2 Optical length and group delay

The atmosphere is first supposed to be a stratified medium without turbulence. The baseline $\vec{B}$ is horizontal, and much shorter than the scale height of the atmosphere. Let $n(\sigma)$ the refractive index of the atmosphere at the interferometer level, and $\sigma$ the wavenumber or inverse of the wavelength, $\vec{s}(\sigma)$ 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 $n(\sigma) \vec{B} \cdot \vec{s}(\sigma)$. 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:

$$OPD_\mathrm{geo} = \vec{B} \cdot \vec{s_{0}}$$ (1)

where $\vec{s_{0}}$ is the source direction outside the atmosphere, or without refraction. The quantity $OPD_\mathrm{geo}$ 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 P₁ and P₂ along a ray path is proportional to the optical length $L_{\phi}(\sigma)$ between this two points with:
  

  $$L_{\phi}(\sigma) = \int_{P_{1}}^{P_{2}} n(\sigma) \mathrm{d}s$$ (2)

• propagation of longitudinal coherence, or of wavepackets. Between two points P₁ and P₂ along a ray path, the group delay is:

$$L_\mathrm{g}(\sigma) = \int_{P_{1}}^{P_{2}} \frac{c}{v_\mathrm{g}(\sigma)} \mathrm{d}s$$ (3)

where $v_\mathrm{g}$ is the group velocity:

$$v_\mathrm{g} = \frac{c}{n(\sigma)}\, \left[1 - \frac{\sigma}{n(\sigma)}\,\frac{\partial n} {\partial \sigma}\right].$$ (4)

In a weakly dispersive medium such as the air, we shall have:

$$\frac{c}{v_\mathrm{g}(\sigma)} \simeq n(\sigma)\,\left[1 + \... ...\sigma}{n(\sigma)}\, \frac{\partial n}{\partial \sigma}\right]$$ (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 10⁶. 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:

$$OPD = OPD_\mathrm{geo} + \epsilon_\mathrm{atmos}$$ (6)

the long term average of the last quantity being zero.

2.1 Evacuated delay line

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

$$\Delta L_{\phi} = OPD - L_\mathrm{del} + C_\mathrm{off}.$$ (7)

As long as the interferometer "offset'' $C_\mathrm{off}$ is achromatic or non dispersive, there exists a fringe with zero phase at any wavenumber. The "white-fringe'' is obtained for $\Delta L_{\phi}=0$. This is also the condition for zero group delay, so that the maximum coherence is obtained at the white-fringe, and all these are well-known results for an optical interferometer with evacuated delay line.

2.2 Non-evacuated delay line

The interferometer offset will not be further considered in this paper, and dropped to zero in the following. A delay line with length $L_\mathrm{del}$ 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 $\Delta L_{\phi}$ and $\Delta L_\mathrm{g}$ with:

$$\begin{eqnarray} \Delta L_{\phi}(\sigma) & = & OPD - n(\sigma) L_\mathrm{del} \\... ...(\sigma) & = & OPD - \frac{c}{v_\mathrm{g}(\sigma)} L_\mathrm{del}\end{eqnarray}$$ (8) (9)

and with (5):

$$\Delta L_\mathrm{g}(\sigma) = OPD - \left[n(\sigma) + \sigma \frac{\partial n}{\partial \sigma}\right] L_\mathrm{del}.$$ (10)

The residual phase being $\phi = 2\pi\,\sigma\,\Delta L_{\phi}$, the relation between residual phase derivative and residual group delay is easily checked:

$$\frac{1}{2\pi}\frac{\mathrm{d}\phi}{\mathrm{d}\sigma}=\Delta L_\mathrm{g}(\sigma).$$ (11)

Coherencing, or group delay tracking with $\Delta L_\mathrm{g}(\sigma)=0$, has to be performed at a peculiar wavenumber determined by the instrumental set-up. Let $\sigma_{0}$ its value, and $\sigma'_{0}$ the wavenumber with zero residual phase, it is solution of:

$$n(\sigma'_{0})=n(\sigma_{0})+\sigma_{0} \frac{\partial n}{\partial \sigma}\vert _{\sigma_{0}}.$$ (12)

The air refractive index being an increasing function wrt wavenumber, $\sigma'_{0} \gt \sigma_{0}$.

2.3 A model for the air refractive index

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 $\sigma$ (see e.g. Owens 1967):

$$n(\sigma) -1 = A + \frac{B}{C-\sigma^{2}} + \frac{D}{E-\sigma^{2}}$$ (13)

where the coefficients are dependent on the partial pressures of the minor components (water vapor, carbon dioxyde, ...), and on temperature and total pressure.

In the near-IR range, with $\sigma < 1~\mu \mathrm{m}^{-1}$, only a few terms in a $\sigma^{2}$ 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):

$$n(\sigma) -1 = (A_{0} + B_{0} \sigma^{2} + C_{0} \sigma^{4})\frac{T_{0}}{T}\frac{p} {p_{0}} \cdot 10^{-6}$$ (14)

with A₀ = 287.604, B₀ = 1.6288, C₀ = 0.0136, and standard conditions T₀=273.15 K and p₀=1013.25 hPa.

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:

$$n(\sigma) = 1 + \alpha + \beta \sigma^{2} + \gamma \sigma^{4}$$ (15)

with $\alpha \simeq 199.329\ 10^{-6}$, $\beta \simeq 1.129\ 10^{-6}~\mu\mathrm{m}^{2}$and $\gamma \simeq 0.009\ 10^{-6}~\mu\mathrm{m}^{4}$.

To better than a few parts in 10³ and for better clarity, we shall neglect the $\sigma^{4}$ 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.

2.4 The dispersive term

  

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 $\beta \sigma^{2}$ term in the simplified expression for the refractive index, and the solution to (12) is:

$$\sigma'_{0} \simeq \sqrt{3} \sigma_{0}.$$ (16)

The residual phase is an odd cubic function of the wavenumber given by:

$$\phi(\sigma) \simeq 2\pi \sigma (3\sigma_{0}^{2} - \sigma^{2}) \beta L_\mathrm{del}$$ (17)

with its maximum at $\sigma=\sigma_{0}$. Equal phase can be measured at two different wavenumbers $\sigma_{1}$ and $\sigma_{2}$ with $\sigma_{1} < \sigma_{0} < \sigma_{2}$,and this property will be used in the proposed fringe tracking system. Residual phase is plotted in Fig. 1 for 1 meter of delay line, and for three different values of $\sigma_{0}$, ranging from the K band to the H band.

The residual group delay is:

$$\Delta L_\mathrm{g}(\sigma) \simeq 3(\sigma_{0}^{2} - \sigma^{2}) \beta L_\mathrm{del}$$ (18)

that is $-6 \sigma_{0}^{2} \beta L_\mathrm{del}$ for zero residual phase, or a few $\mu$m per meter of delay line. The residual group delay is plotted in Fig. 1 for the three different $\sigma_{0}$ values. It is seen also as the slope of the phase curves.