Geometrical Optics /
Mirror and Lenses
Outline
Reflection
Plane Mirrors
Concave/Convex Mirrors
Refraction
Lenses
Dispersion
Geometrical Optics
In describing the propagation
of light as a wave we need to
understand:
wavefronts: a surface passing
through points of a wave that
have the same phase and
amplitude.
rays: a ray describes the
direction of wave propagation.
A ray is a vector perpendicular
to the wavefront.
Reflection and Refraction
When a light ray travels from one medium to another, part of the
incident light is reflected and part of the light is transmitted at the
boundary between the two media.
The transmitted part is said to be refracted in the second medium.
http://www.geocities.com/CapeCanaveral/Hall/6645/propagation/propagation.html
*In 1678 the great Dutch physicist Christian Huygens (1629-1695) wrote a treatise called
Traite de la Lumiere on the wave theory of light, and in this work he stated that the wavefront
of a propagating wave of light at any instant conforms to the envelope of spherical wavelets
emanating from every point on the wavefront at the prior instant. From this simple principle
Huygens was able to derive the laws of reflection and refraction
incident ray
reflected ray
refracted ray
Types of Reflection
When light reflects from a
smooth surface, it undergoes
specular reflection (parallel
rays will all be reflected in the
same direction).
When light reflects from a
rough surface, it undergoes
diffuse reflection (parallel rays
will be reflected in a variety of
directions).
The Law of Reflection
For specular reflection the incident angle θi
equals the reflected angle θr:
θ i = θr
(Known since 1000 BC)
The angles are
measured relative
to the normal,
shown here as a
dotted line.
Forming Images with a Plane Mirror
A mirror is an object that reflects light. A plane mirror is simply a flat
mirror. Plane mirrors are ground to be flat – the flatter the more
expensive. (Typically good ones have - where we use visible radiation
- no hills or valleys larger than 500nm).
Consider an object placed at point P in front of a plane mirror. An
image will be formed at point P´ behind the mirror.
do = distance from object to
mirror
di = distance from image to
mirror
ho = height of object
hi = height of image
For a plane mirror:
ho
do
di
do = di and ho = hi
vertex Q = do + di
hi
Images
An image is formed at the point where the rays of light
leaving the object either actually intersect or where they
appear to originate.
If the light rays actually do intersect, then the image is a real
image. If the light only appears to be coming from a point,
but is not physically there, then the image is a virtual image.
We define the magnification, m, of an image to be:
di
image height hi
m=
=
=−
object height ho
do
If m is negative, the image is inverted
(upside down).
The image is called virtual because it does
not really exist behind the mirror
Real image
Plane Mirrors
A plane mirror image has the following properties:
*The mirror in your bathroom is a piece of plate glass with a coating on the
backside so they are second surface mirrors.
•
•
•
•
•
The image distance equals the object distance.
The image is unmagnified.
The image is virtual.
The image is not inverted.
Left and right are reversed
**The intensity of the reflected beam depends upon the angle of
incidence and the indices of refraction and they type of coating.
To save expenses, you would like to buy the
shortest mirror that will allow you to see your entire
body. Should the mirror be (a) half your height (b)
two-thirds your height, or (c) equal to your height?
Does the answer depend on how far away from
the mirror you stand?
eye
mirror
you
Spherical Mirrors
concave
A spherical mirror is a mirror
whose surface shape is
spherical with radius of curvature
R. There are two types of
spherical mirrors: concave and
convex. **The principal axis (optical axis,
vertex) is the straight line between C and the
midpoint of the mirror
We will always orient the mirrors
so that the reflecting surface is
on the left. The object will be on
the left.
convex
Focal Point
When parallel rays are
incident upon a
spherical mirror, the
reflected rays intersect
at the focal point F.
For a concave mirror,
the focal point is in front
of the mirror.
For a convex mirror, the
focal point is behind the
mirror.
The incident rays
diverge from the convex
mirror, but they trace
back to the focal point F.
Focal Length
The focal length f is the distance from the surface of
the mirror to the focal point. It can be shown that
the focal length is half the radius of curvature of the
mirror.
Sign Convention: the focal length is negative if the
focal point is behind the mirror.
For a concave mirror, f = ½R
For a convex mirror, f = −½R
(R is always positive)
Ray Tracing
We will use three
principal rays to
determine where an
image will be
located.
Optical axis
Curvature point
Curvature point
The parallel ray (P ray) reflects
through the focal point. The focal
ray (F ray) reflects parallel to the
axis, and the center-of-curvature
ray (C ray) reflects back along its
incoming path.
