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4.3. Telescope astigmatism

Similarly to coma, *astigmatism* is an off-axis
point wavefront aberration, caused by the inclination of incident wavefronts relative to
the optical surface. However,
while coma always originates at the optical surface, astigmatism may results
simply from
the projectional asymmetry arising from wavefront's inclination to the
surface. For flat incoming wavefront and stop at the surface, the
diameter of wavefront's projection onto the optical surface varies from the
minimum in the
plane of wavefront tilt - determined by the chief (central) ray and
optical axis (which also defines the
*tangential plane*) - to the maximum in the direction orthogonal to it
(*sagittal plane*), where it equals aperture diameter.

Since the focal length
(i.e. radius of curvature of the wavefront) changes with the square of
the diameter for given sagitta depth, the two orthogonal wavefront
sections
focus at the longitudinal separation of (1-cos2α)ƒi,
which constitutes longitudinal astigmatism (**α**
is the field angle in degrees, and **ƒ****i**
is the image-to-pupil separation, equal to focal length for distant objects).

Another peculiarity of
astigmatism is that a cross-section along any wavefront diameter is
still spherical, but with the radius of curvature varying
with the pupil angle. Thus the wavefront form as a whole deviates from
spherical.

For displaced stop -
either first optical surface significantly separated from the aperture
stop, or secondary and tertiary surfaces (whose stop is formed by a
preceding surface) - other surface properties, such as shape, position and conic, also
can influence the size of astigmatism, due to the displaced stop for
these surfaces adding the element of surface
radial asymmetry. **FIG. 22** illustrate the form of
astigmatic wavefront deformation and the resulting geometric (ray)
aberration.

**
**

**
FIGURE 22**:* To the right*:
Mirror astigmatism as a result of the projected diameter of the incoming
wavefront (**W****i**) varying with the radial orientation. For the
inclination angle shown, the vertical (tangential) wavefront
projection onto the surface is shortest at cosαD,
gradually increasing with the radial angle around the chief ray to the
maximum projection width **D** in the orthogonal
(horizontal, or sagittal) orientation. With the
wavefront sagitta (depth) constant, its radius (the focal length) varies
with the square of diameter. Being of the smallest diameter, the wavefront section** W****t**
in the tangential (vertical) plane focuses closest, and the
wavefront section **W****s**
in* *the* *sagittal plane farthest away from the mirror*. To the left*,
an illustration of the actual wavefront deviation from the
respective reference sphere (red dots): for the reference
sphere **W****P**
centered at the mid point of defocus **M**, and for the two reference spheres centered at either sagittal or tangential focus (**S**
and **T, **respectively; the deviation for the former has
cylindrical form oriented horizontally, for the latter vertically).
The P-V error is identical at all three focus location; however, the
deviation averaged over the wavefront is lower at the mid-focus by a
factor 0.82, making it best (diffraction) focus. Solid blue line in
best focus wavefront deviation** (M)** represents the deviation along
the central cross-section of the actual wavefront vs. perfect reference
sphere centered at the mid-focus point. The dashed blue line is a
projection of the deviation along the wavefront edge, indicating
saddle-like shape of the wavefront deformation.

Gaussian focus for astigmatic wavefront lies on the
Petzval surface of an optical
surface, or system. Balancing defocus aberration for this point -
located on the opposite direction from the sagittal focus, and at
identical distance from it as the best focus - is zero, and the
wavefront error is largest. Between the sagittal and tangential focus, ray disturbance resulting from the astigmatic wavefront deformation
takes on rather peculiar form (**FIG. 23**).

**
****
****FIGURE
23**:
Geometry of the astigmatic defocus produced by a mirror with the
stop at the surface: the wavefront radius at the pupil gradually increases
from the minimum in the
tangential (vertical) plane to the maximum in the orthogonal to it
sagittal plane. Consequently, all wavefront meridians focus at a
different length, producing longitudinal defocus, as an axial separation between tangential and sagittal focus. At the
sagittal plane focus **S** it forms sagittal line, contained in the
tangential plane. And at the tangential plane focus **T**
it
forms tangential line, laying in the sagittal plane. The lines
transform into ellipses of decreasing eccentricity toward the inside
of defocus zone. Midway between the two lines is the circle
of least confusion (**BF**), which is the location of best
astigmatic focus.

