Transcript Image Formation & Optical Instruments

Images
Chapter 35
Mirrors
Lenses
Image Formation
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.
Images Formed by Plane Mirrors
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.
object
image
virtual image
(the ray reaching your
eye doesn’t really come
from the image)
But…. the brain thinks the ray came from the image.
Images Formed by Plane Mirrors
You can locate each point on the image with two rays:
1. A ray normal to the mirror
object
image
Image is reversed
front to back
Images Formed by Plane Mirrors
You can locate each point on the image with two rays:
1. A ray normal to the mirror
2. The ray that reaches the observer’s eye
object
image
Image is reversed
front to back
Images Formed by Plane Mirrors
You can locate each point on the image with two rays.
object
image
Image is reversed
(front to back)
Images Formed by Plane Mirrors
You can locate each point on the image with two rays.
object
image
Image is reversed
(front to back)
Images Formed by Plane Mirrors
The distance from the image to the mirror equals
the distance from the object to the mirror: p = i
p
object
i
image
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 parabola’s axis of rotation, is
reflected to pass through a
common point - the focus.
• What about the reverse?
•
Parabolic Mirrors
• 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.
Spherical 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.
f
c
A ray parallel to the mirror axis reflects through the focal point f
A ray passing through the focus reflects parallel to the axis
A ray that strikes the center of the mirror reflects symmetrically
A ray passing through the center of curvature c, returns on itself
Spherical Mirrors
When the object is beyond c, the image is:
real (on the same side as the object), reduced,
and inverted.
f
c
Spherical Mirrors - Concave
Object between c and f.
f
c
Image is real, inverted, magnified
Spherical Mirrors
Object between f and the mirror.
f
c
Image is virtual, upright, magnified
The Mirror Equation
c
f
i
p
1  1 1
f p i
Here f = R / 2
Magnification
h
f
h’
c
The magnification is given by the ratio M = h’ / h = - i/p
Curved Mirrors
mirror equation
focal length
magnification
Sign conventions:
1 1 1
 
p i
f
f  R /2
h'
i
M  
h
p
Distance in front of the mirror  positive
Distance behind the mirror  negative
Height above center line  positive
Height 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
negative
mirror is
concave
convex
Image With a Convex Mirror
c
f
Here the image is virtual (apparently positioned behind
the mirror), upright, and reduced. Can still use the mirror
equations (with negative distances for f, c=R, and i).
The Simple Lens
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
•
Snell’s Law applied at each surface will show where
the light comes to a focus.
Image Formation in a Lens
Each point in the image can be located using two rays.
f
f
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.
Some Simple Ray Traces
f
2f
f
f
Object between 2f and f
Image is inverted, real
enlarged.
f
2f
2f
2f
Object between f and lens
Image is upright, virtual,
and enlarged.
Some Simple Ray Traces (diverging lens)
f
2f
f
Object beyond 2f.
Image is upright,
virtual, reduced.
f
f
2f
2f
2f
Object between
f 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 positive
image 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 negative
image 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
f
h
2f
h’
f
2f
f
p
M = h’/h = -i/p
1/p+ 1/i= 1/f
i
Magnification
The thin lens equation
About the Thin Lens Formula
1/p + 1/i = 1/f
•
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.
The Lensmaker’s Formula
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.
 1
1 1
1 
   n  1


 
 R1 R 2 
Lensmaker’s Formula
 1
1
1 
  n  1


f
 R1 R 2 
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.
The retina senses the light.
Your eye’s lens is a simple
refracting lens which you can
deform to change focus.
The Eye: Near & Farsightedness
Far-sighted:
Near-sighted:
Eye too short:
correct with a
converging lens
Eye too long:
correct with a
diverging 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
A telescope magnifies the angular size
(or separation) of objects.
The Telescope
“Objective”
fo
a
fo
The “objective” lens first creates an image, in the focal plane,
of a point at infinity.
The Telescope
“Objective”
fo
fe “Eyepiece”
EYE
a
b
fe
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 b.
The Telescope
“Objective”
fe “Eyepiece”
fo
EYE
a
b
h
fe
fo
tan b/ tan a = (h/fe)/(h/fo)
The eye sees an image at infinity, but at an angle of b instead of a.
The magnification is therefore:
M = b/a= fo / fe.