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4.8.2.
Measuring chromatic error in an achromat: polychromatic PSF
Summing it up, an achromat optimized for a particular wavelength, will have spherical aberration canceled for
that wavelength, and chromatic aberration nearly cancelled
laterally, while reduced to nearly ƒ/2000 of the F/C secondary
spectrum. The error at the optimum focus results from other wavelengths
being:
(1) defocused, and (2) affected by spherical aberration, with the
latter being comparatively low or negligible. The main error component,
that of chromatic defocus, can be expressed as a P-V wavefront error:
W= Pr2
(51)
with P=-Dƒ/8F2 being the peak aberration coefficient, equal to the
P-V wavefront error,
and r
the pupil ray height in units of the pupil radius. This error combines
with the error of spherochromatism for that particular wavelength, and
the combined error is finally "measured up" by the
sensitivity factor of
the eye.
With
Df
being, for typical achromats, ~f/2000 at best, the P-V wavefront error
of chromatic defocus for an achromat can be written as W~D/16,000F at
its red and blue foci. For D=100mm and F=10, this gives 0.000625mm, or
1.29 wave P-V of defocus for the blue F-line (λ=0.000486mm), and 0.95
wave P-V of defocus for the red C-line (λ=0.000656mm).
For film/CCD applications, defocused
wavelengths are more to much more detrimental, depending on both,
characteristics of the chromatic defocus and spectral sensitivity of the
detector. For instance, most achromats have defocus error significantly
greater toward the blue/violet end, which would seriously impair
performance with a detector with high sensitivity for that range.
However, if the detector is relatively insensitive in the blue/violet, a
decent to good results can be achieved even with relatively fast
achromats, with significant gross chromatic errors.
However, defocus error is not an accurate
indicator of the level of chromatism, in the sense that they change at a
different rate with the change in either aperture D or focal
ratio number F. For instance, while the defocus error at any
farther (from the optimized) wavelength doubles with either doubling
D or halving F, resulting chromatism - measured by the drop
in peak diffraction intensity - will increase less than half as much.
The reason for this "strange" behavior is that much of defocused light is already out of the Airy
disc, and merely gets spread out wider (for instance, F and C line in the 4"
f/10 above have only a few percent of the energy left within the Airy
disc). The spectral range around the
optimized wavelength also doesn't contribute "new" lost energy, since it
is not affected. It is only a pair of spectral segments on either side
of the optimized wavelength which previously had relatively small amount
of energy out, that adds significant new energy to the outside of the
Airy disc.
For the same reason, chromatism level in
an achromat diminishes slower than the reduction in D, or
increase in F. It is well approximated with the square root of
the change in either aperture D, or focal ratio 1/F.
Hence, doubling the aperture or focal ratio will increase the chromatism
little over 40%, while halving it would result in about 30% reduction.
The graphs bellow (FIG. 33) show how polychromatic PSF (PPSF) peak - and
with it the effect of secondary spectrum on image quality - changes with
the change in F and D (solid gray curve in
both is how the PPSF would change if the chromatism would change in
proportion to F - for D=4" - and D, for F=12). It is calculated by OSLO based on 25 wavelengths
from 440nm to 680nm (each 10nm), weighted for the photopic eye
sensitivity, for a typical C-e-F Fraunhofer doublet (Dƒ~f/1920).
FIGURE 33: Change of polychromatic peak diffraction intensity
(PPSF) in a Fraunhofer-type doublet achromat as a function of aperture size and
relative aperture. While there is no simple expression, the change in
PPSF is approximately in proportion to F1/4
and 1/D1/4.
The corresponding RMS error of monochromatic aberration is obtained from
RMS=0.24(-logPPSF)1/2.
In any instance, the detriment of chromatism with the increase in
aperture size - or reduction in relative aperture - is much smaller than
nominal increase in the defocus error (secondary spectrum) caused by
either of the two.
For a given aperture D in inches,
the peak diffraction intensity (PPSF, or polychromatic Strehl) in
function of the focal ratio number F of the achromat, is
approximated by PPSF~0.8(F/3.4D)x,
with x=Dx',
where x'=-D/y and y=3(D+10/D2)-10.75.
For given focal ratio F, polychromatic Strehl as a function of
the aperture D in inches is approximated from
Eq. 56, with the corresponding
RMS wavefront error ω~0.0745(3.4D/F)x,
with x=Fx'
where x'=-0.75√F/(F-3).
Following table shows nominal
polychromatic PSF peak.
It includes comparable amount of lower spherical aberration, and comparable central
obstruction size (not adjusted for the effect of brighter central disk).
| Achromat |
PPSF peak |
Comparable p-v S.A. |
Comparable C.O. |
| 4" f6 |
0.64 |
1/2.8 wave |
0.44D |
| 4" f/10 |
0.74 |
1/3.4 wave |
0.37D |
| 4" f/12 |
0.77 |
1/3.7 wave |
0.35D |
| 4" f/24 |
0.89 |
1/5.5 wave |
0.24D |
| 4" f/48 |
0.95 |
1/8.3 wave |
0.16D |
| 4" f/15 |
0.82 |
1/4.2 wave |
0.31D |
| 6" f/15 |
0.74 |
1/3.4 wave |
0.36D |
| 8" f/12 |
0.63 |
1/2.8 wave |
0.45D |
| 8" f/15 |
0.67 |
1/3 wave |
0.42D |
| 36" f/10.8 |
0.34 |
1/1.8 wave |
0.63D |
TABLE 1:
Approximate comparative effects of secondary spectrum vs. spherical
aberration and central obstruction
◄
4.8.1. Secondary
spectrum and spherochromatism
▐
5. INDUCED ABERRATIONS
►
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