# Conjugate Ratio: ```Conjugate Ratio:
The ratio of object distance s to image distance s”along the principal axis of a
lens or mirror. For an object at the focal point of a lens, the conjugate ratio is
infinite.
For an object placed at twice the focal length of the lens or mirror, the image
is formed at twice the focal length and the conjugate ratio is 1.
F =50 mm, s = 200 mm, s "= 66.7 mm
Conjugate Points:
The object point and image point of a lens system are said to be conjugate points.
Since all the light paths from the object to the image are reversible, it follows that if
the object were placed where the image is, an image would be formed at the
original object position.
Plano-convex lens
Plano-concave lens
Bi-convex lens
Bi-concave lens
Positive meniscus lens
Negative meniscus lens
Focal lengths and focal point
θ1
F-Number (f/#):
f/# = f /φ
φ
The f-number (also known as the focal ratio, relative aperture, or speed) of a lens
system is defined to be the effective focal length divided by system clear aperture
Numerical Aperture (NA): NA = n sinθ
θ
The numerical aperture of a lens system is defined to be the sine of the angle, θ1, that
the marginal ray (the ray that exits the lens system at its outer edge) makes with the
optical axis multiplied by the index of refraction (n ) of the medium.
Stops in Optical Systems
In any optical system, one is concerned with a number of things
including:
1. The brightness of the image
Two lenses of the same
focal length (f), but
diameter (D) differs
S
S’
Aperture Stop
A stop is an opening (despite its
name) in a series of lenses, mirrors,
diaphragms, etc.
The stop itself is the boundary of the
lens or diaphragm
Aperture stop: that element of the
optical system that limits the cone of
light from any particular object point
on the axis of the system
Aperture Stop (AS)
O
Entrance Pupil (EnP)
is defined to be the image of the aperture stop in all the lenses
preceding it (i.e. to the left of AS - if light travels left to right)
E’
How big does the
aperture stop look
E
to someone at O
L1
E’E’ = EnP
O
F1’
E
EnP – defines the
cone of rays
accepted by the
system
Exit Pupil (ExP)
The exit pupil is the image of the aperture stop in the lenses
coming after it (i.e. to the right of the AS)
E’’
E
L1
E”E” = ExP
F2’
O
E
E’’
Chief Ray
• for each bundle of rays, the light ray which
passes through the centre of the aperture stop
is the chief ray
• after refraction, the chief ray must also pass
through the centre of the exit and entrance
pupils since they are conjugate to the aperture
stop
• EnP and ExP are also conjugate planes of the
complete system
Marginal Ray
• Those rays (for a given object point) that
pass through the edge of the entrance and
exit pupils (and aperture stop).
Field Stop
• That component of the optical system that limits the field of
view
A
d
A = field of view at distance d
= angular field of
view
Optical Aberrations and Diffraction
An ideal lens will focus an input parallel beam to a perfect point
(focal point). The size of the focal point should be infinitesimal.
However, because of lens aberrations and diffraction, the focal
spot of a real lens has a finite size. The size of the focal spot is a
measure of lens aberrations and diffraction. All singlet lenses
have significant amount of aberrations.
Optical Aberrations
The sine functions in Snell's law can be expanded in an
infinite Taylor series:
The first approximation we can make is to replace all sine functions with their arguments
(i.e., replace sin θ 1 with θ 1 itself and so on). This is called first-order or paraxial theory
because only the first terms of the sine expansions are used.
Design of any optical system starts with this approximation.
The assumption that sin θ = θ is reasonably valid for θ close to zero (i.e., high f-number lenses).
With more highly curved surfaces (and particularly marginal rays), paraxial theory yields
increasingly large deviations from real performance because sin θ θ.
These deviations are known as aberrations.
Seidel developed a method of calculating aberrations resulting from the
θ3/3! term.
The resultant third-order lens aberrations are therefore called Seidel
aberrations.
Seidel put the aberrations of an optical system into several different
classifications:
In monochromatic light they are
spherical aberration, astigmatism, field curvature, coma, and distortion.
In polychromatic light there are also
chromatic aberration and lateral color.
In practice, aberrations occur in combinations rather than alone. This
system of classifying them, which makes analysis much simpler, gives
a good description of optical system image quality.
Focusing with aberration-free lens
Focusing with a lens exhibiting spherical aberration
The distance along the optical axis between the
intercept of the rays that are nearly on the optical
axis (paraxial rays) and the rays that go through the
edge of the lens (marginal rays) is called
longitudinal spherical aberration (LSA).
The height at which these rays intercept the
paraxial focal plane is called transverse spherical
aberration (TSA). These quantities are related by
Astigmatism
Coma
Imaging an off-axis point source by a lens with positive transverse coma
A wavefront with coma
Field Curvature
Distortion
Chromatic aberrations
Lognitudinal chromatic aberration
Lateral Color
Aberrations of positive singlets as a function of shape for infinite conjugate ratio
Coddington shape factor, q, defined as
Three possible systems for use at the unit conjugate ratio
Cooke Triplet Lenses
All above lens configurations provide improved performance on-axis only. To
achieve good performance both on- and off-axis, more complex lens forms are
required. Cooke triplet is a well-know lens form that provides good imaging
performance over a field of view of +/- 20-25 degrees. Many consumer grade
film cameras use lenses of this type.
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