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Chapter 5: Geometric Optics
• Introduction
– When the dimensions of the system are large compared to the wavelength of the light and the effects of diffraction become less important,
we have the domain of Geometric Optics.
• A lens is a refracting device ie. a disconinuity in the prevailing medium
that reconfigures the transmitted energy distribution.
• A lens is made up of a some material with two curved surfaces, each with
possibly different radii of curvature. To analyze this, consider refraction at
a spherical surface.
• Refraction at a spherical surface
– Look at figure 5.6, p. 153 in the book.
– C is the center of curvature of the spherical surface, with vertex V. The
line joining CV is th eoptical or principal axis. The index of refraction
of the medium is n2 whilst that of the initial medium is n1 .
– A point source is placed at s0 , with s0 called the object distance.
– Table 5.1, p. 154 gives the sign convention used when looking at
spherical refracting surfaces and thin lenses.
– An image is formed at si , where si is called the image distance.
– OPL stands for optical path length.
– By considering the optical path length for a ray, can show that at the
spherical interface,
n1 n2
n2 − n1
+
=
,
so
si
R
where n1 and n2 are the refractive indices in the incident and ”second”
(ie. the medium out of which the apsherical surface is made) medium
respectively. so and si are the object and image distances and R is the
radius of curvature of the spherical interface.
– A paraxial ray is a ray that arrives at a shallow angle to the optical
axis.
– Assuming small values of φ, the angle between the transmitted ray
and the optical axis leads to the above equation and what is known as
first order, paraxial or Gaussian Optics.
– If the object is placed such that the image is at infinity ie. si = inf,
then so = fo , the first focal length or the object focal length so that
n1
fo =
R.
n2 − n1
1
– Similarly if so = inf, then si = fi , or the second or image focal length
and is given by
n2
fi =
R.
n − 2 − n1
– An image is real when the rays converge to it.
– An image is virtual when the rays diverge from it.
– An object is virtual when the rays diverge to it.
• Thin Lenses
– Lenses are essentially two spherical refracing surfaces, each with perhaps different radii of curvature, stuck together.
– Lenses are convex, converging or positive lenses or they are concave,
diverging or negative lenses.
– Each surface of the lesn has a radius of curvature. Denote quantities
relating to the interface facing the object with a subscript 1 and the
interface facing away from the object with a subscript 2.
– Thus we have C1 , V1 and C2 , V2 . The quantitiy d is the thickness of
the lens.
– Look at figure 5.14, p. 157 and apply the equation derived above to
each interface.
– This leads to
1
1
nl d
nm nm
+
= (nl − nm )(
−
+
.
so1
si2
R1 R2 (si1 − d)si1
– Here, nm is the refractive index outside of the lens and nl is the refractive index in the lens.
– Here, so1 , so2 are the object distances for the first and second interfaces
and si1 , si2 are the image distances for the first and second interfaces.
– R1 , R2 are the radii of curvature for the first and second interface
respectively, that is the side of the lens facing to and away from the
object respectively.
– Applying the thin lens approximation (d ≈ 0) and assuming the surrounding medium is air (ie nm = 1), we get the thin lens equation,
1
1
1
1
+ = (nl − 1)(
−
),
s0 si
R1 R2
where we let so1 = so and si2 = si .
– This is the thin lens equation or the lens maker’s formula.
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– As before, in the limit as so tends to infinity, then si = fi , and in the
limnit as si tends to infinity, then so = fo .
– Then it can be shown that
1
1
−
),
1overf = (nl − 1)(
R1 R2
1
1
1
+ = .
s0 si
f
– This is the Gassian lens formula.
– The magnification is
yi
si
MT =
=− .
yo
so
– A positive/negative MT denotes an erect/inverted image. If MT has
magnitude greater than or less than equat to 1, the image is magnified
or minified respectively.
– Table 5.2, p. 163 denotes sign conventions used in this field. Table 5.3
gives a summary of various quantities.
– In considering thin lens combinations, apply the this lens equation to
each lens so that the image of one lens is the object of the next lens in
the system. Ray 2 leaves the object and is parallel to the optical axis.
Ray 3 goes through an object focus and strikes the lens.
– In geometric construction, use 3 types of rays: Ray 1 is through the
optical center of the lens and is undeflected.
• Mirrors
– For all mirros, angle of incidence equals angle of reflection.
– For spherical mirrors,
1
2
1
+ =− ,
so si
R
where all quantities have theor usual meanings.
– Again by considering what happens when first the object and then the
object are placed at infinity, we get
−R
,
fo = fi = f =
2
1
1
1
+ = .
so si
f
– Table 5.4, 5.5, p. 184-185 give sign conventions and summary properties of different types of mirrors.
– The remaining part of chapter 5 describes some applications of these
laws in optical systems such as the eye, telescope, camera and fibreoptics.
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