Re: Mirror's Zerikes at Angle

1/
The mirror shape can be determined by using the Zygo at normal
incidence.
You'll need to ensure the shape is measured over an aperture at least
as big as the elliptical footprint of the intended oblique beam.
Depending on the complexity of the shape, it may be accurately
described by a few low order Zernikes, otherwise you'll need to
consider whether you need to store the wavefront pixel data over the
mirror surface.

So, assuming the mirror shape is not too complex, you will have a set
of Zernike coefficients from the Zygo, given in whatever units are
convenient. You might prefer radians, or waves, or nm. It doesn't
matter as long as you are consistent.

2/
In my formulae, Zernike[n, r, theta] represents the nth Zernike
polynomial.
It does not represent the coefficient of that Zerniike in a wavefront
or surface.

The inner product integral gives the coefficient of Z_n in the
mirrorShape, relative to the incident oblique beam:

a[n] = Integrate[ mirrorShape[R(r, theta, d), Theta(r, theta, d)]
Zernike[n,
r, theta] r dr dtheta]

You need to work out the required coordinate transformation functions R
and Theta (or let ZEMAX do this---see below).

Alternatively, just use the Zygo to measure the mirror at the oblique
incident angle of the intended beam in your system!

3/
I included the term "d" just to represent the need to take it into
account.
I was thinking of it as a ratio as you suggest.

By the way, my tools of choice for doing this work would be ZEMAX
and/or Mathematica. ZEMAX would be more direct since it has all the
required functionality built in. With Mathematica, you'd need to use
something like my Zernike package, available at MathSource.

Good luck!

Regards,
Brett

.

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