1. Autocollimation is an optical setup where a collimated beam of parallel light rays leaves an optical system and is reflected back into the same system by a plane mirror. It is used for measuring small tilting angles of the mirror or for testing the quality of the optical system. 2. Five common types of optical aberrations are spherical, chromatic, field curvature, comatic, and astigmatic aberration. Spherical aberration occurs due to the increased refraction of light rays striking the edge of a lens. Chromatic aberration is caused by different wavelengths of light having different focal lengths. 3. Principal planes in an optical system are hypothetical planes where the refraction of light

1. Geo Optics
LEOT 330
Exam 2
Michelle Schroth
2. Autocollimation is an optical setup where a collimated beam (of parallel light
rays) leaves an optical system and is reflected back into the same system by a
plane mirror. Collimated.
It is used for measuring small tilting angles of the mirror, see autocollimator, or
for testing the quality of the optical system or of a part of it. One special applica-
tion is to determine the focal length of a diverging lens.
3. a. The equivalent power and focal length of the following system is 8 m-1 and
12.5 cm.
F1sys = f1× f2 ∕f1 + f2 - (d∕n)
= 20 cm × 20 cm ∕20 cm + 20 cm - (8 cm∕1)
= 12.5 cm
F1sys = F2sys = 12.5 cm
Pequ = P1 + P2 - P1 × P2 × d
= 5 m-1 + 5m-1 - (5 m-1) (5 m-1) (0.08 m)
= 8 m-1
3. b. The front vertex power and the front vertex focal length are 13.34 d (m-1)
and 7.5 cm. The back vertex power and the back vertex focal length are
13.34 d (m-1) and 7.5 cm.

2. P = 1∕f f1 = 20 cm = .20 m
P = 5 m-1 f2 = 20 cm = .20 m
P = 5 m-1 d = 8 cm = 0.08 m
n=1
PFv = P1 + P2∕1 - P2 (d∕n)
= 5 m-1 + 5m-1 (0.08 m-1∕1)
= 13.34 d (m-1)
fFv = 1∕PFv = 1∕13.34 m-1 = 0.075 m = 7.5 cm
PFv = PBv = 13.34 d (m-1)
fFv = fBv = 1∕13.34 m-1 = 0.075 m = 7.5 cm
3. c. The location of the principal planes H1 and H2 are shown below.
4. Five common types of optical aberrations are:
Spherical
Which is an optical effect observed in an optical device (lens,mirror, etc.) that oc-
curs due to the increased refraction of light rays when they strike a lens or a re-
flection of light rays when they strike a mirror near its edge, in comparison with
those that strike nearer the center. It signifies a deviation of the device from the
normal operation, i.e., it results in an imperfection of the produced image.

3. For single lens, spherical aberration can be controlled by bending the lens into its
best form. Also for multiple lenses, spherical aberrations can be canceled by
overcorrecting some elements. The use of symmetric doublets greatly reduce
spherical aberrations.
Chromatic Aberration
Chromatic aberration or "color fringing" is caused by the camera lens not focus-
ing different wavelengths of light onto the exact same focal plane (the focal
length for different wavelengths is different) and/or by the lens magnifying differ-
ent wavelengths differently. These types of chromatic aberration are referred to
as "Longitudinal Chromatic Aberration" and "Lateral Chromatic Aberration" re-
spectively and can occur concurrently. The amount of chromatic aberration de-
pends on the dispersion of the glass. A lens will not focus different colors in ex-
actly the same place because the focal length depends on refraction and the in-
dex of refraction for blue light (short wavelengths) is larger than that of red light
(long wavelengths). The amount of chromatic aberration depends on the disper-
sion of the glass.

4. One way to minimize this aberration is to use glasses of different dispersion in a
doublet or other combination.
The use of a strong positive lens made from a low dispersion glass like crown
glass coupled with a weaker high dispersion glass like flint glass can correct the
chromatic aberration for two colors, e.g., red and blue.
Field Curvature
Is where the sharpest focus of the lens is on a curved surface in the image space
rather than a plane. Objects in the center and edges of the field are never in fo-
cus simultaneously.

5. We can correct this aberration by using specially designed objectives. These
specially-corrected objectives have been named plan or plano (for flat-field)
and are the most common type of objective in use today, providing ocular fields
ranging between 18 and 26 millimeters, which exhibit sharp detail from center to
edge.
Comatic Aberrations
They are seen mainly with off-axis light fluxes and are most severe when the mi-
croscope is out of alignment. With a comatic aberration, the image of a point is
focused at sequentially differing heights producing a series of asymmetrical spot
shapes of increasing size that result in a comet-like shape to the Airy pattern.

