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▪ CONTENTS
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4.5. Image distortion
▐
4.7. Chromatic aberration
► 4.6. IMAGE Field curvature
PAGE HIGHLIGHTS Ideally, every point imaged by a telescope objective would be contained in the focal plane. More often than not, this is not the case. Most telescope types form images over a curved surface symmetrical around the optical axis. Radius of this surface, which can be approximated as spherical, is usually called "field curvature". In paraxial approximation, with zero astigmatism, an optical surface forms curved image surface of a radius given by:
with n and n' being the index of incidence and reflection/refraction, respectively, and R the optical surface radius of curvature (FIG. 60). It is called Petzval radius of curvature, or Petzval surface, in honor of Joseph Petzval, Hungarian mathematics professor from the 19th century who was the first to analyze it. For mirror surface oriented to the left, n=1, n'=-1 and RP=R/2; this means that Petzval surface for a concave mirror is concentric with its surface, with the radius numerically equal to its focal length. In a multi-surface optical system, the curvatures induced by each surface combine into a final system's Petzval surface RPS, that can be either curved or flat. The curvatures generated by each surface simply add up as:
nf being the
final index
of reflection/refraction at the last surface. For instance, a bi-convex lens with R1=1000mm,
R2=-1000mm and n=1.5,
thus n1=1,
n1'=1.5
and n2=1.5,
n2'=1
for the front and rear surface, respectively (for light traveling from
left to right), would have Petzval radius of curvature RP=-1500mm.
As the two relations imply, Petzval radius is independent of object and
image distance.
For a single lens, Eq. 28.1 can be simplified to:
RP
= -nf
(28.2) where n is the
glass refractive and
f
the lens focal length.
For a doublet objective, Petzval radius of curvature is given by:
with 1 and 2 marking the front and rear
lens, respectively. With
astigmatism corrected it is the actual field curvature.
The presence of astigmatism in a doublet will make the best image surface more strongly
curved than the Petzval. This best - or median - image surface is
formed by best astigmatic foci, midway between the astigmatic line foci.
As mentioned, if the objective is an aplanat (w/corrected spherical
aberration and coma), its astigmatism is not influenced by the position
of aperture stop, and so is not its actual field curvature. Rough but
useful approximation is that best image curvature of a typical contact
doublet objective is ~1/3 of
the focal length.
Petzval curvature for a pair of mirrors
is given by:
with R1/2
being the surface radius of curvature for the first and second mirror,
respectively. Such a pair is likely to have a fair amount of
astigmatism, resulting in a significantly stronger best (median) image
surface. This holds true for a single concave mirror as well. However,
while change in the aperture stop position
affects two-mirror systems little in regard to astigmatism, it is
entirely different story with a single mirror. It is already mentioned in
2.3. Astigmatism how it changes for a concave
mirror with the aperture stop position. There are consequences of it for
the actual field
curvature as well.
For
mirror conics of 0 and smaller, the stop position for zero astigmatism, and best field curvature
coinciding with the Petzval, is given by σ=[1-√|K|]/(K+1),
with α being the stop separation in units of the mirror
radius of curvature, and K
the mirror conic. It gives σ=1 for a sphere (K=0), and σ=0.5
for a paraboloid (K=-1, which makes the relation formally undefined, but
the value of σ
is found for K approaching -1). The field is astigmatic but the best surface is flat for the stop
position σ=[1-√0.5(1-K)]/(1+K),
obtained by setting the right site of Eq. 39 to zero. It is
also undefined for K=-1 (paraboloid), but as K approaches -1,
σ approaches 0.25.
Evidently, the field can't be formally flattened for K>1, or K<-1. For
these conic ranges, median field curvature is only minimized with this
stop position (for all practical purposes, it is flat for K of about -1.5
and greater, as well as for K~1.1 and smaller).
Finally, best image surface curvature for the stop at the focal plane is given
by Rm=R/(1-K),
R being the mirror radius of curvature. As mentioned, for
K<0 this is also the stop position cancelling astigmatism, so the best
surface coincides with the
Petzval surface. As long as the image is
observed on the Petzval surface, there is no optical path difference
between the chief ray and the other rays, and field curvature doesn't cause image degradation.
The wavefront
is still spherical, thus the effect is merely displacing the point image
from the focal plane. The displacement very seldom becomes as large as to affect
visual observing by exceeding accommodation power of the eye. But it
becomes a source of image degradation with flat
detectors in imaging, when the longitudinal point image displacement
results in defocus.
From Eq. 24, peak
aberration coefficient (the P-V error) for defocus is given by P=Ld/8F2.
In the case of field curvature, the longitudinal aberration Ld
closely approximates the depth z of the image surface curvature.
For the Petzval surface of the radius RP
at the height h in the image plane, it is given by z=h2/2RP.
