Notes on focal plane arrays

Some notes on the usable size of the focal plane of a telescope and the number of feeds that can be packed in. 
  - let f be the effective focal length.
        i.e, if Cassegrain or Gregorian, the effective focal length of 
          the combined optics.
        e.g., for the GBT at Gregorian, f=190m; at prime focus f=60m.

  - h is the displacement in the focal plane from the axis of symmetry.
       (in cm or mm, e.g.)

  - θ is the beam deviation from the boresight corresponding to 
      the displacement h.
  - Thus: θ = h/f  in radians.

  - let hmax be the radius of the usable area of the focal plane,
        i.e, for  h < hmax optical abberations are acceptably small.
In a paper by Padman (ASP conf ser vol75, 1995), optical aberrations are given by a quantity Φ the wave-front error, which is a series of terms of the form:
   Φi = hq F(f,Dp,Ds,etc)

   where i represents the type of aberration; there is a term for coma, a term
   for curvature, etc.  h is raised to the power q, 
   F is a combination of focal length f, Diameters of the primary and secondary
   reflectors, and perhaps other terms, depending only on the geometry and optics
   of the telescope.
   
   For example, the Coma term is

   Φcoma = h (Dp/f)3 /32    (Dp is the diameter of the primary mirror)

 - So if you set Φ to a criterion for the allowable aberration, 
   for example λ/20 or whatever, then you can solve for hmax

 - in general lets say the criterion is Φ <= λ/G

 So for a particular aberration term we would write:
   Φ = λ/G = hq F(...)
The usable radius, h, of the field of view, given a criterion that the wavefront errors must be less than λ/G would be: 
    hmax = [ λ /(G * F) ](1/q)

The corresponding beam offset is θ = hmax/f

The HPBW θHPBW = λ/Dp

So the size of the field of view in terms of the HPBW is:
   (θ/θHPBW) = (hmax/f) * Dp/λ

Substituting the expression for hmax gives: 
(θ/θHPBW) = (Dp/f) λ1/q-1 (1/GF)1/q

If we just think about the dependence on λ then we can just write that 

  (θ/θHPBW)  ∝ λ1/q-1

Or in terms of frequency ν : (θ/θHPBW)  ∝ ν1-(1/q)


The maximum number of feeds that can be packed in the field of view is proportional to :
    Nfeeds = (θ/θHPBW)2 ∝ ν2(1-(1/q))
So, given an aberration with a dependence on hq then the number of feeds is proportional to observing frequency to the power 2( 1-(1/q) )