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I Closure Phase - Interferometry - Recurrence Relation

  1. May 21, 2016 #1
    Hello,
    I'm trying to calculate a recurrence relation of the phases of 3 telescopes in a closure phase.
    Usually in a stellar interferometer we have 3 telescopes, located in a triangle, measuring intensity of light in 3 points on a far field plane. I found an article, describing how the phase is reconstructed out of the closure phase. However, in this article all 3 telescopes are located in a line, and not on a plane. I'm trying to find a similar derivation as in this article, but for case where 3 telescopes are not necessary on a straight line, but can form a triangle on plane.
    Article is here:
    https://www.osapublishing.org/ao/abstract.cfm?uri=ao-17-13-2047

    Their derivation is as following: we have 3 telescopes at points
    [tex](x_0), (x_0-\Delta x), (x_0+n \Delta x)[/tex]
    where n=...-1,0,1,..., i.e. the first two are fixed, with a distance of Δx between them, and the third one is "mapping" the line at points with a "step" of Δx.
    Their write a following equation, that satisfy the relation of measured phases around the closed loop of the 3 telescopes:
    [tex]G\left ( n \Delta x \right ) \equiv \phi \left [ \left ( n+1 \right )\Delta x \right ]-\phi\left ( n \Delta x \right )-\phi \left ( \Delta x \right )[/tex]
    where G is some value, measured for each position of 3 telescopes, and is known.
    From this, they write a solution of the above equation as difference equation (recurrence relation) for a phase at any point:
    [tex]\phi\left ( n \Delta x \right ) = \sum_{k=1}^{n-1}G\left ( k \Delta x \right )+n \phi \left ( \Delta x \right )[/tex]

    I'm trying to find a similar recurrence relation, but for a case where all 3 telescopes are not in line. Actually, from what I know from the stellar interferometry (VLA, VLBI, CHARA, etc.), generally, telescopes are not located at the same line, so there should be equation for a general case of a plane, and not a line.

    So, I start from a similar setup, where 3 telescopes are located at points:
    [tex](x_0,y_0), (x_0-\Delta x,y_0), (x_0+n \Delta x,y_0+k\Delta x)[/tex]
    i.e. again, first two are fixed in place and the third one is "mapping", but this time a plane, and not a line.

    I get the following equation for the phases around a closed loop:
    [tex]G\left ( n \Delta x, k \Delta y \right) \equiv \phi \left [ \left ( n+1 \right )\Delta x, k \Delta y \right ]-\phi\left ( n \Delta x , k \Delta y \right )-\phi \left ( \Delta x , 0 \right )[/tex]

    Now, what is a recurrence relation solution of this equation? I guess it should be something like:
    [tex]\phi\left ( n \Delta x , k \Delta y \right ) = \sum_{m=1}^{n-1} \sum_{p=2}^{k}G\left ( m \Delta x , p \Delta y \right )+n \phi \left ( \Delta x ,0 \right )+k \phi \left ( 0, \Delta y \right )[/tex]
    but I'm very unsure in this result.
    What wonders me is that in article (1D case), a phase at some point is a linear sum of a phase at some starting point and corrections G along this line. In my guess above, a phase at some point is a linear sum of a phase at some starting point and corrections G at all points in plane, not a line.
    I'm very skeptic for my solution above and will be very happy If someone could correct me. Thank you!
     
  2. jcsd
  3. May 26, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. May 26, 2016 #3
    Already found a solution. Thread can be closed.
     
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