We will use geometrical optics: light propagates in straight lines until its direction is changed by reflection or refraction.
When we see an object directly, light comes to us straight from the object.
When we use mirrors and lenses, we see light that seems to come straight from an object but actually doesn’t.
Thus we see an image (of the object), which may have a different position, size, or shape than the actual object.
Image Formation
Images Formed by Plane Mirrors
(the ray reaching your eye doesn’t really come from the image)
object image
virtual image
But…. the brain thinks the ray came from the image.
When we use mirrors and lenses, we see light that seems to come
straight from the object but actually doesn’t. Thus we see an
image, which may have a different position, size, or shape than
the actual object.
Images Formed by Plane Mirrors
You can locate each point on the image with two rays:1. A ray normal to the mirror
Image is reversedfront to back
object image
Images Formed by Plane Mirrors
You can locate each point on the image with two rays:1. A ray normal to the mirror2. The ray that reaches the observer’s eye
Image is reversedfront to back
object image
Images Formed by Plane Mirrors
object image
Image is reversed(front to back)
You can locate each point on the image with two rays.
Images Formed by Plane Mirrors
object image
Image is reversed(front to back)
You can locate each point on the image with two rays.
Images Formed by Plane Mirrors
object image
The distance from the image to the mirror equals the distance from the object to the mirror: p = i
p i
Also, the height of the image equals the height of the object
Parabolic Mirrors
• Shape the mirror into a parabola of rotation (In one plane it has cross section given by y = x2).
• All light going into such a mirror, parallel to the para- bola’s axis of rotation, is reflected to pass through a common point - the focus.
• What about the reverse?
•
• These present the concept of a focal point - the point to which the optic brings a set of parallel rays together.
• Parallel rays come from objects that are very far away (and, after reflection in the parabolic mirror, converge at the focal point or focus).
• Parabolas are hard to make. It’s much easier to make spherical optics, so that’s what we’ll examine next.
Parabolic Mirrors
To analyze how a spherical mirror works we draw
some special rays, apply the law of reflection where
they strike the spherical surface, and find out where
they intersect.
Spherical Mirrors
cf
A ray parallel to the mirror axis reflects through the focal point fA ray passing through the focus reflects parallel to the axisA ray that strikes the center of the mirror reflects symmetricallyA ray passing through the center of curvature c, returns on itself
When the object is beyond c, the image is:
real (on the same side as the object), reduced,
and inverted.
Spherical Mirrors
c
f
Curved Mirrors
mirror equation
focal length
magnification
€
1
p+
1
i=
1
f
f = R /2
M =h'
h= −
i
pSign conventions:
Distance in front of the mirror positiveDistance behind the mirror negativeHeight above center line positiveHeight below center line negative
Positive and Negative Mirrors
• You can fill a positive mirror with water.• You can’t fill a negative mirror.
positive mirror is concave
negative mirror is convex
Image With a Convex Mirror
c
f
Here the image is virtual (apparently positioned behindthe mirror), upright, and reduced. Can still use the mirror equations (with negative distances for f, c=R, and i).
A simple lens is an optical device which takes
parallel light rays and focuses them to a point.
This point is called the focus or focal point
The Simple Lens
Snell’s Law applied at each surface will show where the light comes to a focus.
•
Each point in the image can be located using two rays.
Image Formation in a Lens
Ray tracing:
1. A ray which leaves the object parallel to the axis, is refracted to pass through the focal point.
2. A ray which passes through the lens’s center is undeflected.
3. A ray passing through the focal point (on the object side) is refracted to end up parallel to the axis.
f
f
Some Simple Ray Traces
2f
2f
f
f
2f
2f
f
f
Object between 2f and f
Image is inverted, realenlarged.
Object between f and lens
Image is upright, virtual,and enlarged.
Some Simple Ray Traces (diverging lens)
2f
2f
f
f
2f
2f
f
f
Object beyond 2f.
Image is upright,virtual, reduced.
Object betweenf and lens.
Image is upright, virtual, reduced.
A ray parallel to the axis diverges such that its extension passes through the focal point.
Sign Conventions
1. Converging or convex lens
focal length is positiveimage distance is positive when on the other side of the lens (with respect to object)height upright is positive, inverted is negative
2. Diverging or concave lens
focal length is negativeimage distance is always virtual and negative(on the same side of the lens as the object)height upright is positive, inverted is negative
The Thin Lens Equation
2f
2f
f
f
fp i
h’
h
M = h’/h = -i/p Magnification
1/p+ 1/i= 1/f The thin lens equation
About the Thin Lens Formula
• When p = f, i = infinity• When p = 2f , i = 2f and the magnification is 1.• When f > p > 0, i is negative
– This means that the image is virtual, and so it is on the same side of the lens as the object.
• If f < 0, i is always negative– A negative lens can not produce a real image. It
always produces a virtual image.
1/p + 1/i = 1/f
The lens formula gives the image distance as a function of the object distance and the focal distance [1/f -1/p = 1/i]
The lensmaker’s formula gives the focal distance f as a function of R1 and R2, the radii of curvature of the two surfaces of the lens.
The Lensmaker’s Formula
( )
( )
1 1n 1
1
R
1
R
1
fn 1
1
R
1
R
1 2
1 2
l l+
′= − −
⎛
⎝⎜
⎞
⎠⎟
= − −⎛
⎝⎜
⎞
⎠⎟
Lensmaker’s Formula
The Eye: A Simple Imager
• A simple lens focuses the light onto the retina -- the photosensor
• The retina sends signals to the brain about which sensor is illuminated, what color the light is, and how much of it there is.
• The brain interprets.Your eye’s lens is a simple refracting lens which you can deform to change focus.
The retina senses the light.
The Eye: Near & Farsightedness
Far-sighted: Near-sighted:
Eye too short:correct with aconverging lens
Eye too long:correct with adiverging lens
Image Forming Instruments
• Cameras are much like the eye except for having a shutter and better lenses.
• Binoculars and telescopes create magnified images of distant objects.
• Microscopes create magnified images of very small objects.
The UCF Robinson Observatory
The telescope here is a 66 cm (26 in) dia. Schmidt-Cassegrain reflecting telescope.
Telescopes are described by their diameter since that describes its ability to collect light from distant stars and glaxies; and that’s more important than magnification.
Florida is not ideal for most observational astronomy. Why?
Image-Forming Instruments: the Telescope
When we look at distant objects, we judge their sizes from their angular size. In the case of stars, what we observe is their angular separation.
A telescope magnifies the angular size (or separation) of objects.
The Telescope
fo
“Objective” fo
The “objective” lens first creates an image, in the focal plane,of a point at infinity.
The Telescope
fo
“Objective” fo
The “objective” lens first creates an image, in the focal plane,of a point at infinity. The “eypiece” is placed fo + fe from the objective, so that it produces parallel rays into the eye, at angle .
fe
fe
“Eyepiece”
EYE