### Comparing Versions of Numeric NULL

For a paraboloid tested at its COC with the test surface image onto the detector the OPD measured in the pupil has the following amplitudes for the primary and secondary spherical aberration Wyant Zernike terms:

C8 = (D/lambda)*(1/(3072*(F/D)^3) - 1/(7*2^20*(F/D)^7))

C15 = (D/lambda)*(1/(5*2^22*(F/D)^7))

NOTES

1) These values were calculated using a reference sphere centred at the paraxial CoC and a radius of curvature equal to the paraxial RoC of the test surface

2) These values do not correspond to an interferogram with a Zernike defocus term (C3) amplitude of zero.

3) The amplitudes of the various Zernike terms depends on the amount of defocus used in obtaining the interferogram. This effect is particularly important for highly aspheric wavefronts and large amounts of defocus.

D is diameter of parabolic mirror, F is focal length, lambda is wavelength of light. They can be in any unit but must be in the same unit e.g. meters.

For anyone implementing this who wants to test their code (or spreadsheet formula) here is a sample using mm. For a 10 inch F/4 mirror tested with a 650nm laser: D=254, F=1016, lambda=0.00065, C8 then becomes 1.987552 waves (with only first term of formula C8 is 1.987555) and C15 is 8.8E-9 waves.

For a conicoidal test surface with conic constant k the corresponding null terms are

C8 = -k*(D/lambda)*(1/(3072*(F/D)^3) - (1+k)/(65536*(F/D)^5))

C15 = -k*(1+k)*(D/lambda)*(1/(327680*(F/D)^5))