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Lenses and Images
• Lenses are used in -
• eyeglasses
• magnifying glasses
• cameras
• telescopes
• eye lens
LAW OF REFRACTION • Light going from a substance of small n to a substance of large n is bent TOWARD the normal
• Light going from a substance of large n to a substance of small n is bent AWAY from the normal
normal to surface
glass or water
diagram works in either direction
incident rays
refracted rays
How do rays behave at spherical convex interfaces?
• Convex glass surfaces are focusing
• Concave glass surfaces are defocusing
convex interface
C F
glass
Rays parallel to the axis are deflected through the focus
How do rays behave at spherical convex interfaces?
convex interface
C F
glass
F
Rays parallel to the axis are deflected through the focus
• Convex glass surfaces are focusing
How do rays behave at spherical convex interfaces?
convex interface
C F
glass
Rays from the focus are made parallel to the axis
• Convex glass surfaces are focusing
How do rays behave at spherical interfaces?
• Concave glass surfaces are defocusing
concave interface
F
glass
C F
Rays parallel to the axis are deflected as if they emerged from the focus
How do rays behave at spherical convex interfaces?
concave interface
F
glass
C F
Rays parallel to the axis are deflected as if they came from the focus
• Concave glass surfaces are defocusing
Ray Tracing for Thin Lenses
• RULE #1: A ray through the center of the lens is not bent
• RULE #2: A ray parallel to the axis is deflected through F (or as if it came from F)
• RULE #3: A ray from F is deflected parallel to the axis (this rule is the reverse of #2)
Notice how we approximate what should be 2 bends with 1 bend
Unlike a mirror there are two focal points of a lens, one on
either side
Lens Simulations
http://micro.magnet.fsu.edu/primer/java/lenses/converginglenses/index.html http://micro.magnet.fsu.edu/primer/lightandcolor/lenseshome.html http://micro.magnet.fsu.edu/primer/java/lenses/simplethinlens/index.html http://micro.magnet.fsu.edu/primer/java/lenses/diverginglenses/index.html http://micro.magnet.fsu.edu/primer/java/components/perfectlens/index.html
Why are Lenses curved anyway?
• Parallel rays parallel to the axis are imaged at the focus of the lens • The Arabian physicist and mathematician Ibn Sahl (c.940–c.1000) used what is now known as Snell's Law to calculate the shape of lenses
Why not just take two prisms stacked one above the other?
Review: Ray Tracing for Thin Lenses
• Where will the images be? Which ray tracing rules can we apply?
Lenses - examples of Ray Tracing
• Parallel rays parallel to the axis are imaged at the focus
• Parallel rays parallel to each other but not parallel to the axis are focused in the focal plane
Images
F
object
F
To locate the image - 1. Draw at least 2 rays: here is a third
Where is the image? Is it upright or inverted?
Concept question
F
object
F
Where is the base of the arrow in the image?
A. Above the axis
B. Below the axis
C. On the axis
D. There is no image
Concept question
F
object
F image height Si
Answer: Notice that with a convex lens, the image is flipped
Imaging with lenses
Concept question - if the person walks towards the lens, his image will become
A. Bigger
B. Smaller
C. Stay the same
Imaging with lenses
Concept question - if the person walks towards the lens, his image will become
A. Bigger
B. Smaller
C. Stay the same
Imaging with lenses
Concept question - if the person walks towards the lens, his image will become
A. Bigger
B. Smaller
C. Stay the same
D. Upright
E. Inverted
Imaging with convex lenses
http://micro.magnet.fsu.edu/primer/java/lenses/converginglenses/index.html
magnifying glass case: virtual image
photography case: real images
(can be projected on film)
real images
real images
virtual image of man
real image of tree
Virtual images in thin convex lenses - magnifying glass
thin convex lens
F F
object
image
when the object is closer to the lens than the focal point, we see a magnified, virtual image beyond it
Virtual images in thin convex lenses - magnifying glass
simple magnifying glass applet http://micro.magnet.fsu.edu/primer/java/lenses/simplemagnification/index.html
Object
Virtual images in thin concave lenses – ”demagnifying glass”
Is there one more ray that you can draw here?
thin concave lens
F F
object
image
diverging lens applet http://micro.magnet.fsu.edu/primer/java/lenses/diverginglenses/index.html
Magnification
Magnification = Size of image = Image distance Size of object Object distance
From similar triangles So = - Si => Si = - Xi Xo Xi So Xo
NOTE: Negative magnification means that the image is upside down
object height So
xo xi
image height Si
Magnification
Magnification = Size of image = Image distance Size of object Object distance
F
object height So
image height Si
xo xi
Finding images: the Thin Lens formula Say that I know F and the distance of the object from the lens
Can I find out where the image will be?
