Interferometry for pedestrians - a very brief introduction and basic concepts 1

Transcription

1 Interferometry for pedestrians - a very brief introduction and basic concepts 1 Erik Bertram 1 Article concerning the lecture Perspektiven Moderner Astrophysik, Lecturer: Priv.- Doz. Dr. Silke Britzen

2 1 Introduction During the last tens of years, interferometry has become a very powerful and important tool for observations with high angular resolution. The basic physical effect which is used by an interferometric measurement is interference. As we know the size of the telesope sets the angular resolution of our images. That means that because there is a practical limit for building bigger telescopes we are limited in our resolution for a given diameter. This is where the concept of interferometry sets in. The principle of interferometry in general is very simple: collect light by more than one telescope and let it interfere as if the telescopes were one big telescope mirror. As one might expect and as we will se later, this is of course a very naive way of looking at interferometry. The devil is in the details. (In principal one can say that every telescope is an interferometer because every lens collects light and focusses it on a focal plane where interference occurs. But normally we only talk about interferometry if light of more than one telescope is interfered in a so called interferometer ) Today interferometry is widly used in many fields of astronomical research, e.g. in radio and infrared astronomy. Well known telescopes for interferometric measurements are the VLTI ( Very Large Telescope Interferometer ) at Paranal-Observatory in Chile, the VLBA ( Very Long Baseline Array ) in the US or the VLA ( Very Large Array ) in New Mexico. With an interferometer called LIGO ( Laser Interferometer Gravitational Wave Observatory ) one also tries to use the effects of interference to directly prove the existence of gravitational waves. [1] To understand the basic concepts of interferometry we first have to deal with the coherence of an electromagnetic wave in space and time. 2 Coherence in space and time Let us first, as an example, consider waves at a beach. Although on the ocean there are a lot of smaller waves which seems to move into every direction (e.g. orthogonal and perpendicular to the beach line) we observe nice big waves at the beach. How is that possible? The waves on the ocean are of course far away from beeing perfectly correlated and coherent. There may be for example the case where waves interfere destructively and cancel each other, so that we would never observe the situation of perfect waves at the beach. 1

3 The answer lies in Poisson statistics. Let us assume that each wave on the ocean can be described by a complex vector with length L. If very much waves, e.g. N ones, with such a vector fall together statistically then the average length of the resulting vector at the beach (in analogy to the electric field vector) is E = NA and the intensity I = EE = NA 2. As we can see, at the beach we get a coherent wave due to poisson statistics. The intensity is proportional to the number of individual waves. [2] This is, as one might expect, not always the case. It can be that the individual phases break out and destroy the original coherent wave. It seems that the wave looses memory about it s original shape. This effect is called memory loss and concerns the stability of the phase of the wave. It can be that, e.g. when we observe some light from a star, the wave becomes instable and looses it s shape after a time τ c (the critial time after which memory loss occurs). This can be translated into a critical spatial distance s = c τ c, where c is the speed of light. Over such a distance s we can assume phase stability of our wave. Mathematically this concept of coherence can be described by an autocorrelation function Γ AB [3] and is defined as: Γ AB (τ) = E( r A, t)e( r B, t + τ) = + E( r A, t)e ( r B, t + τ)dt (1) This (complex) integral compares two wave amplitudes with each other at given spatial positions r A and r B at time τ. Furthermore we can define a contrast function: C AB (τ) = 2Γ AB (τ) Γ AA (0) + Γ BB (0) (2) In a special case where we measure at the same positions r A = r B we get C(τ) = Γ(τ) Γ(0), (3) which is a normalized function which reaches values between 0 and 1. In general we distinguish between three different cases: if C = 1 we have completely coherent waves, if C = 0 the waves are totally incoherent. For 0 < C < 1 we have partial coherence (or incoherence...). With this mathematical tool we can compute the degree of coherence of a wave. 2

4 3 Van-Cittert-Zernike theorem We will now discuss an easy and fast way of understanding the Van-Cittert- Zernike theorem [4], which is the basic theorem for doing interferometry. Therefore let us assume a celestial source at position P 1(x 1, y 1, z 1) which emits radiation, that can be described by an electric field ɛ(p 1). We observe this source from a position P 1 (x 1, y 1, z 1 ), which is shown in the following sketch: The same for another source at P 2(x 2, y 2, z 2) which is observed from Fig. 1: Illustration of the coordinates used to derive the Van-Cittert-Zernike theorem. a position P 2 (x 2, y 2, z 2 ). At our position P 1 (x 1, y 1, z 1 ) we measure an electric field E which is given by E(P 1 ) = ɛ(p 1) e ikd(p 1,P 1 ) D(P 1, P 1) dω 1, (4) where D(P 1, P 1) is the distance between the two points P 1 and P 1 and k is the wavenumber. We get the same result for the other two points P 2 and P 2. Let us now compute the crosscorrelation function between those two electric fields at the same time: E(P 1 )E (P 2 ) = ɛ(p 1)ɛ (P 2) e ik[d(p 1,P 1 ) D(P 2,P 2 )] D(P 1, P 1)D(P 2, P 2) dω 1dΩ 2 (5) We now assume that the emission from the source is spatially incoherent, such that ɛ(p 1)ɛ(P 2) = 0 for every position P 1 and P 2, except when P 1 = P 2. 3

