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Star Test

Ref: Chapter 11 of Malacara

The careful visual examination of the image of a point source formed by a
lens being evaluated is one of the most basic and important tests that can
be performed. The interpretation of the image in terms of aberrations is
to a large degree a matter of experience, and the visual examination of a
point image should be a dynamic process. The observer probes through focus
and across the field to determine the type, direction, and magnitude of
aberrations present. For ease of carrying out the test, the magnifying
power should be such that the smallest significant detail subtends an
easily resolvable 10 to 15 minutes of arc at the eye. It is also important
that the numerical aperture of the viewing optics is large enough to
collect the entire cone of light from the optics under test.

If the lens being tested is perfect, the image of a point source as seen at
best focus is called the Airy disk. The Airy disk consists of a bright
circular core surrounded by several rings of rapidly diminishing
brightness. The diameter of the central core is equal to 2.44?f#, where f#
is the f/number of the converging light beam. Note that in the visible,
the diameter of the central core is approximately equal to the f# in
microns. The central core contains approximately 84% of the total amount
of light, while the total amount of light contained within the first,
second, and third rings is approximately 91%, 94%, and 95%, respectively.
If the microscope is moved back and forth along the axis, the image will be
seen to go in and out of focus. The change in the pattern is rather
complex, consisting first of a redistribution of light from the core to the
rings, then with larger focus shifts the diameter of the image will appear
to grow. A perfect image will appear totally symmetrical on opposite sides
of focus as shown in Fig. 8.2.10-1.

Spherical aberration, coma, and astigmatism are also easily observed using
the star test. The presence of spherical aberration is most easily
inferred by examination of the symmetry of the image through focus. As one
focuses on the image, starting from well inside the marginal image plane
and moving toward paraxial focus, the following set of images shown in Fig.
8.2.10-3 is noted for undercorrected spherical. First, a diffuse, fairly
uniform blur is seen. As the region of the marginal focus is approached,
the beginning of the outer spherical caustic is reached. Here, a "hollow"
or ring image is observed. Next, the ring diminishes in size and intensity
and gives way to a core with a rather bright set of surrounding diffraction
rings. Eventually, the size of this structure reaches a minimum and then
becomes a small, intense core surrounded by a diffuse halo. Beyond the
paraxial plane a growing diffuse flare is observed. The best focus
(minimum spot size) occurs at ѕ the distance from paraxial to marginal
focus. The minimum spot size is ј the spot size at paraxial focus.

Off-axis images are complex. Almost always, a mixture of coma and
astigmatism of various orders is obtained. For third-order coma, the image
looks much as indicated in Fig. 8.2.10-5, while the line foci for third-
order astigmatism appears as indicated in Fig. 8.2.10-6. Fig. 8.2.10-7
shows the diffraction pattern for third-order astigmatism in the
neighborhood of the circle of least confusion.

It is useful to obtain a rough estimate of the geometrical spot size
produced by the different aberrations. Let ?W be the maximum aberration
for third-order spherical, coma, and astigmatism and f# be the f/number of
the converging light beam. At paraxial focus, the blur radius ?y, for
third order spherical is given by


The minimum radius of the blur due to third-order spherical would be ј of
this.

The tangential coma, ?y, is given by


The sagittal coma is 1/3 this value and the width of the coma image is 2/3
of this.

The length of the line focus for astigmatism is given by

The blur for astigmatism halfway between the sagittal and tangential focus
would be Ѕ of this value.

Therefore, the minimum spot diameter for third-order spherical, the width
of the coma image (2/3 the tangential coma), and the diameter of the blur
for astigmatism that falls halfway between the sagittal and tangential
focus are all given by

where again ?W is the maximum wavefront aberration due to third-order
spherical, coma, or astigmatism at the edge of the pupil.

It is of interest to look at the ratio of geometrical blur to the Airy disk
diameter.



That is, the ratio of the geometrical blur diameter to the Airy disk
diameter is approximately equal to 1.64 times the amount of aberration in
units of waves.

The star test is very useful for detecting chromatic aberration. The
testing is carried out by observing the color changes in the image as the
focal position is varied toward and away from the lens. In a perfectly
apochromatic system a symmetrical "white" image is obtained for all focal
positions. Chromatic aberration provides an image whose color is a
function of focal position. In moving away from the lens through the
paraxial focal plane, a sequence of images is observed. Well away from
focus, a white flare is observed. As the blue focus is reached, the color
balance is seen to change as blue light appears to be removed from the
flare and is concentrated in a core. Farther away from the lens a similar
color effect is observed as the foci for green and red are reached. For
overcorrected color, the colors appear in the opposite order.

The chromatic errors in an off-axis image are most spectacular in visual
testing. The lateral separation of the images in red and blue light gives
directly the amount of lateral chromatic aberration. If the red image is
found to lie at a greater distance from the axis than the blue image,
negative or undercorrected lateral color is present, while for
overcorrected lateral color, the blue image is a greater distance from the
axis than the red image.

The following pictures are from "Atlas of Optical Phenomena" by Cagnet,
Francon, and Thrierr.



Fig. 8.2.10-1. Diffraction by a circular aperture as a function of defocus
for no aberration


Airy Disk



1 wave defocus Less than 1 wave defocus

Fig. 8.2.10-2. Diffraction by a circular aperture in the presence of
defocus.


Fig. 8.2.10-3. Diffraction by a circular aperture as a function of defocus
for third-order spherical aberration




Paraxial Focus Small distance
inside paraxial focus



Moderate distance from marginal focus Immediate neighborhood of
marginal focus


Fig. 8.2.10-4. Diffraction by a circular aperture in the presence of third-
order spherical aberration.



6 ?



2.5 ? 1 ?

Fig. 8.2.10-5. Diffraction by a circular aperture in the presence of third-
order coma.



7 ?


1.5 ? 0.23 ?

Fig. 8.2.10-6. Diffraction by a circular aperture in the presence of
astigmatism.



7 ?


1.6 ? 0.23 ?

Fig. 8.2.10-7. Diffraction by a circular aperture in the presence of
astigmatism in the neighborhood of the circle of least confusion.

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