Ray Tracing – Examples
concave
Real image
applet mirror/lens
convex
Virtual image
Theorem of intersecting lines
h0
d0 − R
=
− hi R − d i
h0
d0
=
− hi d i
R − di di
=
do − R do
with
f= ½ R
Mirror Equation
1 1 1
= +
f d0 di
The Mirror Equation
The ray tracing technique
Sign Conventions:
shows qualitatively where the
do is positive if the object is in front of
image will be located. The
the mirror (real object)
distance from the mirror to the
image, di, can be found from
do is negative if the object is in back of
the mirror (virtual object)
the mirror equation:
1 1 1
+ =
do di f
do = distance from object to
mirror
di = distance from image to
mirror
f = focal length
di is positive if the image is in front of
the mirror (real image)
di is negative if the image is behind
the mirror (virtual image)
f is positive for concave mirrors
f is negative for convex mirrors
m is positive for upright images
m is negative for inverted images
The Refraction of Light
The speed of light is different in different materials. We
define the index of refraction, n, of a material to be the ratio
of the speed of light in vacuum to the speed of light in the
material:
n = c/v
When light travels from one medium to another, its velocity
and wavelength change, but its frequency remains
constant.
http://www.geocities.com/CapeCanaveral/Hall/6645/propagation/propagation.html
Snell’s Law
In general, when light enters a new material its direction will
change. The angle of refraction θ2 is related to the angle of
incidence θ1 by Snell’s Law: sinθ1 sin θ 2
v1
=
v2
= constant
where v is the velocity of light in the medium.
Snell’s Law can also be written as:
n1sinθ1 = n2sinθ2
The angles θ1 and θ2 are
measured relative to the line
normal to the surface
between the two materials.
Normal line
θ1
Air
Glass
θ2
Example: Which way will the rays bend?
n = 1.4
n=2
n = 1.6
n = 1.2
Which of these rays can be the refracted ray?
Total Internal Reflection
When light travels from a medium with n1 > n2,
there is an angle, called the critical angle θc, at
which all the light is reflected and none is
transmitted. This process is known as total
internal reflection. The critical angle occurs
when θ2= 90 degrees:
n
sin θ c = 2
n1
The incident ray is both reflected and
refracted.
Total Internal Reflection
A pencil in a glass of water looks
bent due to the light refraction
A mirage is created due to
the bending of light. The
index of refraction of the
hot air near the ground is
lower than the n of the
colder air on the top.
Object in the sky appear to be shifted towards the zenith by a small amount.
This is due to the refractive effect of the atmosphere. This has been known
since the time of Ptlomey in Egypt in 150 BC.
ASTRONOMICAL REFRACTION: The displacements of astronomical objects by
atmospheric refraction.
These effects are many orders of magnitude larger than the accuracy of the best
astronomical position measurements, and so large that the mountings of most
astronomical telescopes are adjusted to minimize the effects of refraction.
http://www.sundog.clara.co.uk/rainbows/primrays.htm
Refraction in a Triangular Prism
n=1
n>1
Light always bends toward the base of a triangular prism (if n of
the prism is higher than the ambient n).
Different colors bend differently. It means that n is different for
different colors. The separation of colors is called light
dispersion. http://www.wolles-website.de/teste_taeuschungen/taeuschungen_uebersicht.htm
Lenses
A lens is an object that
uses refraction to bend
light and form images
Light is reflected from
a mirror. Light is
refracted through a
lens.
Focal Point
The focal point of a lens is the place where
parallel rays incident upon the lens converge.
converging lens
diverging lens
Ray Tracing for Lenses
Just as for mirrors we use
three rays to find the image
from a lens. The lens is
assumed to be thin.
The P ray propagates parallel to the principal axis until it encounters the
lens, where it is refracted to pass through the focal point on the far side of
the lens. The F ray passes through the focal point on the near side of the
lens, then leaves the lens parallel to the principal axis. The M ray passes
through the middle of the lens with no deflection.
Ray Tracing Examples
The Thin Lens Equation
The ray tracing technique shows qualitatively
where the image from a lens will be located.
The distance from the lens to the image, di,
can be found from the thin-lens equation:
Sign Conventions:
1 1 1
+ =
do di f
do is positive for real objects (from which light diverges)
do is negative for virtual objects (toward which light converges)
di is positive for real images (on the opposite side of the lens from the
object)
di is negative for virtual images (same side as object)
f is positive for converging (convex) lenses
f is negative for diverging (concave) lenses
m is positive for upright images
m is negative for inverted images
Lens maker’s formula
The equation in the box is the thin lens equation. The
focal length is given by the lens maker’s formula:
⎛1
1
1 ⎞
= (n − 1) ⎜ − ⎟
f
⎝ R 1 R2 ⎠
This expression is good for a lens in air. The R-values
are the radii of curvature of the first and second
surfaces of the lens. n is the refraction index. So f is
determined by construction: n and curvature R’s
ARE fixed by construction.
http://www.phy.ntnu.edu.tw/java/Lens/lens_e.html
**Not all lenses are thin lenses - Thick lens equation:
Dispersion
In a material, the velocity of light (and therefore the index of
refraction) can depend on the wavelength. This is known
as dispersion. Blue light travels slower in glass and water than does
red light. (The shorter wavelengths are refracted by the greatest
amount)
As a result of dispersion,
different colors entering a
material will be refracted at
different angles.
Dispersive materials can be
used to separate a light
beam into its spectrum (the
colors that make up the light
beam). Example: prism