6. The distinct shape displayed by images with comatic aberration is a result of re-
fraction differences by light rays passing through the various lens zones as the
incident angle increases. The severity of comatic aberration is a function of thin
lens shape, causing meridional rays passing through the periphery of the lens to
arrive at the image plane closer to the axis than do rays passing nearer the axis
and closer to the principal ray.
When the Off-Axis Distance slider is moved to the far right position, the ray trace
diagram shows several skewed light ray paths representing those rays involved
in the aberration. Off-axis light rays often interfere with each other near the focal
plane to generate malformed images seen in the microscope. The image point
produced by a comatic aberration is actually a complicated three-dimensional
asymmetrical diffraction pattern that departs from the classical Airy pattern.
What is formed is an elongated structure composed of arcs and ellipsoidal inten-
sities that only vaguely resemble the disk-ring arrangement from which the point
spread function evolved.
The severity of Coma is heavily dependent upon the shape of the lens. A strong-
ly concave positive meniscus lens will demonstrate substantial negative comatic
aberration, whereas plano-convex and bi-convex lenses produce comas that
range from slightly negative to zero. Objects imaged through the convex side of
a plano-convex lens or a convex meniscus lens will have a positive coma.
Coma can be corrected by using a combination of lenses that are positioned
symmetrically around a central stop. In order to completely eliminate coma, the
Abbe sine condition must be fulfilled:
d' × n(sinβ') = d × n(sinβ')
where d' and d are the distances from the optical axis in the image space (prime
values) and object space, n is the refractive index, and β is the viewing angle. A

7. lens system, such as a microscope condenser or objective, which is free of co-
matic aberration is referred to as aplanatic.
Astigmatic Aberration
An objective lens for which spherical and coma aberrations have been corrected
may not be able to converge object points off the axis to a point, separating those
points into a segment image in a concentric direction and that in a radial direc-
tion. This aberration is known as "astigmatic aberration". An objective with any
astigmatic aberration will change the blur orientation of a point image to longitudi-
nal or lateral with respect to before or after the focal point.

8. When an object lies an appreciable distance from the optical axis, the incident
cone of rays will strike the lens asymmetrically, giving rise to the aberration
known as astigmatism. To describe it, picture the plane which contains both the
chief ray, which is the ray which passes through the center of the lens, and the
optical axis. This plane is knows as the meridional, or tangential, plane. The
sagittal plane is defined as the plane containing the chief ray which is also per-
pendicular to the tangential plane.
When the object is on the optical axis, the cone of rays is symmetrical with re-
spect to the spherical surfaces of the lens. In this case the meridional and the
sagittal planes are the same, and the ray configurations in all the planes contain-
ing the optical axis are identical. In the absence of any spherical aberration, all
of the focal lengths are the same and all of the rays arrive at a single focus.
When the object is located off axis, the rays come into the lens at an oblique an-
gle. Now the configuration of the ray bundle will be different in the meridional
and sagittal planes. Because of this, the focal lengths in these planes will be dif-
ferent as well. Basically, the meridional rays are tilted more with respect to the
lens than the sagittal rays, and thus have a shorter focal length. Using Fermat's
principal, we find that the focal length difference depends effectively on the pow-
er of the lens and the angle at which the rays are inclined. This is known as the
astigmatic difference, and it increases rapidly as the rays become more oblique.

9. Since there are two distinct focal lengths, the incident conical bundle of rays
changes after being refracted. The cross section of the beam as it leaves the
lens is initially circular, but it gradually becomes elliptical with the major axis in
the sagittal plane, until at the tangential focus, FT, the ellipse degenerates into a
line (at third order). All the rays from the object traverse this line, which is known
as the primary image. Beyond this point the beam's cross section rapidly opens
out until it is again circular. At that location the image is a circular blur known as
the circle of least confusion. Moving further from the lens, the beam's cross sec-
tion again deforms into a line, called the secondary image. This time it is in the
meridional plane at the sagittal focus, FS.
5. Find the location of the image in the following diagram. If the object is 2 cm in
height, determine the image height. Indicate whether the image is upright of in-
verted.
Lens 1
do = -20 cm
1∕do + 1∕f = 1∕di M = hi∕ho
1∕-20 cm + 1∕10 cm = 1∕di = di∕do