5 Performing the integration yields V = E(P 1 )E (P 2 ) = I(P 1) e ik[d(p 1,P 1 ) D(P 2,P 1 )] D(P 1, P 1)D(P 2, P 1) dω 1 (6) Here, V is called the complex visibility and I(P 1) = ɛ(p 1)ɛ (P 1) is the intensity of our source. We now approximate D(P 1, P 1) D(P 2, P 1) := d (for a source very far away) in the denominator and define dω 1 = d 2 dω n. Putting all together we end up with V = I n (P 1)e ik[dn(p 1,P 1 ) Dn(P 2,P 1 )] dω n, (7) where the D n are new quantities that depend somehow on a new set of coordinates (which are unimportant for us for the moment because we just want to show a more general dependency). That means: the correlation function between the electric field vectors of two points (e.g. two telescopes) is connected to the intensity of the source (e.g. the image on the sky). It is it s Fourier-transform! This is called the Van-Cittert-Zernike theorem, after Pieter Hendrik van Cittert ( 1889) and Frits Zernike ( ). Now we know that we can reconstruct an image on the sky by measuring the correlation between signals. The most important question for us right now is: How can we meaure correlations of signals? 4 Measuring correlation functions Measurements of the correlation function between two signals reaching for example two telescopes are done with delay lines. Imagine that our telescopes do not lie exactly on a plane perpendicular to the source where the signals reach each telescope at the same time, but the baseline of the telescopes is inclinated with respect to this plane by an angle θ, as it is shown in Figure 2 (that is of course the normal situation when two telescopes stand next to each other on the ground). To get interference we have to delay the right signal by D sin(θ) (see Figure 2) such that there is no phase shift between the two signals anymore. This can be done e.g. by introducting an artificial (longer) way that one signal has to travel to observe interference. One big problem is that the earth rotates. That means that the phases of the signals change with changing spatial positions of the telescopes with respect to the source. To solve this problem one has to introduce movable 4

6 Fig. 2: The baseline D has to be projected with an angle θ such that both telescopes would lie on a plane perpendicular to the source on the sky. This is called the projected baseline. (Picture taken from 08/11/2011) delay lines, e.g. by introducing different movable mirrors in between, which can be adjusted with the earth s rotation (Figure 3). These mirrors have to be adjusted very precisely on a scale of about 1µm, since the delay path has to be quite accurate. Another problem lies in the atmospheric turbulence, which induces fluctuations of the order of milliseconds. The question now is: How can we get good images of the sky? And what limits the image quality? The answer is, that we need a lot of different correlation points (telescopes) to reproduce the image on the sky. How one can see in Figure 4, the telescopes seem to be distributed randomly, with a few telescopes being very near and other telescopes being far away from each other. Assume that we have N telescopes in total for our observation. Then we can compute the correlation function between every possible N(N 1)/2 combinations of telescopes that can measure the signal s correlation from different positions. The more telescopes and thus the more correlation points we have, the better one can reproduce the image at the end. That also means that for very short exposures we need more telescopes and thus more correlation points to get better images. The longer we expose with an telescope-array 5

7 Fig. 3: Underground delay lines of the VLT. (Picture taken from 08/11/2011) Fig. 4: ALMA-telescope configuration: For nice pictures of the sky one needs a lot of different correlation points. (Picture taken from 08/11/2011) 6

8 Fig. 5: An example for VLBI: two telescopes are one earth diameter away from each other. (Picture taken from 08/11/2011) configuration like ALMA the more correlation points we get due to the rotation of the earth, which changes the relative positions of our telescopes and the source. 5 Very Long Baseline Interferometry (VLBI) We already discussed that we can obtain a better angular resolution if we put our telescopes far away because α λ/d, where λ is the wavelength in which we observe and D the distance between the telescopes. We could now also think of a situation where we use the maximum distance on earth that is possible, e.g. when both telescopes are located exactly one earth diameter away from each other (Figure 5). In this situation it is of course impossible to construct delay lines and to interfere two signals in-situ. Therefore the signals need to be saved with a computer with very accurate time references. The cross correlation is then computed afterwards. In this way it is possible to do interferometry over very large distances! Another very useful application of the VLBI is to measure the intercontinental drift on the surface of the earth. Since the interferometric measurements are very exact and have to be done carefully, also very small shifts of the continents with respect to each other become measurable. This is only possible because reference stars, that are very far away from us, seem to be pointlike and static so that we can easily observe small deviations in the distance of the continents. For several years it is possible to measure drifts in the cm and mm regime. The result: the continents move with a speed of 2-20cm per year. [5] 7

9 Literature [1] A. Unsöld, B. Baschek, Der neue Kosmos, 2002 [2] C. P. Dullemond, Interferometry: The basic principles (from his lecture Observational Astronomy, SS 2011) [3] Wikipedia, 08/11/2011 [4] GMRT, 08/11/2011 [5] Wikipedia, 08/11/2011 8