Chapter 26 Geometrical Optics
Section 26-1 Waves, geometrical optics Wave optics analyzes the passage of light through an optical system in
terms of electromagnetic waves. Geometric optics analyzes the passage of light through an optical system in
terms of light rays, i.e. the direction in which light travels as it passes through an optical system. Light rays
are perpendicular to wave fronts.
•
Reflection Mirrors (or mirror-like surfaces) reflect light so that the angle of incidence is equal to the angle of
refraction ( θi = θr ). Both of these angles are defined with respect to the normal to the mirror surface. The
normal makes a 90° angle with respect to the mirror surface.
Section 26-2 Plane mirror Plane mirrors reflect light and form a virtual image. The virtual image is located at a
point determined by light rays that are traced back to the point from which they appear to originate. No light
rays actually meet or originate at a virtual image. To describe an optical image requires three descriptors: type
(real or virtual), orientation (right-side-up, inverted, perverted), and relative size (magnified, same size, or
minified). The image formed by a plane mirror is thus: virtual, same size, and perverted (left-right symmetry
is inverted).
Section 26-3 Spherical mirrors The focal length of a concave spherical mirror is given by f = R/2, where R is the
radius of curvature of the "sphere". It is assumed that only light rays near the principal axis will be
considered, in order to avoid spherical aberration.
•
Convex spherical mirror Convex mirrors diverge light, so they can only form virtual images. The focal length
is given by f = –R/2, thus f is negative. Since f < 0, di will always be negative for a convex mirror (thus only
virtual images). Aside from that fact, convex mirrors can be analyzed using the mirror equation and ray traced
using the same techniques as for concave mirrors. One has to be careful with the ray tracing since light rays will
be aimed at points on the opposite side of the mirror surface.
Section 26-4
Ray tracing, the mirror equation A ray diagram consists of: the object, the mirror, and three light
rays emitted from the part the object that is farthest from the principal axis. One ray should initially be
parallel (P) to the principal axis and will reflect through the focal point. One ray should initially pass
through the focal point (F) and will then be reflected parallel to the principal axis. Finally, a ray should
point through the center of curvature (C) and will reflect back along itself.
The focal length, object distance, and image distance can be found numerically using the mirror equation (1/f =
1/do + 1/di ) if any two of the three values is known (see formulas on back). All three of these values have a
sign convention (see the table on the back), and images may be either real or virtual. The magnification of an
di
hi
object is given by m = –
=
. The minus sign is included to insure that the results agree with the sign
do h o
conventions object and image distances.
Section 26-5
Refraction Light is bent when it passes from one type of transparent material (medium) into
another. The amount of bending depends on the type of material and the wavelength of the light. The
information about the type of material is included in the index of refraction, which is defined by comparing the
speed of light in vacuum to the speed of light in a given material (n = c/v). The index of refraction has its
lowest value for vacuum (=1), thus light travels fastest in vacuum and slower in transparent materials. The law
of refraction (Snell's law) quantifies the connection between the incident and refracted light ray: ni sin θi = nt
sin θt .
•Total internal reflection This is a phenomenon only observed when light passes from a region of higher index
of refraction into a region of lower index of refraction. There exists a critical angle above which incident light
n2
is totally reflected back into the incident medium (Figure 26-25) where: sin θc =
n1
Section 26-6
Ray tracing The lens considered are assumed to be "thin" lenses (where light that passes through
the center of the lens exits along the original line of sight). It is also assumed that only light rays near the
principal axis (paraxial) will be considered, in order to avoid problems such as spherical and chromatic
aberrations.
A ray diagram consists of: the object, the lens, and three light rays emitted from the part the object that is
farthest from the principal axis. For an object outside the object focal point of a convex lens, One ray should
initially be parallel (P) to the principal axis and will refract through the focal point. One ray should initially
pass through the focal point (F) and will then be refracted parallel to the principal axis. Finally, a ray
pointing through the center of the lens (M) will travel back along the same direction, unrefracted. An object
inside the focal point of a convex lens will create a virtual image, and a concave lens will always create a
virtual image.
Section 26-7 The focal length, object distance, and image distance can be found numerically using the lens equation
(1/f = 1/do + 1/di ) if any two of the three values is known (see formulas on the back side). All three of these
values have a sign convention (see the table on the back side), and images may be either real or virtual. The
di
hi
magnification of an object is given by m = –
=
. The minus sign is included to insure that the results agree
do
ho
with the sign conventions.
Section 26-8 Dispersion Dispersion can be physically observed as the effect where light of different wavelengths
is bent by different amounts when passing through the same piece of glass (this is called chromatic aberration
in lenses). This is due to the fact that the speed of propagation depends on the frequency of the light. The index
of refraction is roughly inversely proportional to wavelength. Thus, longer wavelength light (red) is bent less
than shorter wavelength light (violet).
Optics Formulas
1
1
1
Lens (or mirror) equation: f = d + d
o
i
hi
di
Magnification: m = – d = h
o
o
do di
f = d + d
o
i
di = – m do
do f
di = d – f
o
di f
do = d – f
i
di
do = – m
hi
ho = m
h i = m ho
Optics Sign Conventions (for single lens or mirror)
Focal
length = f
+ (positive)
Object
distance = do
+ (do> f)
+ (positive)
+ (do < f)
– (negative)
+ (do< f and do
> f)
Object height
= ho
+ (above
principal axis)
– (below
principal axis)
+ (above
principal axis)
– (below
principal axis)
+ (above
principal axis)
– (below
principal axis)
Image Image height =
distance
hi
= di
+ (real)
– (below
principal axis)
+
+ (above
principal axis)
–
+ (above
(virtual) principal axis)
–
– (below
principal axis)
–
+ (above
(virtual) principal axis)
–
– (below
principal axis)
Magnification =
hi
di
m = h =– d
o
o
– (inverted image)
–
+ (right-side-up
image)
+
+ (right-side-up
image)
+