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 RefractionWhen a light ray travels from one medium to another, part of theincident 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 wavefrontof 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 ReflectionFor specular reflection the incident angle iequals the reflected angle r:

i = r(Known since 1000 BC)

The angles are measured relative to the normal, shown here as a dotted line.

Forming Images with a Plane MirrorA 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 diFor a plane mirror:do = di and ho = hi

ho hi

do = distance from object to mirror

di = distance from image to mirror

ho = height of objecthi = height of image

vertex Q = do + di

ImagesAn 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:

o

i

o

idd

hh

m ===heightobject height image

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 MirrorsA 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

youmirror

Spherical Mirrors

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.

concave

convex

Focal PointWhen 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 LengthThe 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 = R

For a convex mirror, f = R

(R is always positive)

Ray Tracing

We will use three principal rays to determine where an image will be located.

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.

Curvature point

Curvature point

Optical axis

Ray Tracing Examples

concave convex

Virtual imageReal imageapplet mirror/lens

Theorem of intersecting lines

ii dRRd

hh

=

00

ii dd

hh 00 =

iddf111

0

+=

Mirror Equationi i

o o

R d dd R d

=

with

f= R

The Mirror EquationThe ray tracing technique shows qualitatively where the image will be located. The distance from the mirror to the image, di, can be found from the mirror equation:

fdd io

111=+

do = distance from object to mirror

di = distance from image to mirror

f = focal length

Sign Conventions:do is positive if the object is in front of

the mirror (real object)

do is negative if the object is in back of the mirror (virtual object)

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 LightThe 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

Snells LawIn general, when light enters a new material its direction will change. The angle of refraction 2 is related to the angle of incidence 1 by Snells Law:

where v is the velocity of light in the medium.

Snells Law can also be written as:

n1sin1 = n2sin2

constantv

sin sin

2

2

1

1 ==

v

1

2

Air

GlassThe angles 1 and 2 are measured relative to the line normal to the surface between the two materials.

Normal line

n = 1.2n = 1.6

Example: Which way will the rays bend?

Which of these rays can be the refracted ray?

n = 1.4 n = 2

Total Internal ReflectionWhen light travels from a medium with n1 > n2, there is an angle, called the critical angle c, at which all the light is reflected and none is transmitted. This process is known as total internal reflection. The critical angle occurs when 2= 90 degrees:

The incident ray is both reflected and refracted.

Total Internal Reflection

1

2sinnn

c =

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

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

n=1

n>1

Refraction in a Triangular PrismRefraction in a Triangular Prism

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 PointThe focal point of a lens is the place where parallel rays incident upon the lens converge.

converging lens diverging lens

Ray Tracing for Lenses

The P ray propagates parallel to the principal axis until it en