Aberration function for the **
wavefront error **of astigmatism at best
focus is given by:

**
Wa= Ar2(cos2q
- 0.5) **
(18)

with **A** being the astigmatism peak aberration coefficient,
**r**
the height in the pupil, and **q**
the pupil angle. It shows that the the
wavefront error peaks for
ρ=1
and cos2θ=1
and 0 (that is, for θ=0,
π/2,
π, 3π/2
and 2π),
which is, every 90 degrees, and orthogonally to the orientations of the
minimum wavefront deviation, occurring for
cosθ=**√**0.5
(for θ=π/4,
3π/4, 5π/4
and 7π/4). It
clearly outlines saddle-shaped wavefront deviation, as illustrated on **
FIG. 22 **left.

Note that the maximum wavefront error given by **
Eq. 18** - which gives
± wavefront deviations, not the peak-to-valley error - is one half of
the peak aberration coefficient, which equals the P-V error. Numerically, it is identical to the P-V
error at either sagittal or tangential focus, but its RMS value is
smaller by a factor of 1/**√**1.5.

When the point of maximum deviation in tangential
(vertical) plane is closer to the center of reference sphere than its
perfect reference point, the wavefront error of astigmatism is negative.
That is the sign of astigmatism in concave mirror, illustrated on **
FIG. 22**. There is no difference in appearance between positive and
negative lower-order astigmatism, since the pattern is merely rotated by
90°, and has inherent 90-degree rotational symmetry
at best focus location (**FIG. 24**).

The
**peak**** aberration coefficient** **A**,
which equals the peak-to-valley wavefront error, is given by:

**
A=a**α**2d2** (19)

with **a** being the
astigmatism aberration coefficient, **
α** the field angle and **d**
the pupil (aperture) radius. The **aberration coefficient** **a** for a concave mirror and stop at the
surface is given by:

** a**M** = **
** ****n****/R** (20)

with **R** being the mirror radius of
curvature. For mirror
in air oriented to the left, n=1 and the aberration
coefficient is **a**M**=1/R**.
The sign of aberration coefficient indicates the tangential wavefront
radius shorter than sagittal, and the sagittal line farther away from
the mirror, as shown on **FIG. 22-23**. Positive astigmatism has this order
reversed. From another perspective, the astigmatism wavefront error is
negative when the optical path difference from the point of peak
deviation in the tangential plane is smaller than the radius of a
perfect reference sphere (the one centered at the mid point between
tangential and sagittal focus).

**Shift to the best focus location**
is half the longitudinal aberration from either of the two line foci. In
terms of the peak aberration coefficient **A**, needed wavefront
error of defocus from either
tangential or sagittal focus to best focus location is ±(A/2).
While the P-V error remains unchanged for all three focus locations -
sagittal, tangential and midway between these two - the latter has the
RMS wavefront error smaller by a factor of 2/**√**6.
As a result, this focus location has has the highest peak diffraction
intensity, making it the best focus location.

The
best focus RMS wavefront error in terms of the
peak aberration coefficient - or P-V wavefront error - is given by:

**
ωa=
A/√24**
(21)

Form of the aberration coefficient shows that the astigmatism
wavefront error, unlike coma and spherical aberration, doesn't change
with object distance. This is expected consequence of astigmatism of a
mirror - as well as that of a lens (contact) objective - being result of
the projectional wavefront asymmetry itself, rather than a product of
the wavefront/surface interaction.

Astigmatism
**ray aberrations**
can also be expressed
in terms of the peak aberration coefficient **A** as:

** L=8AF2**,** T= 4FA
** and** T****a=4A/D
** (22)

for the *longitudinal*, *transverse*
and *angular* astigmatism, respectively. After substituting for **
A**, the transverse aberration - as the circle of least confusion
diameter - can be also expressed as T= -Dα2/2
= -h2/2DF2
for object at infinity, with **h** being the point height in the image plane
(note that aperture **D** needs to be in the metric used for the
coefficient calculation, which also becomes the metric of transverse
aberration). Since focal lengths of the astigmatic wavefront do not
change with the height in the pupil (i.e. the zonal height), transverse
astigmatism changes in proportion to the normalized pupil ray
height **ρ**. With h=αƒ,
**ƒ** being the focal length, angular astigmatism **T****a=T/ƒ=**α**2/2F**.
Longitudinal astigmatism **L=-ƒ**α**2**.

There are simple geometric
relationships between the *circle of least confusion* diameter and the *sagittal* and
*tangential line length*, as well as between the three and
the *longitudinal aberration*. The line length is double the circle
diameter, and the longitudinal aberration is greater than either line by a factor of
**F**
(the focal ratio denominator), as illustrated in **FIG. 23**.

