Mapping the rays scattered by comatic
wavefront deformation follows a simple, elegant symmetry. As illustrated on
FIG. 20,
every zone at the pupil focuses not into a point, but into a circle of
the diameter
R=hp2/22 (11)
with h
being the height in the image plane, p the zonal height in the pupil
and
the focal length. The center of
each circle is
also shifted by a length equal to the circle diameter from the Gaussian image point, in the image
plane, along the axis of aberration
(determined by the chief ray and optical axis).
FIGURE 20: Geometry of the coma
ray spot diagram, created by the rays projected from a comatic
wavefront. The spot consists entirely from circles of varying sizes,
with the exception of the very central ray, projecting into the
central point (the tip). Both, the diameter of the comatic circles formed in the
focal zone and their shift from the Gaussian focus increase with the square of the
pupil (zonal) height p, expanding around the axis of
aberration a, in the direction
of the axis. Diameter of the largest comatic circle, formed by the
very edge of the aperture, is 2/3 of the blur
length. The ray circle formed by the 0.5 (p=D/4) wavefront zone is only 1/4 of
the diameter of of the edge circle (p=D/2), and has 1/4 of its shift. Tangential coma (T)
is the entire blur length, with the sagittal coma (S=T/3)
being the "V" shaped tip oriented toward Gaussian image point. About
80% of all rays are contained within the sagittal part of the comatic blur. It is the sagittal part of the comatic image that is
actually visible as a deformation of the point-source image in a
telescope.
An interesting, and rather unique consequence
of only the mid area of the comatic wavefront focusing near the central
ray, and the rest of them being spread out radially in the image plane
is comatic image as a
whole being quite insensitive to defocus, whether axially or one caused
by image curvature.
Aberration function (i.e. the peak
wavefront error as optical path difference) for coma is given by
Wc= C(r3-2r/3)cosq
(12)
with
C being the coma peak
aberration coefficient, r
the pupil radius normalized to 1 and q
the pupil angle. Note that
Wc
represents 1/2 of the P-V wavefront error. The aberration peaks for ρ=1,
with the sign of aberration changing symmetrically on both sides of the
wavefront. The factor in brackets makes this error 1/3 of the P-V error
at paraxial focus, as a result of correction for wavefront tilt (FIG.
19, left).
On FIG. 19, the peak error on the top radius (edge point)
is positive, being farther to the center of a perfect reference sphere
than its perfect reference point, while the peak error on the bottom radius is negative.
This is so
called "positive" coma, with the top section of the wavefront flattened
with respect to the reference sphere, and the bottom portion more
curved. Its comatic tail is oriented outward, away from field center. Presence of the pupil angle factor
(cosq) indicates that the
aberration is not rotationally symmetrical. The peak aberration coefficient is given by:
C=cαd3
(13)
with c being the coma aberration
coefficient, α
the field angle (in radians) and d the pupil radius. Coma
aberration coefficient c for either refractive or
reflective surface, for object at infinity, and aperture stop at the surface is given by:
c =
-n2/4n'2 (14)
with being the focal length, and n and n' being the
index of incidence and refraction/reflection, respectively.
For
mirror surface, the coma coefficient for any object distance is given by:
c = -n(m+1)/(m-1)/42 (15)
with m being the transverse image magnification (explained with
Eq. 9.). For distant objects mg0,
n=1 for mirror oriented to the left,
and the coefficient reduces to:
c =1/42 =
1/R2 (15.1)
As the aberration function shows,
deviation from the perfect reference wavefront is largest for ρ=1 and
either cosθ = 1 or
cosθ
= -1 (i.e. θ=0
or θ=π
radians, respectively). With
q
being the position angle in the pupil, measured to the plane of the axis of
aberration, the maximum
deviation occurs at the marginal points of the wavefront in the plane of
aberration (FIG. 19,
left). The
deviation is zero for cosθ=0
(i.e. θ=π/2
or θ=3π/2),
which is the wavefront diameter orthogonal to the plane of aberration.
For
r
smaller than |~0.8|, the deviation changes the sign, as a result
of the two lobe-like deformations, illustrated on FIG. 19, left.
Due to the peak
wavefront deviations with coma affecting relatively small area, the
error averaged over the entire wavefront is smaller for coma than for
spherical aberration for any given p-v wavefront error (FIG. 21).
FIGURE 21: Coma ray aberration as a ray spot (bottom) and the actual
diffraction pattern (top). Top: from left to right, perfect
diffraction pattern of a relatively bright star, diffraction pattern
affected with
0.42 wave P-V wavefront error of coma (corresponding to 0.074 wave RMS, for 0.80
Strehl, thus comparable to 1/4 wave P-V of primary spherical
aberration), and diffraction pattern affected by 0.84 wave P-V
wavefront error of coma (~0.15 wave wavefront error RMS, comparable
to 1/2 wave P-V of spherical aberration). As the
error increases, best focus location shifts from the middle of the
Airy disc in the direction of intensity shift. Unlike spherical
aberration, there is no gain from axial defocus, which is pretty
much obvious from the shape of wavefront deformation. Bottom:
corresponding ray spot plots for identical amounts of the aberration. The "V" shaped
form obvious in the ray spot doesn't become apparent in the actual
diffraction pattern until the error exceeds ~1 wave P-V, provided
there is sufficient magnification to make it visible to the eye.