10. 20 cm = di = 20 cm∕-20 cm
M = -1 (Inverted)
hi = (ho) × (m)
= (2 cm) × (-1)
hi = -2
Lens 2
do = 10 cm
1∕do + 1∕f = 1∕di M = hi∕ho
1∕10 cm + 1∕-30 cm = = di∕do
15 cm = di = 15 cm∕10 cm
M = 1.5 (Upright)
Lens 3 is a Mirror.
do = 12 cm
1∕do + 1∕f = 1∕di M = hi∕ho
1∕12 cm + 1∕10 cm = = di∕do
5.5 cm = di = 5.5 cm ∕12 cm
M = 0.458 or .46 (Upright)
hi = (ho) × (m)
= (-3 cm) ×(0.458)

11. hi = 1.374 or 1.4 cm
The image is upright and the image height is 1.4 cm.
6. See attachment.
7. Using the Lens-makers equation, design a double-convex glass lens with a fo-
cal length of 60 cm. Assume the refractive index of the glass is 1.50.
P = 1∕f
= 1∕60 cm
= 1∕.6 m
P = 1.67 d (m-1)

12. R2 = -2R1
Use the Lens makers formula
P = 1.67 m-1 = (n-1)(1∕R1 - 1∕R2)
= (1.50 - 1)(1∕R1 - 1∕-2R1)
= (0.5)(1∕R1 - 1∕R1)
= (0.5) 1∕R1 [1 + 1∕2]
= (0.5) (1∕R1) [1.5]
1.67 m-1 = (.75)∕R1
R1 = .75∕167 m-1
= 0.45 m
R1 = 45 cm
R2 = -2R1
= -2 × (45 cm)
R2 = - 90 cm
8. a. Optical dispersion is the phenomenon in which the phase velocity of a wave
depends on its frequency, or alternatively when the group velocity depends on
the frequency. Media having such a property are termed dispersive media. Opti-
cal dispersion is sometimes called chromatic dispersion to emphasize its wave-
length-dependent nature, or group-velocity dispersion (GVD) to emphasize the
role of the group velocity.
The most familiar example of chromatic dispersion is a rainbow, in which disper-
sion causes the spatial separation of a white light into components of different
wavelengths i.e. different colors.

13. Chromatic dispersion is especially important to researchers who are designing
optical equipment like cameras, optical microscopes, and telescopes. When a
lens system is not carefully designed, the system will focus different colors of
light at different spots – and this doesn’t give a very good image! By planning the
system carefully and using a combination of lenses made out of different materi-
als with different indices of refraction, these chromatic aberrations can be greatly
minimized.
The effect of chromatic dispersion is also important to people who send short
pulses, which are made up of many different wavelengths, through optical
waveguides, like optical fiber. Short pulses of EM Rad are used as a way of en-
coding data, like voices during a telephone call and the information on this web-
site, so that the data can be sent from one place to another. As the pulse travels
in the waveguide, some wavelengths of light travel faster than others. As the
pulses travel down the waveguide, they increase in width and overlap with one
another. If they spread too much, it is difficult to tell where one pulse begins and
the other ends, and this results in information being lost. Researchers who work
in the Communications and Fiber Optics fields of optics are developing devices
to combat the effects of dispersion.
The dispersion of light by glass prisms is used to construct spectrometers and
spectroadiometers. Holographic gratings are also used, as they allow more ac-
curate discrimination of wavelengths.

14. 8. b. Principal planes in optical systems
The two principal planes in a lens system are hypothetical and have the property
that a ray emerging from the lens appears to have crossed the rear principal
plane at the same distance from the axis that that ray appeared to cross the front
principal plane, as viewed from the front of the lens. This means that the lens can
be treated as if all of the refraction happened at the principal planes.
The principal planes are crucial in defining the optical properties of the system,
since it is the distance of the object and image from the front and rear principal
planes that determines the magnification of the system.
If the medium surrounding the optical system has a refractive index of 1 (e.g., air
or vacuum), then the distance from the principal planes to their corresponding fo-
cal points is just the focal length of the system. In the more general case, the dis-
tance to the foci is the focal length multiplied by the index of refraction of the
medium.
For a thin lens in air, the principal planes both lie at the location of the lens. The
point where they cross the optical axis is sometimes misleadingly called the opti-
cal center of the lens. However, that for a real lens the principal planes do not
necessarily pass through the centre of the lens, and may not lie inside the lens at
all.

15. For a given set of lenses and separations, the principal planes are fixed and do
not depend upon the object position. The thin lens equation can be used, but it
leaves out the distance between the principal planes. The focal length f is that
given by Gullstrand’s equation. The principal planes for a thick lens are illustrat-
ed. For practical use, it is often useful to use the front and back vertex powers.