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3.1. Wavefront aberrations
▐
3.3. Conics and aberrations
►

#
**3.2. Ray (geometric) aberrations**

Wavefront deformations are inevitably
accompanied by disturbance of rays - by definition straight
lines, or light paths - orthogonal to the wavefront surface. Disturbance
of rays caused by wavefront errors manifests as angular deviation of the
aberrated ray, producing linear ray deviation in the image plane,
usually either from
Gaussian image point*, *or from* *
best (diffraction) focus*.*

Note that the condition for light rays to
remain straight is homogenous media. This is the consequence of
*
Fermat's principle*, which in its elementary form states that light
ray follows a path that requires the least amount of time to connect two
separate points. For instance, a ray that connects point-source with its
conjugate point (focus) formed by a positive lens will follow different
paths, depending on its initial orientation. A ray arriving at the front
lens surface along the path coinciding with its optical axis, will
proceed straight toward the focus. On the other hand, a ray arriving at
the front lens surface at an angle, will bend (refract) entering the
lens, according to the law of refraction, reach the opposite lens surface
along the straight line, refract again when
exiting the lens, and arrive at its intersection with the chief ray
(i.e. Gaussian image point)
following a straight line.

Whether the point of intersection of all
the rays will coincide with the lens focus, is determined by the optical
properties of the lens. In order for that to happen, **
optical path length** for all rays needs
to remain constant. Note that optical path length differs from path's
geometric length, being subject to the media refractive index: optical
path length through a media of refractive index 2 is, for instance,
twice its geometric length. This is why rays having different geometric
path lengths traveling through a lens objective, still can have
identical optical path lengths, enabling wave points in phase to arrive
at the focal point simultaneously.

Variations in optical path length of
individual rays, on the other hand, result in **
optical path difference** for wave points in phase, wavefront
deformation, and wave phase mismatch at the focal point. While optical
path length and optical path difference resulting from it are obtained
from the geometry of rays in a given optical system, the focal zone ray
geometry alone is not directly related to these categories, hence it is
not a reliable predictor of the level of optical quality. Ray
aberrations are merely linear ray deviations from the focus point, thus
not conveying any information on the actual energy distribution around
it.

Yet, ray geometry offers convenient way
for determining basic properties of an optical system.
Gaussian, or "paraxial" approximation,
introduced by Carl Friedrich Gauss, follows rays close enough to the
axis that sines and tangents can be replaced by the angles themselves;
this is so called "first-order" optics, used to determine paraxial focus location.
Gaussian focus coincides with the focus of a perfect optical surface, or
system. On the other hand, best or diffraction focus is the location of
best actual image, which is in the presence of aberrations often
displaced from Gaussian focus.

Linear ray deviation from the focus
point has two forms: longitudinal, measured along the chief (central)
ray, from where it intersects with aberrated ray to the focus location,
and transverse, measured from the focal point to the ray height above or
below it in the focal plane. The former defines
**longitudinal ray aberration**, and the latter
**transverse ray aberration** (**FIG.
22**,
center).

Transverse ray aberrations can be
presented graphically by plotting points of intersection of a selected
number of rays with the focal plane. These are referred to as ray spot
or **ray spot plot** (**FIG. 22**,
center) also, *ray spot diagram, *or* geometric blur*. Another
form of presenting transverse ray aberration is **
ray fan plot**
(**FIG. 22**, left); similarly, longitudinal aberration can be plotted
as a function of the ray height in the pupil (**FIG. 22**, right).
Transverse ray deviations can be expressed in **
angular** form, given by the size of transverse aberration
relative to the system focal length.

