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1.3. Optical system of a
telescope
 Optical system
of a telescope consists from: (1)
objective, which captures light
from faraway objects and uses it to form their real image, and (2)
eyepiece, a sophisticated magnifying glass that uses
image formed by the objective to project even larger image onto the
retina. Telescope objective can be a single concave mirror; it can
also consist from two or more mirrors, or lenses, or of the two
combined. It gathers light and forms images of distant objects. While
the image it forms can be observed directly, the eye would only
receive a small fraction of the light emerging from it (FIG. 5).
It is the role of the ocular - or eyepiece - to make all the light
from the image formed by the objective available to the eye and, by increasing angles
of incidence, to add a significant magnification factor to the final
image formed on the retina (FIG.
6).
FIGURE 5: Image
formed by a telescope objective, as seen without the eyepiece. Wavefronts emitted from
distant object of height h become practically flat over
the aperture of a telescope, here the concave mirror M.
The mirror
changes the shape of the incoming flat wavefront section (W) into spherical (W')
by delaying reflection of the points
in phase
belonging to the wavefront's inner area. Point of
convergence (c) - or focal point - is at the center of curvature
R of the wavefront. Thus, with the stop at mirror surface and
exit pupil plane at the mirror vertex, mirror focal length
equals the radius of the wavefront in the pupil. The top
point of distant object h is imaged into
(reversed) top of the image h'
by off-axis wavefront Wa,
originating at
the object's top point. If observed directly, from the least
distance of distinct vision d (approx. 25cm, or 10
inches) most of the light from the
object image h' misses the eye pupil (more so for the
points farther off-axis, with no light from the image's reversed
top reaching the eye); also,
magnification is limited to ƒ/d,
ƒ
being the objective's focal length.
FIGURE 6:
Image formed by a telescope, as seen through the eyepiece. Since
the light is slowed down in glass, the
in-phase points of
the incoming axial wavefront W are retarded the most in the
center of the lens objective L (for simplicity, both
objective and eyepiece are shown as a single lens), and the least
at its edges. Properly made lens objective will re-shape the flat
wavefront W into spherical
(W') after exiting the lens. That is a goal for off-axis
point wavefronts (Wa)
as well, although some form of deviation due to the tilt-created
asymmetry is usually present. In terms of rays, the change in
direction of straight lines orthogonal to the
wavefront (rays), resulting from its new shape, is called refraction.
The lens objective focal length
ƒ is the distance
between its second principal point P2
and the focal point F. The eyepiece (EP), placed at a distance of its focal length ƒ' from the
object image h' formed by the objective, converts diverging
spherical wavefront into flat, for which the eye has preference.
It also
increases the apparent incidence angle (α
vs.
ε), making
the object imaged at the retina (E) appear larger by a
factor ~ƒ/ƒ'.
The very basic element of a telescope is
the diameter of its
aperture.
Given optical quality, it is the main determinant of
telescope's capabilities with respect to light gathering and resolution, thus
also of its limits in useful magnification. If well made, the eyepiece has
no appreciable effect on the light gathering or inherent resolution of a telescope.
Its main function is magnification of the real image formed by
the objective. Consequently, the main optical parameters of a
telescope relate to its objective. They are:
● aperture
diameter, hereafter denoted by D
● focal length
ƒ and
● relative aperture
D/ƒ=1/F, with F being the focal ratio number
Thus, telescope consist from a single or
or multi-element objective, and
an eyepiece centered around the optical axis of the objective. The objective
forms the focal point - a point
of wave energy or, geometrically, ray convergence on its optical axis - which determines the focal length of a telescope.
Telescope focal length is a
distance from the objective to where it focuses collimated light.
That is, when the light arrives
from objects far enough that the wavefront
entering the objective is practically flat, and the light rays
are practically parallel. For closer objects, the focus forms
farther away from the objective.
In the paraxial, or Gaussian approximation,
the focal length is obtained from this basic equation relating object
distance O, single optical surface radius of curvature R
and image-to-surface separation I:
(n1/I)-(n0/O)=(n1-n0)/R, (1)
with n0
and n1
being the refractive indici of incident medium, and that of reflection
or refraction at the surface, respectively. Hence, as the object
distance O approaches infinity, image distance reduce to the focal
length ƒ=R/(1-n0/n1).
The appropriate focal point is called paraxial, or
Gaussian focus.
A few words about Gaussian
approximation: it is based on ray geometry for angles small enough that
the angle (in radians) doesn't differ appreciably from its sine or
tangent. Thus, it is strictly valid only for rays very close to the optical axis - paraxial rays -
and used to determine their points of convergence. In principle, these points
coincide with the points of convergence of a perfect
(aberration-free) system.
For thin lens in air, first surface
indici are n0=1
for light traveling from left to right, and n1=n,
n being the glass refractive index, so the relation applied to
the first surface becomes (n/I1)-(1/O)=(n-1)/R1,
I1
being the front surface to (its) image separation, and R1
being the front surface radius of curvature. For very distant object,
1/O is infinitely small, and its image forms at a distance I1=nR1/(n-1)
from the first surface. This is now the object distance O for the
second surface (n0=n
and n1=1),
which will form the final image at a distance I2,
equal to the lens'
focal length ƒ according to (1/I2)=1/ƒ=[(n-1)/R1]-(n-1)/R2,
which comes to:
(1/f)=(n-1)[(1/R1)-(1/R2)]
(1.1)
This expression
is referred to as thin lens equation, or
lensmaker's formula. It can be also written as
ƒ=R1R2/(n-1)(R2-R1).
Note that in the standard right-hand
Cartesian coordinate
system distances to the left are negative, and those to the right
positive; consequently, biconvex lens has the front radius positive and
the rear radius negative.
Obtaining
mirror focal length from Eq. 1 is direct:
with n0=1
and n1=-1
(for incident light traveling from left to right, according to the
sign
convention), (1/I)=-2/R=1/ƒ, which gives
the mirror focal length
ƒ as:
|ƒ|=R/2
(1.2)
with R
being the mirror radius of curvature. According to the sign convention,
both, mirror radius of curvature and its focal length are numerically
negative. While it is usually applied to the radius, mirror focal length
is often given positive, for practical reasons.
Relation between object distance O,
image-to-objective separation I and objective's focal length
ƒ can be expressed in general form given by
the Gaussian lens formula (also valid for mirrors and objectives
in general) as:
(1/ƒ)=(1/I )+(1/O),
or ƒ=IO/(I+O)
(1.3)
For very distant objects, 1/O
gravitates to zero, and O/(I+O) to 1, with the focal length
ƒ and image separation I practically
coinciding. Evidently, the relation directly implies that the closer
the object, the farther from the objective its image. Thus,
telescope focal length can also be described as the smallest
distance at which telescope objective can form a real image.
Note that the above formula applies when the
imaging medium is air. In general, focal length is defined by ƒ=nIO/(I+O), with n being the
refractive index of the imaging medium. Hence, the effective focal
length increases with the medium refractive index.
◄
1.2. Reflection and refraction
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2. MAIN FUNCTIONS OF A TELESCOPE
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