The transverse aberration in
terms of the RMS wavefront error is **T****a= ω√384/2.44**,
and in terms of the peak-to-valley error **T****a=4A/2.44**
(for
**ω**
and **A** in units of the wavelength), both expressed in
units of the Airy disc diameter. That makes astigmatic blur significantly
smaller for given amount of wavefront aberration than
geometric blur for
either spherical aberration or coma (**FIG. 24**). It is a stark
remainder that optical criteria can

**
FIGURE
24**: The ray spot size (top) and actual diffraction patterns for 0.37 wave
P-V wavefront error of primary astigmatism (resulting in 0.80
Strehl, thus comparable to 1/4 wave P-V of primary spherical
aberration). Perfect diffraction pattern is to the left. Geometric
blur diameter at the best focus location (balanced primary
astigmatism) is only 0.6 Airy disc
diameters. There is no rays outside the Airy disc, yet considerable
amount of energy has spread out from the spurious disc - result of
the complex wave interference around best focus point (not
unexpected, considering that the rays focused to a point still
produce a pattern) . Compared to
spherical aberration and coma, the energy spread is concentrated
closer to the disc.

not be reduced to geometrical considerations; it is the underlying realm
of electromagnetic field that determines the properties of the point
object image.

As expected due to its uniformly dense
geometric blur, the smallest RMS blur radius for astigmatism is at the location
of the circle of least confusion. It is given by rRMS=FA**√**2,
or smaller by a factor of **
√**0.5
than the radius of the circle of least confusion.

**EXAMPLE**: A 200mm ƒ/5 concave mirror,
d=100, R=-2000. Setting θ=0
and ρ=1,
the **peak wavefront error** at h=1.4mm off-axis, giving the field angle α=1.4/1000=0.0014,
is **W**=A/2=α2d2/2R=-0.0000049mm.
The **P-V wavefront error** is twice greater - equal to the peak
aberration coefficient - or -0.0000098mm. In units
of the 550nm (0.00055mm) wavelength, it is 0.0178, or 1/56 wave.
Consequently, the **RMS wavefront error**
**ω**=A/**√**24=0.000002mm or, in
units of the 550nm wavelength, 1/275 wave. The **transverse
astigmatism** (circle of least confusion diameter) is
**T**=4FA=20α2d2/R= =0.000196mm,
or 0.03 Airy disc diameters, and** angular astigmatism**
**Ta**=T/ƒ
is 0.000000196 radians, or
206.265x0.000000196=0.04 arc seconds..

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.

For the stop at the surface, the
astigmatism error doesn't change with **object
distance**. For relatively close objects,
transverse astigmatism increases as ƒi/ƒ,
**ƒ****i**
being the image-to-pupil separation. However, it doesn't affect the wavefront
error: since the wavefront radius is also longer by the same ratio, identical nominal wavefront error results in proportionally greater
longitudinal and transverse aberration.

For the **aperture**** stop displaced** from mirror
surface, the aberration coefficient of astigmatism changes in
proportion to [Kσ2+(1-σ)2],
with **σ**
being the mirror-to-stop separation (positive in sign) in units of the
mirror radius of curvature. Needed stop separation for zero astigmatism
is given by σ=[1-**√**|K|]/(K+1). Thus, astigmatism is canceled for
σ=0.5
with a paraboloid and σ=1
with a sphere. The relation
is not defined for K=-1 (parabola), but implicates σ=0.5 limit for K"-1. For positive values of
the conic **K**, the aberration coefficient cannot be zero regardless of the
stop position, due to the right side of the aberration factor being squared (always positive).

Unlike coma, change in astigmatism caused by
the aperture stop position is independent of object distance.

Aberration coefficient of primary astigmatism for a
**lens** with the aperture stop at the surface is identical to the one given for
concave mirror (**Eq. 20**). For a contact doublet, it gives the peak
aberration coefficient as a sum of the aberration coefficients at the first
and second lens, respectively, as:

**
Ad=A1+A2=(-**α**2D2/8ƒ1)+(-**α**2D2/8ƒ2) **(23)

with **
α** being the field angle, and **
ƒ****1**,**ƒ****2**
the respective lens focal lengths (keep in mind that focal length of a
negative lens is numerically negative in the left-to-right Cartesian coordinate
system). Change of the stop position results in change of the
aberration coefficient only with systems not corrected for spherical
aberration, or coma, or both. Since modern refractor objectives commonly
are aplanats, their astigmatism is not affected by the stop
position. As already mentioned, wavefront error of astigmatism of the
contact doublet doesn't change with object distance.

◄
4.2. Coma
▐
4.4.Defocus
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