Dr. Biretta/Group -

A careful further examination of the data in the comparison table of post # 673,
plus some investigation into how the performance numbers are calculated by the
two applications, WinRoddier2.2 and OpenFringe, has produced some further
insights into the question of comparing results between the two.

The reported Strehl ratios in the table are calculated from the value of the
normalized Zernike coefficient according to the algoritm:

Strehl ratio = exp(-(2*pi*sigma)^2) Eq. 1

where sigma is the rms value of the wavefront or surface error.

If only the single value of primary spheric is considered, sigma, the rms error,
is numerically equal to the normalized coefficient value; for WinRoddier2.2 this
is:

(Primary Spheric Zernike Coefficient)/(Reference Wavelength) Eq. 2

this value is then used in Eq. 1 to obtain the reported Strehl ratio for Primary
Spheric alone.

For WinRoddier2.2, the mathmatical methodology for obtaining the Primary Spheric
Zernike coefficient is not known.

In the case of interferometry and OpenFringe, the rms value to be used to
calculate Strehl ratio must be obtained from the reported Zernike coefficient by
a consdierably more complex calculation.

First a "software null correction" (necessary because a parabolic mirror is
being tested at its center of curvature, not is focal point) must be subtracted
from the listed Primary Spheric Zernike coefficient value:

sc = Zernike value minus Null Correction Eq. 3

Then this difference is squared and divided by a scaling factor used from
Zernike theory:

S = sc^2/(Scale Factor) Eq. 4

Next, the rms error sigma is calculated:

Sigma = Sqrt(Sigma)* (wavelength of test laser)/(Reference wavelength) Eq. 5

This final sigma value is then the one used in Eq. 1 to obtain the Strehl ratio
of Primary Spheric coefficient. In the case of the table in post # 673, the rms
value used to produce the Primary Spheric only Strehl ratio (0.983) is 0.021,
not the 0.0396 value listed on the last line. Roddier and OpenFringe then do in
fact calculate quite different values of Strehl ratio, 0.934 versus 0.983 in
this example.

From the above it is readily apparent that the Strehl values from interferometry
are obtained using a number of factors from the interferometry test (Null
correction, laser wavelengths, etc.); further, the end results are descriptors
of a surface, not a wavefront. So direct and specific comparison of Roddier and
interferometric results may not be really meaningful, as the underlying
processing methods may be different. This then makes comparing the two tests in
a quantitative way rather difficult, particularly since interferometry measures
a mirror alone while Roddier measure a complete telescope. As of now, it seems
that comparisons need to be used mostly as sanity checks, raising alerts if
differences seem grossly out of proportion; comparing numbers in the third
decimal place may not mean much.

There are probably many ways in which my interferometric measurement Strehl
ratios in the mid 0.90s (mirror) can evolve into Roddier measured results in the
low 0.80s (telescope).

Mick Hollimon
11 June 09
8:53 PM PDT USA