% coherence loss with 0.1 relative bandwidth. The different choices investigated for the guiding pairs ($\sigma_{1}$, $\sigma_{2}$) are shown

{ l l r@{.}l r@{.}l l l} \hline
\multicolumn{2}{l}{IR band} & 
 ...\space & 
2&77 & 
21&25 & 
$5.79$\space & 
$13.24$\space \\ \hline \end{tabular}

The residual group delay (18) is a linear function of the delay length. It is given in Table 1, for the J, H, $K_\mathrm{s}$ and M bands, per meter of delay length at Paranal. It has to be kept much smaller than the coherence length for fringes to be observed, and the effect will be more stringent at shorter wavelength. The coherence is given by $\gamma_{\rm c}(l)$ where l now stands for the residual group delay $\Delta
L_\mathrm{g}(\sigma)$. The air delay length L95 responsible for a 5% visibility loss, with relative bandwidth of 0.1, has been estimated in the 4 bands, and is given in Table 1. The three just mentioned choices for the guiding wavenumbers have been considered.

3.4 Compensation of the air dispersion

Glass compensation plates will be needed with delay lengths larger than about 20 meters and the $(K_\mathrm{s},H)$ pair for fringe tracking (Fig. 2). The problem has been investigated by Lévêque (1997) for the VLTI delay lines, and the same kind of results is obtained with our approximation for the air dispersive component. Let $N(\sigma)$ the glass refractive index, N'0 and N''0 respectively its first and second derivatives wrt wavenumber at $\sigma=\sigma_{0}$. The addition of a glass plate with thickness e in one of the interferometer arms gives a group delay change $E(\sigma)$:
E(\sigma) \simeq 
N(\sigma) - 1 + \sigma\frac{\partial N}{\partial \sigma}
\cdot e\end{displaymath} (22)
and the compensation of the group delay dispersion reduces to:
E(\sigma) - E(\sigma_{0}) = \Delta L_\mathrm{g}(\sigma).\end{displaymath} (23)
In the useful wavenumber interval $[\sigma_{1},\, \sigma_{2}]$, the glass refractive index is supposed to be correctly described by its series expansion, at order 2:
N(\sigma) \simeq N(\sigma_{0}) + (\sigma - \sigma_{0}) N'_{0} + \frac{1}{2}(\sigma-
\sigma_{0})^{2} N''_{0}.\end{displaymath} (24)
With (22), it is shown that group delay compensation at $\sigma_{1}$ and $\sigma_{2}$,and then in the whole range $[\sigma_{1},\, \sigma_{2}]$, can be obtained if and only if $N'_{0} = \sigma_{0} N''_{0}$, that is:
N(\sigma) = N(\sigma_{0}) + \frac{1}{2} (\sigma^{2} - \sigma_{0}^{2}) N''_{0}.\end{displaymath} (25)

The thickness of the glass plate will be $e = 2\beta \, L_\mathrm{del}/N''_{0}$.

The glass plate will add an additional delay at $\sigma_{0}$, given by (22) with $\sigma=\sigma_{0}$, so that the air delay line has to be moved accordingly. This jump has not to be known exactly if we accept a different interferometer offset $C_\mathrm{off}$ for each added compensation plate, the offset being determined through astrometric calibration of the instrument, with the observation of reference stars.

3.5 Effect of a non flat emission spectrum

\includegraphics [width=8cm,clip]{ds1718f3.eps}\end{figure} Figure 3: Effect of the source emission spectrum, characterized by its color temperature $T_{\rm c}$, on the phase measured in each band, and on the relative delay offset due to the spectral shape. The curve parameter ranging from 1 to 5 is respectively for $T_{\rm c}$ = 2 300 K, 2 800 K, 4 000 K, 6 000 K and 10 000 K

With the proposed scheme for fringe tracking and air-filled delay lines, phase measurements are performed in two wavebands at non zero residual group delay. The measured phase $\phi(l)$ is then dependent on the shape of the observed spectrum, and given by:
\gamma_{\rm c}(l) 
{\rm e}^{i\phi(l)} = \frac{\int_{\rm band...
 ... d}\sigma}}{\int_{\rm band} {g(\sigma)I(\sigma) {\rm d}\sigma}}\end{displaymath} (26)
where $\gamma_{\rm c}(l)$ is the module of the coherence factor, $g(\sigma)$ the shape of the bandpass filter (supposed to be square-shaped), and $I(\sigma)$ the source intensity spectrum.