The RMS
wavefront error for coma at best focus, in terms of the peak aberration
coefficient C is:
ωc=C/√72 (16)
Shift from the Gaussian image
point to the best focus location is transverse, in the image plane and along
the axis of aberration, given by 4FC/3. That places best focus at
1/4.5 of the blur length away from the Gaussian image point (the tip).
The shift effectively
corrects for the wavefront tilt element. Unlike spherical aberration,
the peak aberration coefficient C for coma does not equal
the p-v wavefront error. At the location of best focus, the P-V
wavefront error is 2C/3, with the peak wavefront error being ąC/3.
The error is smaller by a factor
of 3 than at the location of Gaussian image point, for which the tilt is
not corrected.
Coma
transverse
aberration also can be expressed in terms of the peak aberration
coefficient C. Tangential coma is given by T=6iC/n'D,
i
being the objective-to-image separation and D the aperture
diameter. For mirror oriented to the
left, n'=-1 and, for
distant objects ig,
resulting in T=6FC. With the peak aberration coma
coefficient C=αD/32F2
for distant objects, tangential coma can be expressed directly as:
T = 3Dα/16F
= 3fα/16F2
= 3h/16F2 (17)
as transverse aberration
(h=α,
the height in the image plane), and
Ta=T/=
3α/16F2
as angular tangential coma (in radians). From Eq. 11, tangential
coma changes in proportion to the square of the pupil (zonal) height, or
with
r2
for the pupil radius normalized to
1. As already mentioned, sagittal coma S=T/3;
it is larger than the P-V wavefront error by a factor of 3F.
Geometric coma blur (tangential coma) can also be expressed in terms of the RMS
wavefront error ω,
as t=ω√2592/2.44,
in Airy disc diameters; also, in terms of the peak aberration coefficient
as t=6C/2.44 (in Airy disc diameters), for the RMS wavefront error
ω
and peak aberration coefficient C in units
of the wavelength. For any given RMS
wavefront error, tangential coma is smaller than spherical aberration
blur at best focus by a factor (81/360)1/2.
Finally, the
RMS ray spot radius for balanced coma
is given by rRMS=2FC(7/9)1/2,
which in terms of the tangential coma becomes (7/9)1/2T/3.
In other words, it is smaller than tangential coma blur by a factor
0.294. The coma RMS blur radius is smallest when measured around the PSF
centeroid (at 1/3 of the tangential coma, along the axis of aberration),
given by r'RMS=FC(8/3)1/2,
or smaller by a factor of 0.926 than the RMS blur radius measured around
diffraction focus. However, due to the increase in wavefront tilt error
component, the P-V error of coma for this location is at the minimum for
ρ=1/√3,
and the aberration function Wc= C(ρ3-ρ)cosθ.
This gives the peak wavefront error Wc=-0.3849C,
greater by a factor of 1.15 than the peak wavefront error - and RMS
wavefront error - at best focus.
As mentioned, the above relations for coma
aberration are for a concave mirror with the aperture stop at the surface.
For displaced stop, the coma aberration coefficient c
changes with a factor
c=1- (1+K)σ,
with K being the mirror conic and
σ
the stop-to-mirror separation in units of the mirror radius of
curvature.
Obviously, stop position doesn't influence coma of a paraboloid (K=-1),
while cancels coma of any conic K>-1 for the stop separation
σ=1/(1+K).
On the other hand, it increases coma for any K<-1 (hyperboliod).
This applies for objects at infinity; for
close objects for which the mirror object-image magnification m
significantly differs from zero, the change factor in the coma coefficient caused
by aperture stop displacement is obtained by replacing both unit figures
in the "infinity" object distance factor
c with (m+1)/(m-1). The wavefront error changes
with the coefficient. Considering that the magnification m is
negative, coma diminishes with object distance, dropping to zero for
m=-1 (for object at the mirror center of curvature) and stop at the
surface (σ=0), regardless of the conic.
EXAMPLE: For a 200mm /5 paraboloid,
thus d=100 and R=-2000, the peak wavefront error of coma at 1.4mm
off-axis (with the field angle
α=1.4/1000=0.0014),
setting ρ=1
and θ=0
in Eq. 12, comes to C/3. With C=cαd3=αd3/R2=0.00035mm,
the peak wavefront error of coma is Wc=0.0001167mm.
The P-V wavefront error is twice as much, or 0.000233mm. In units of
the 550nm (0.00055mm) wavelength, it is 0.424 (1/2.36 wave). The RMS wavefront
error is ω=C/721/2=0.000041mm
or, in units of the 550nm wavelength, 0.075 (1/13.33 wave). The RMS
wavefront error can also be obtained from the P-V wavefront error as
ω=2Wc
/321/2.
Either way, the aberration is at the conventional
"diffraction-limited" level.
The transverse tangential coma
T=6FC=0.0105mm, or 1.56 Airy
disc diameters. Since both, wavefront error and geometric (ray)
aberrations are directly proportional to the aberration coefficient,
it implies that they are in a constant proportion themselves. In
other words, doubling the wavefront error also doubles the geometric
aberration.
A single thin lens
is coma-free if its shape factor
q and position factor p
relate as q=-(2n+1)(n-1)p/(n+1). Lens doublet can have both, coma and spherical aberration cancelled. In
regard to the aperture stop position, it does affect coma only with
objectives not corrected for spherical aberration. With an aplantic
(corrected for spherical aberration and coma) lens system, stop position
doesn't affect neither coma, nor astigmatism and field curvature.
◄
4.1. Spherical aberration
▐ 4.3.
Astigmatism
►