FIGURE 22: Main forms of ray (geometric) aberration
and its presentation. RIGHT, aberrated wavefront deviates from
its perfect reference sphere by being more strongly curved toward the
edge. As a result, marginal rays focus shorter than central rays, with
the distance from the aberrated to proper focus constituting **
longitudinal aberration**, and the radial
height of the aberrated ray in the image plane constituting **
transverse aberration** (in this
illustration, both are negative, the former being oriented to the left,
and the latter having the radius on the opposite side of the pupil
radius of origin). **Ray spot plot**
presents ray distribution in the image plane; in the square, ray spot
plots for the conventional "diffraction limited" level (o.074 wave RMS,
0.80 Strehl) of coma, spherical aberration and astigmatism (0.42, 0.25
and 0.37 wave P-V, respectively) vs. Airy disc (black circle). The spot size changes in proportion to
the P-V/RMS wavefront error.
LEFT,
**ray fan plots** representing
transverse aberration as a function of the ray height in the pupil. The
top
graph is a ray fan plot for transverse primary spherical aberration at
the best
focus location (at the illustration in the center, that would be
mid-point of the longitudinal aberration). The plot shows that rays from
below 70% zone focus before best focus location, thus intersecting the
image plane at the opposite side of the axis; the 70.7% zone ray focus
coincides with the best focus location (thus the transverse aberration
is zero), while rays from the higher zones focus longer, intersecting
the image plane at the same side of the axis where is their pupil radius
of origin. Bottom plot is a ray fan plot for paraxial focus location,
for which all rays other than paraxial focus shorter, the higher zone,
the more so (paraxial blur is larger than best
focus blur by a factor of 2, which would show on the vertical numerical scale).
CENTER,
**longitudinal aberration plot** (also
for primary spherical aberration) shows focus location as a function of zonal
height in the pupil; since longitudinal aberration here changes with the
square of zonal height, 70% zone ray focuses at just short of the mid
point of the longitudinal aberration span.

As already mentioned,
ray aberrations are not reliable indicator of energy re-distribution
within diffraction pattern, which is the key information needed to
access the effect of aberration on image quality. There is simply no
direct relation between either geometric or angular form of ray
aberrations with the physical aspect of imaging - that is, wave
interference. However, they can give a rough impression of imaging
quality and, when resulting from known forms of wavefront aberrations,
their relation to the RMS wavefront error is also known and, thus, can
be associated with a specific degree of image deterioration.

As an initial indicator of optical quality, optical designers often
consider a system close to "diffraction-limited" if its ray spot radius
doesn't exceed the Airy disc radius. Called the *
Golden rule of optical design*, it is still a very loose
indicator of optical quality which, depending on the type of aberration,
can be associated with aberration levels of
anywhere from ~0.5 to 0.999
Strehl. Ray spot diagrams at left are good
illustration of how unreliable is geometric blur size as a criterion of
optical quality. If coma spot would have been made large as Airy disc,
it would still be significantly lower aberration level than the
"diffraction limited" 0.80 Strehl. At that level, coma blur is 1.56
times, primary spherical 3.4 times, astigmatism 0.6 and defocus 0.8
times the Airy disc diameter.

A more recent
form of the geometric blur presentation, the **RMS
spot size**, gives statistical spot size expressing average
distance of a large number of individual rays from the central spot,
according to R_{rms}=(A_{x}+A_{y})^{1/2},
where **A** is the averaged squared spot deviation in **x** and
**y** direction from central point. It
is not to be confused with the RMS wavefront error, a much more accurate
indicator of optical quality. While the RMS spot size is probably
somewhat less uncertain as an optical quality indicator than a plain geometric blur size, it is still mainly determined by the
geometric blur size, hence not significantly more reliable. For
instance, geometric blur size for paraxial, marginal and best focus for
spherical aberrations relate as 1:0.385:0.5, respectively; their RMS
spot sizes relate (in the same relative units) as 0.5:0.289:0.204, while
the respective RMS wavefront errors relate as 1:1:0.25. It can be
helpful in converting ray spot plots of different shapes and
distributions to a circle radius, hence making them comparable in
that respect.

As shown below, discrepancy between the RMS spot size for different
aberrations of the same magnitude is just as large as for the standard
ray spot plot. At left is the schematics of the RMS spot components
for rotationally symmetrical (top) and asymmetrical (bottom) aberrations.
The "diffraction limit" is the Airy disc radius (586nm wavelength),
and the "RMS R size" is the RMS spot radius.

All aberrations are shown at the level of 0.80 Strehl, for an f/8 system.
Yet the RMS spot size varies threefold. Note that all spots have
for the reference center image centroid, which for rotationally symmetrical spots
coincides with the chief ray point of intersection with image surface,
but for asymmetrical spots, such as coma spot (centroid-centered left,
chief ray centered, right), does not.

Directly related to the RMS spot size is
**geometric PSF** (Point Spread
Function), determined by the distribution of rays in the image plane, so
called *ray* or *geometric irradiance* (as opposed to actual
energy distribution given by
diffraction PSF). The point around
which the rays are balanced quantitatively is called *centroid*
(center of gravity). For radially symmetrical aberrations like
spherical, or those with a central symmetry, like astigmatism, it
coincides with the position of the chief (central) ray; for asymmetric
aberrations, such as coma, it is shifted from it according to the form
of ray distribution.

◄
3.1. Wavefront aberrations
▐
3.3. Conics and aberrations
►

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