I. B. Interferometry Tutorial
| HOME | INDEX | SEARCH | HELP | NEWS |
|
|||||
|---|---|---|---|---|---|
I. B. Interferometry Tutorial |
|||||
|
|
The very large telescope (VLT) has been constructed with a network of telescopes and an infrastructure to allow interferometrics observations. When it is working in interferometric mode, we talk about the very large telescope interferometer (VLTI). To understand the different capabilities of PRIMA and the kind of measures and results in its modes of work, we will recall here some general explanations about the interferometry theory. PRIMA will always practice interferometry with two telescopes. That's why the results take care only on one baseline and not more (like for example in the case of phase enclosure technics). Moreover we will directly apply the results and the schemes to the VLTI disposition and the general principles of Interferometry will be given without considering the influence of the terrestrial atmosphere (see atmospheric constraints). We will thus recall some results of interferometry for a monochromatic source then for a polychromatic source, then explain phase referencing. Afterwards we will detail the interferometric results with extended source, then how to synthesize images by discrete visibility measurements, before approaching the physical notion of contrast and the visibility of common celestial sources morphologies. |
||||||||||||||||||||||
|
|
|||||||||||||||||||||||
| Warning! The Netscape and Mozilla navigators don't recognize the greek letters, that could disturb the good reading of this page. |
Let's consider a monochromatic point source located at infinity and two telescopes that are observing the source in the direction S. The telescopes are on a reference axis, with each an x-coordinate x1 and x2 (cf fig.1).
Fig. 1: Observation of a source S by two telescopes
They are also separated by a baseline B given by: B=x2-x1 . The source being very far, the wavefronts can so be considered as plane. The waves observed by the telescopes are given by:
f1 ~ exp(-ik*x1.s)exp(-ivt)
f2 ~ exp(-ik*x2.s)exp(-ivt)
If we take for example x1=0, the expressions become:
f1 ~ exp(-ivt)
f2 ~ exp(-ik*B.s)exp(-ivt)
N.B: The negative sign of the first exponential of f2 is due to the fact that the plan wavefront arrives later on the telescope 1.
The interferometer is equipped with delay lines that offer the possibility to add a supplementary OPD. If we consider that the first delay line adds the OPD1=d1 to the travel of the beam observed by the first telescope and d2=OPD2 added by the other delay line, the expressions become:
f1 ~ exp(ik*d1)exp(-ivt)
f2 ~ exp(ik*d2)exp(-ik*B.s)exp(-ivt)
The beams are then combined by a beam combiner. The resulting interferences have the following form:
f = f1+f2 ~ exp(-ivt)[exp(ikd1)+exp(ikd2)exp(-ik*B.s)]
The resulting intensity is not merely the simple addition of the intensities 1 and 2 but is modulated by the phenomenon of interference: Imoy ~ f*fconj = 2(1+cos(k*[s.B+d1-d2]))
Typically we have the general form:
Imoy = 2Io(1+cos(k*D))
that gives the traditionnal fringe scheme: I(D) (shown in fig.2), with a sinusoidal form, varying between 0 and 2Io, with Io proportionnal to A* F (and not equal because of some loss in the VLTI optical train concerning F) where A is the collecting area of the apertures and F the incident source flux power in units of incident energy per units time per units cross-sectionnal area.. Two peaks are separated by: Ds=l/b.
Fig. 2: Theoretical fringes ( I / (2Io) = f(D) )
The interferometers have a limited bandwidth Dl , have a transmissivity depending on the frequency given by h(n): F(n)=F*h(n)*dn . Then the brightness has the following form:
Concerning the coherence, it can be divided in two aspects, the spatial and the temporal coherence. With astronomical sources, the spatial coherence is always verified because the star is very far from the telescopes, thus the beams that interferate have a null phase difference. On the other side the temporal incoherence comes from the fact that a source is never monochromatic: always a spectral width Dl centered on lo, the interferences will appear under the condition: lo^2/Dl=lcoh.
By integrating the general equation above, we get:
That gives the previous fringe pattern (monochromatic case) modulation with an envelope function. This fringe pattern is characteristic of a polychromatic source (cf fig.3). The envelope function can be written as: 1+M(lcoh,D)cos(ko*D) where M is the modulation of the envelope (its value is given by the Fourier transform of the system band pass).
Fig. 3: The fringe pattern of a polychromatic source
In general a source point is given this equation for the brightness modulated by: ko*D=ko*(so.B+d1-d2).
If we define d2-d1=so.B, we know that we will be at maximum of the fringes envelope function for a source at location so.
Now we can see what would be the response for s=so + Ds :
I = 2*A*F(1+M(lcoh,DD)cos(ko(s.B-so.B)))
i.e. I = 2*A*F(1+M(lcoh,DD)cos(ko*DD))
where DD = Ds.B. In this construction the sky position so (reference) as defined by the relative delay d2-d1 is the phase reference of the interferences fringes on the sky.
Let's describe the source intensity spatial distribution with the function F(s) with the units of power incident per area per solide angle on the sky. We work with a wavelength l.
The efficiency of the telescope is a function of the sky position and we will use A(s,so). The extended source, the star, can be decomposed in a incoherent sum of infinitesimal source points, the brightness can so be written like:
Let's simplify by returning to a monochromatic source (a given l). The optical interferometers are measuring fringes by introducing a small added phase in one of the two delay lines d. We have also:
We introduce here the complex visibility:
Thus the brightness takes the following form: I(so,B,d)= 2Io + 2Re{V*exp(ikd)}
where 2Io is the maximum brightness of the two apertures. We will directly apply this form on the case of PRIMA ,a bright guide star and a fainter star, far from the telescopes, separated by a narrow angle. Let's assume that so has the following coordinates (0,0,1) and Ds perpendicular to so: (a,b,0). Then the complex visibility takes the form:
It is further conventionnal to introduce spatial frequencies u= B*x/l and v= B*y/l , implying the final form of the complex visibility:
that is complex and has the dimension of a power.