Thus, the P-V wavefront error of defocus caused by field curvature with
respect to the focal plane is given
by:
with ρ being, as before, the ray height in the pupil for pupil radius
normalized to 1. Note that defocus
error caused by field curvature normally doesn't affect visual
observing, due to eye accommodation. In the presence of astigmatism,
point-images on best (median) image curvature combine the above defocus
error with the error of astigmatism; in effect, defocus error is
measured from best astigmatic focus to the image plane.
As
mentioned, the presence of astigmatism significantly alters the image field curvature
properties. The Petzval surface becomes fictitious, and the actual best image
surface becomes the one containing best astigmatic foci. Due to the
longitudinal extension of astigmatism, this image is split into
sagittal,
tangential and best, or median image surface
sandwiched in between the first two. Relation between the Petzval on one side, and sagittal/tangential/median field curvature on the other is constant,
as given by:
with RP' Rs, Rt
and Rm
being the Petzval, sagittal,
tangential and median field curvature, respectively. Sagittal surface is
always between the Petzval and tangential, with the tangential surface
being three times farther away from the Petzval
than sagittal (FIG. 81).
FIGURE 61: Illustration of image
field curvatures of a concave mirror with the stop at the
surface: tangential (T), median (M),
sagittal (S) and Petzval (P) image surface are
in a constant relationship with respect to the image plane, and to
each other. Best, or median
image surface contains best astigmatic foci (i.e. circles of least
confusion). For a concave mirror with the stop at the surface, sagittal surface is
flat, coinciding with the Gaussian (paraxial) image plane.
Tangential (vertical) and sagittal planes are perpendicular to each
other, and the plane in which rays focus at the median surface
is midway between them. Both, longitudinal astigmatism and field
curvature increase with the square of the field angle
α. All image surfaces are paraboloidal. This implies that a system with given longitudinal defocus due to
its Petzval curvature will have flat median astigmatic surface when
the longitudinal astigmatism equals longitudinal Petzval defocus.
Field flattening by introduction of astigmatism is often used in
optical systems. Since for given longitudinal aberration astigmatism
produces half as large blur as defocus, with its RMS error
smaller by a factor of 0.68, there is substantial gain in the
quality over flat detectors. For a
singlet lens, sagittal and tangential image curvature, respectively, are
given by:
f being the singlet's focal length
and n the refractive index.
Obviously, median curvature is (1/Rm)=(1+2n)/nf. Since astigmatism doesn't change with object distance, the
image surface radii scale with the image distance (as mentioned, astigmatism in aplanatic doublets
also doesn't change with the stop position).
For a doublet, the respective image curvatures are simply a sum for the
two lenses:
1/RsD=[(1+n1)/n1f1]+[(1+n2)/n2f2], 1/RtD=[(1+3n1)/n1f1]+[(1+3n2)/n2f2], and
1/RmD=[(1+2n1)/n1f1]+[(1+2n2)/n2f2].
Note that the focal length of a negative
lens is numerically negative for incident light traveling left to right.
For a mirror
of radius of curvature
R with the stop at the surface, sagittal and tangential image curvatures,
respectively, are given by:
which, with RP=R/2,
can be written as 1/Rs=0
(implying Rs=
∞), and 1/Rt=-4/R.
The median image surface, given by Eq. 33 as one half of the sum
of sagittal and tangential curvatures, is 1/Rm=-2/R,
equaling mirror focal length (as absolute value; the plus sign implies
that it is concave toward mirror, for mirror oriented to the left). In
other words, the median surface has the same curvature radius as the
Petzval, but of opposite sign.
Similarly to the doublet objective, astigmatic curvatures for a pair of
mirrors are given by a sum of the respective curvatures for each mirror.
With the aperture stop axially
displaced from the mirror to a distance σ
in units of the mirror radius of curvature, the astigmatic image
curvatures become:
with K being the mirror conic,
and 2/R=1/RP.
For a pair of mirrors, the
curvatures are combined: the sum of Petzval curvatures results in a
system Petzval, which is then used to obtain sagittal and tangential
curvatures. The former from
1/Rs=(1/RP)+2as/n'α2,
for
the system astigmatism coefficient (primary oriented to the left),
as=(α2/R1)+α2[(K2/R2)+(1-σ2)2]/R2,
with α,
K2,
σ2,
R2
and n' being the field angle, secondary mirror conic, relative stop
separation, radius of curvature and index of reflection at the final
surface, respectively. For the latter, from
1/Rt=(1/RP)+6as/n'α2.
For the secondary, the relative stop distance
σ2 is
the primary-to-secondary separation (since the primary acts as an
aperture stop - or entrance pupil - for the secondary) in units of the
primary mirror radius of curvature.
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FIGURE 60: Illustration of
the Petzval field surface (Ps)
in lens objective (left) and a concave mirror (right). Actual
objectives usually do suffer from some astigmatism, resulting in altered best image surface. Note that the two curvatures, although
of the same sign, are oriented differently with respect to the
eyepiece. Even with the identical Petzval and astigmatism in both, they
would still have field curvatures of different properties. In other
words, the eyepiece optimized for one system, wouldn't be optimized
for the other one.