The phase measured in the two guiding bands can be written:
\phi_{i}(l_{i}) = 2\pi\sigma_{i}l_{i} + \psi_{i}(l_{i}) \end{displaymath} (27)
with i=1,2, and $\psi_{i}(l_{i})$ the "spectrum phase'', due to the non-flat observed spectrum. The residual delay is $l_{i} = \Delta L_\mathrm{g}(\sigma_{i})$, given by (18) in terms of the air delay length.

The residual group delay has to be much smaller than the coherence length $1/\Delta\sigma$ for a large enough coherence module, so that fine structures in the source emission spectrum will have negligeable effects on the residual phase. As we are concerned with astrometric measurements and large bandwidth observations, the observed spectrum is supposed to be characterized by a single parameter, the color temperature $T_\mathrm{c}$ of the equivalent black body radiation spectrum:
I(\sigma) \propto \frac{\sigma^{3}}{\exp(\sigma/\eta)-1}\end{displaymath} (28)
with $\eta = 6.94 \ 10^{-5} T_\mathrm{c}~\mu$m-1.

The "spectrum phase'' is shown in Fig. 3 in terms of the air delay length for the $K_\mathrm{s}$ and H bands, and for different values of the color temperature parameter, ranging from 2 300 K to 10 000 K. The pic of the black body emission spectrum is found at 2.16 $\mu$m for $T_\mathrm{c} \simeq 2 365$ K, so that the "spectrum phase'' is nearly zero in the $K_\mathrm{s}$ band for the first curve. In the H band, a similar situation occurs for $T_\mathrm{c} \simeq$ 3 095 K, that is in between curves labelled number 2 and 3. The coherence module does not show any significant dependence on the temperature of the black body spectrum.

For air delay much larger than L95, Fig. 3 shows that the spectrum phase deviation is not linear with the delay length, particularly in the H band. The consequence is a dependence of the relative delay offset $\epsilon$ with the delay length:
\epsilon (L_\mathrm{del}) = \frac{\psi_{2}(L_\mathrm{del}) -...
 ...L_\mathrm{del})}{2\pi\, \Delta\sigma_{12}\,L_\mathrm{del}}\cdot\end{displaymath} (29)
This quantity is plotted in Fig. 3 for the different color temperatures as parameter. The curve with $T_\mathrm{c}=2 \, 800$ K is the less dependent on the delay, as the maximum of the emission spectrum is then located between the two guiding wavenumbers.

\includegraphics [width=8cm,clip]{ds1718f4.eps}\end{figure} Figure 4: Relative delay offset due to the non uniformity of the observed spectrum, characterized by its color temperature. Zero delay offset is for a flat spectrum, and the three curves are for the different pairs of guiding wavenumbers (Table 1)

Relative to flat emission spectra with guiding wavenumbers $\sigma_{1}$ and $\sigma_{2}$, the mean observed wavenumbers are now $\tilde \sigma_{1}$ and $\tilde
\sigma_{2}$, and the true leading wavenumber $\tilde \sigma_{0}$ is now dependent on the source color temperature. Let $\delta \sigma_{0} = \tilde \sigma_{0} -
\sigma_{0}$. With the derivative of the residual group delay wrt leading wavenumber, the chromatic delay departure is:
\delta L_\mathrm{g} \simeq 6 \sigma_{0} \cdot\delta\sigma_{0} \cdot\beta\,
L_\mathrm{del}\end{displaymath} (30)
and its relative value $\epsilon_{0} =\delta L_\mathrm{g}/L_\mathrm{del}$, or relative delay offset for small delay lengths, is plotted in Fig. 4 in terms of the color temperature of the observed source. This departure has no significant impact on the (single-field) astrometric calibration of the interferometer, which is supposed to be at a precision level of a few parts in 108, or 1-10 mas. But for dual-field astrometry and small angle measurements, the situation is different, and $\epsilon_{0}$ gives the chromatic error on the measured angular distance between two sources with different color temperatures. It is particularly significant for stars cooler than about 4 000 K. Nevertheless, a precision of a few 10-10, or about 50 $\mu$as, should be reached with simple modelisation of the emission spectra.

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