We can see that the precedent form of the complex visibility is a two-dimensional fourier transform of the distribution of the brightness F. Thus the visibility (provided we can measure it) can be used to compute the brightness distribution of the observed object:
Thus in order to egt an image, we have to collect a set of visibility measurements in the (u,v) plane before doing mathematical operations to define F and then the brightness. In practice a set of discrete baselines Bi tracking the same object allows to get a set of discrete visibility measurements Vi(ui,vi). Typically the informations are collected in maps (u,v), as you can see on the figure 4:
Fig. 4: Set of baselines to realize a map (u,v)
The accuracy of the image synthesis will of course depend on the coverage of the (u,v) plane (cf fig.5).
Fig. 5: (u,v) coverage with a UT
But to reconstruct the brightness distribution we must define a sampling function to take into account the discrete measurements:
S(u,v) = Sd(u-ui)*d(v-vi)
With this sampling function, we write again the expression of F :
We have written Fd because it refers to the dirty brightness distribution that is apparently related to the true brightness distribution by a convulution with an effective point-spread function (PSF) or synthesized beam:
We have thus the true brightness distribution from the deconvolution product of Fd(a,b) = F(a,b) * p(a,b) with mathematical methods. This allows reconstituting the image, as an example is shown on the fig.6:
Fig. 6: Reconstructing images
N.B: If the object is not symmetrical (as a spiral nebula for example), its fourier transform is complex instead of being purely real.
The definition we have given for the complex visibility gives it a dimension of power whose amplitude measures the difference of intensity between the fringes (V is capturing the coherent response of the interferometer to the astronomical source). In reality the optical interferometer measures the normalized fringe power, the fringes power relative to the total power collected from the source, this is quantified as the Michelson fringe visibility nM:
nM = (Imax-Imin)/(Imax+Imin)
that has no dimension and varies in the [0,1] interval. We can link the visibility of Michelson with the complex visibility:
nM =V(k,B)/2Io
We have thus the following expression: I = 2Io(1+Re{nexp(ikd)})
and nM = abs(n).
With all these considerations we will finally study the more common source morphologies to illustrate in practice the theory.
a) Point source:
Many astronomical sources are sufficiently distant and isolated to be considered as point-like. Thus the brightness distribution of a source with the coordinates (ao,bo) relative to the phase reference is simple: Fo*d(a-ao)*d(b-bo) that enables writing the total collected power:
is given by the complex visibility:
implying the following value for the normalized visibility: n = exp(-i2p[aou+bov]) (a pure phase).
With it, we can compute the interferometer detected power:
or more simply:
I = 2Io(1+cos(k(Ds.B)))
where we have identified ao,bo to Ds and set d = 0.
b) Uniform Disk:
With an high angular resolution, as offered by the VLTI, we can resolve some sources. One of the most important subjects of their class are nearby stars. A reasonable approximation to the brightness distribution of a resolved star is the model of an uniform disk. In polar coordinates, we can write the model as: F(r) = Fo (r<q/2) , where r is an angular offset on the celestrial sphere from the center of the source and q the diameter of the source. Knowing it we determine the total collected power: Io = Ao*Fo*p(q^2)/4 (the crossing area of the source is known).
Then comes the complex visibility:
To compute the visibility for the disk we need to consider the two-dimensional Fourier tranform of an axisymetric function, let's take f=f(r) to evaluate:
with a = rcos q and b = rsin q
To perform the estimation, it's convenient to take: u=nrcosF and v=nrsinF and to work with the phases and the amplitudes (nr is the radial spatial frequency). We write F again with these notations:
F is by construction an axisymmetric function, we can take F = 0 without losing the generality. The q -integral can be performed with:
where Jo is the zeroth-order Bessel function. This allows us finally to write:
Back to the uniform disk, the visibility can be written:
N.B: 
,
where the Ji (i = 0 to 1) are the Bessel functions.
Recall nr is a radial spatial frequency: nr^2 = u^2 + v^2 = (B^/l)^2, making the visibility:
The normalised visibility is also:
Because of the noise properties, interferometers typically measure the squared normalised visibility, and the squared normalised visibility amplitude is: n(B^,l,q)^2 = [2J1(pqB^/l)/(pqB^/l)]^2
c) Multiple stellar systems:
Most ot the stars often belong to a multiple star system. To study such systems we can consider the response of the interferometers as a collection of quasi uniform stellar disks. The total received power is the sum of the powers of the visibles sources: Po = S(j) Pj . As the Fourier transform is linear we can compute the complex visibility as the sum of the complex visibilities of the constituent disks af the positions (aj,bj)
and have directly the expression of the normalised visibility:
e.g. The best known example concerns binaries (cf picture below).
The normalised visility is written:
with r = I2 / I1, Da= a2 - a1, Db=b2 - b1
when the observable is nbinary^2 ,
|
with sbinary = (Da,Db) noticed that these corrections are due to finite bandwidth effects when B.sbinary is more that a few fringe spacings. The exacts forms of these corrections depend on the fringe tracking process. In the limit of a point-like source, the previsou expression simplifies to:
When sbinary -> 0 , the two components of the binary system are not resolved by the interferometer. With increasing sbinary, B.sbinary-> l/4, nbinary^2 -> 1/2 and the binary system becomes resolved by the interferometer. Finally, nbinary^2 the modulus of the visibility exhibits sinusoidal variations as a function of B.sbinary varying..
|
|
|
|||
|
|||
