The ray goes along line A (blue),hits the dish, reflects back along ray B (magenta), and then intersects the line feed a distance y (red line) from the paraxial point (R/2 feet below the center of curvature). The distance P=A+B+y must be a constant for all values of x along the plane wave. Remember that there is an axis of symmetry so x needs to be rotated (or integrated) around a ring of 360 degrees.

A summary of the computations are:

- A=Rcos(za)
- B=R/(2cos(za))
- y=R/2 * (1./cos(za) - 1)
- zamax=asin(500./870(radius)) = 35 degrees.

- Fig top. Intersection of linefeed at position y versus zenith angle za.
- Fig 2nd. Intersection of linefeed at position y versus horizontal distance x from center of curvature.
- Fig 3rd. The integrated phase needed from a position y on the linefeed to y=0 so that a plane wave is brought to a focus at the paraxial surface. This distance is P-(A+B) (see above image). A ray leaving the center of curvature travels P=R+R/2 so P=1.5R. The plot has y versus (1.5R-(A+B)). This value is less than the physical distance, so the phase velocity of the wave guide must be greater than c.
- Fig Bottom. The 3rd figure was the integral of the phase from the y intersection of the ray to y=0. This can be written as

differentiating this with respect to y gives the c/waveGuide velocity along the wave guide. This determines the taper you need on the wave guide to make things work. The taper starts with v=c at the top and goes down to about .4 (velWg=2.5c).

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