**
EXAMPLE**: Zernike wavefront analysis of a 6"
f/8 sphere at 0.25° off-axis, for 550nm wavelength. OSLO (which, as mentioned, uses Wyant's
expansion scheme for term notation) gives the following non-zero values
in the first four decimals (omitting the zero term, piston, which is not
an error term with a single aperture): **Z**1=0.008024
for the tilt, **
Z**3=0.007947
for defocus, **
Z**4=0.041248
for the lower-order (primary)
astigmatism, **Z**6=-0.196736
for lower-order coma,
**Z**8=0.175973
for lower-order spherical,
**Z**13=-0.000245
for higher-order (secondary)
coma, and **Z**15=0.000181
for the secondary spherical
(as mentioned, this coefficient is for the balanced form of secondary
spherical aberration, or 6th/4th order higher spherical aberration; pure
unbalanced secondary
spherical will be quantified mostly through the balanced primary
spherical coefficient - **Z**8
- with a smaller residual in **Z**15,
because its wavefront deformation more closely resembles that of the
former).

Multiplying the coefficients with the corresponding unit variance RMS values from
TABLE 5
(column 6, with the unit variance RMS being 1/N) gives the RMS wavefront error (in units of the wavelength) as **Z**1 /2
or 0.004012 for the tilt, **Z**3/**√**3
or 0.004588 for defocus, **Z**4/**√**6
or 0.016839 for primary astigmatism, **Z**6/**√**8
or 0.069557 for primary coma, **Z**8/**√**5
or 0.078698 for primary spherical, **Z**13/**√**8
or 0.000087 for secondary coma, and **Z**15/**√**7
or 0.000068 for secondary spherical. The RMS values for lower-order
spherical, astigmatism and coma agree with those obtained with standard
calculation, given with Eq.
68.2,
70.1 and
71.1,
respectively. The combined RMS OPD error on the bottom is a square root
of the sum of all individual RMS wavefront (OPD) errors squared.

The secondary spherical RMS value does not correspond
to the actual magnitude of this aberration form in a 6" f/8 sphere. As
mentioned, the corresponding Zernike coefficient refers to the fully
balanced 6th order spherical aberration, which combines pure 6th order
spherical, pure 4th order spherical and defocus in order to minimize the
aberration (elsewhere in this site it is referred to as balanced 6th/4th
order spherical aberration). The P-V error of pure 6th order spherical
aberration for mirror surface at paraxial focus is given by Ws6p=(K+1)2d6/8R5,
so its magnitude in a 6" f/8 sphere is 0.000000283mm, or 0.000515 in units
of 550nm wavelength. Its RMS wavefront error is smaller by a factor of
~0.3, or ~0.000155. When balanced with defocus, the RMS value of the
aberration is minimized by a factor of ~0.4, to ~0.000062 (if now combined
with a similar amount of 4th order spherical, the RMS reduces to
~0.00001 wave of fully minimized 6th order spherical). Since the above
set of Zernike coefficients does not contain one corresponding to
balanced 6th order aberration, it is substituted with a similar RMS
value of balanced 6th/4th order spherical aberration, Since the latter
is a product of similar amounts of 6th and 4th order aberrations, each
about six times greater in their RMS magnitude than that of the fully
minimized 6th order spherical, the calculation adds as
much to the mirror's 4th order component. This, in effect, produces a
combination of wavefront forms that corresponds to the mirror's inherent
ratio of 4th and 6th order components, resulting in the identical
combined wavefront.

That implies that the above conversion relations for
spherical aberration will give accurate result for the lower-order only
when the higher order term is comparatively negligible, and vice versa,
which is usually the case in amateur telescopes (similar limitations may
apply to other aberrations as well). Also, Zernike terms for other
aberrations need to be comparatively low; if they are not, it indicates
asymmetrical wavefront deformation, with the overall RMS value possibly significantly
different from that for the spherical aberration component alone.

In addition, the above conversions are valid for clear aperture. In the presence of
central obstruction, the RMS error changes somewhat, depending on the
relative size of obstruction *o* and type of aberration, as
noted earlier. Both, Zernike aberration term and expansion coefficient
should change in proportion. For instance, with 0.25D c. obstruction (*o*=0.25),
all three, the RMS error, **Z**S
and **z**S, are reduced by a (1-*o*2)2=0.88
factor.

Neglecting
higher-order terms, the actual wavefront of a 6" f/8 sphere 0.25° off
axis (leftmost) can be decomposed to its two most significant Zernike terms: (1) primary
spherical aberration term, with the wavefront map (middle) also
including negligible defocus term (it has reversed peak compared to the
pattern above because it shows undercorrection), and
(2) primary coma term, whose wavefront map is slightly altered by low primary astigmatism and tilt
components (right).

Note that the Zernike coefficients given by OSLO were for 550nm
wavelength. Their values change in inverse proportion to the
wavelength, hence these same values would imply larger error with longer
wavelengths, and vice versa. Same as with the standard P-V and RMS wavefront errors,
determining specific error level from any given set of Zernike coefficients requires
knowledge of the wavelength for which they were calculated.

Another example of interpreting Zernike coefficients is given with
wavefront analysis of an off-axis segment
mirror.