- 2.1 Evacuated delay line
- 2.2 Non-evacuated delay line
- 2.3 A model for the air refractive index
- 2.4 The dispersive term

The atmosphere is first supposed to be a stratified medium
without turbulence. The baseline is horizontal,
and much shorter than the scale height of the atmosphere. Let the
refractive index of the atmosphere at the interferometer level,
and the wavenumber or inverse of the wavelength, the source
direction as seen by the instrument. The Optical Path
Delay of the rays at the two entrance pupils, or geometrical *OPD*, is the scalar
product . With the Snell's law on ray
propagation, this quantity keeps to be the same all along the ray path through the
atmosphere, and it equals its value outside the atmosphere:

(1) |

where is the source direction outside the atmosphere, or without refraction. The quantity is also the geometric group delay for the propagation of wavepackets.

For wave propagation through an isotropic dispersive medium, two features have to be considered:

- propagation of lateral coherence, or of wave surfaces normal to the ray paths.
The phase variation between two points
*P*and_{1}*P*along a ray path is proportional to the optical length between this two points with:_{2}(2) - propagation of longitudinal coherence, or of wavepackets. Between two points
*P*and_{1}*P*along a ray path, the group delay is:_{2}

(3) |

(4) |

(5) |

and in the visible and near-IR range, the difference between optical length and group
delay is as small as a
few parts in 10^{6}. Nevertheless the use of *OPD* for the whole optical path within
the interferometer with non-evacuated pipes is ambiguous. Does it stand for optical
length or for group delay? so that we shall use it only for the geometrical delay
outside the interferometer, eventually affected by the fluctuations of the atmospheric
piston:

(6) |

With an evacuated delay line with length ,the basic equation for the residual optical length of the interferometer is:

(7) |

The interferometer offset will not be further considered in this paper, and dropped to
zero in the following. A delay line with length located in an
air-filled tunnel will compensate exactly the *OPD* at a single wavelength only.
Furthermore two types of compensation are obtained, either optical length compensation
or group delay compensation, the residual lengths being respectively given by and with:

(8) | ||

(9) |

(10) |

(11) |

(12) |

There is no simple analytical form for the air refractive index in a wide optical
range. Fits to index measurements lead to the following general expression in terms of
the wavenumber (see e.g. Owens 1967):

(13) |

In the near-IR range, with , only a few terms in a
power expansion of the preceeding expression are to be considered. With
the additional assumption of the dry component and wet air component obeying the ideal
gas law, both separately and for the combined mixture, a dry air refractive index
expansion is given by Gubler & Tytler (1998):

(14) |

with *A _{0}* = 287.604,

In this paper, the numerical applications will be performed for the atmospheric
conditions prevailing in the VLTI interferometer tunnel, i.e. *T*=289 K and
*p*=743 hPa (Lévêque 1997), so that (14) reduces to:

(15) |

To better than a few parts in 10^{3} and for better clarity, we shall neglect the
term in the analytic developments of the air dispersive effect. The
contribution of water vapor to dispersion effects may be difficult to modelize in the
near-IR transmission bands, that is in the vicinity of water vapor absorption bands.
With the conditions prevailing in the VLTI tunnel, its contribution, although small,
is not known exactly and should be measured and monitored during astrometric
observations.

Figure 1:
Residual phase and residual group delay per meter of
air-filled delay line, and for the 3 pairs of guiding wavenumbers (see
Table 1). Zero group delay is shown as a (*) on each curve. The near-IR
wavebands with 15% relative width are also shown (full segments) |

The departure to a white-fringe, with zero residual phase and group delay, is due to
the term in the simplified expression for the refractive index, and
the solution to (12) is:

(16) |

(17) |

The residual group delay is:

(18) |

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