Yes from - 1/xo + 1/xi = 1/F
=> 1/xi = 1/F - 1/xo
where xi = distance to image, xo = distance to object, and F = focal length
Example of finding images Say that F = 0.5m, and xo = 0.8m 1/xo + 1/xi = 1/F
=> 1/xi = 1/0.5m - 1/0.8m = 2/m - 1.25/m = 0.75/m = 3/4m
=> xi = 1.33m
Raindrop lenses
A water droplet can act as a convex lens. Can you see the inverted image of the house in this picture?
Icicle drop lens
A water droplet can act as a convex lens. Can you see the inverted image of the tree in this picture?
Raindrop retroreflectors
A water droplet and a blade of glass can act as a retroreflector. Only one incident ray is shown, and only a few of the many diffusely reflected rays due to this one incident ray are shown.
Fresnel lenses • For lighthouses and stage lighting (spotlights) for example, to be efficient in collecting and focusing the light, one wants a lens that is large and that has a short focal length
• This suggests a very thick lens - but these are heavy, lossy, and are difficult to make and maintain
• So use a Fresnel lens instead (developed by Augustine Fresnel 1788-1827)
In a fresnel lens, most of the glass is removed.
Uses of Fresnel lenses
• Overhead projectors
• Theatre spotlights
• Focusing of x-rays
• Lighthouses
making fire with a Fresnel lens http://www.youtube.com/watch?v=zJsFCET-TuY&NR=1
More uses of Fresnel lenses
• Traffic lights
• Aircraft carriers plane landing on aircraft carrier http://www.youtube.com/watch?v=ecoeU2OugSg
Aberrations - Chromatic
Fblue Fred
Chromatic aberration happens because blue light is bent more than red light
It causes color separation to appear at the edges of images
F
Chromatic aberration can be eliminated by using two different kinds of glass in a lens so that the effect cancels in the two different lenses
http://micro.magnet.fsu.edu/primer/java/aberrations/chromatic/index.html
Adding lenses
Ftotal
Where is the combined focal length of two convex lenses
A: Closer to the lens than either focus?
B: Between the two focii?
C: Further from the lens than either focus?
F1 F2
Lens #1 Lens #2
Formula for adding lenses
F
To combine lenses, add their DIOPTERS (not their focal lengths)
where diopter (D) = 1/F, and F is the focal length
So Dtotal = D1 + D2
or 1/Ftotal = 1/F1 + 1/F2
Example of adding lenses
F
Question: If F1 is 25 cm, and F2 is 1 m, what is Ftotal?
Answer: D1 = 4D and D2 = 1D
=> Dtotal = D1 + D2 = 5D
=> Ftotal = 1/5 m = 20cm
Another example of adding lenses
Question: If F1 is 0.5m, and F2 is -1m, what is Ftotal?
Answer: D1 = 2D and D2 = -1D
=> Dtotal = D1 + D2 = 1D
=> Ftotal = 1 m
Spherical Aberrations – in Mirrors and Lenses
Spherical aberration happens because a sphere does not have a perfect focus for light rays
Rays at the edge focus closer than rays from the center
Spherical aberration can be eliminated by -
1. using a parabolic mirror (headlights or telescopes)
2. using an aperture to stop the edge rays
Spherical Aberrations in Lenses
Spherical aberration happens because a sphere does not have a perfect focus for light rays
A perfect lens (top) focuses all incoming rays to a single point on the optic axis, but a lens ground with spherical surfaces (bottom) focuses different rays to different points along the optic axis, depending on the radial position of each incoming ray.
Rays at the edge focus closer than rays from the center
Concept Question: Spherical Aberration
http://micro.magnet.fsu.edu/primer/java/aberrations/spherical/index.html http://en.wikipedia.org/wiki/Hubble_Space_Telescope#Flawed_mirror
When spherical aberration is present, what part of the image is blurred?
A. Center
B. Edges
C. All
Hubble Space Telescope Main Mirror
Off-Axis Aberrations • Spherical and chromatic aberrations are on-axis aberrations i.e., these aberrations happen to rays that are close to the axis
• Off-axis aberrations include CURVATURE OF FIELD, COMA, ASTIGMATISM, and DISTORTION
Curvature of field aberration
• CURVATURE OF FIELD: Here the image of a flat object does not lie in a plane. The center is in focus while the extremes are blurred. This aberration can be compensated for by curving the viewing screen (e.g. large TV screens or drive-in theatres)
Off-Axis Aberrations
• COMA: Coma is a modification of spherical aberration for off-axis objects. The blur due to spherical aberration moves to one side when the object is off axis, so that the image seems to have a tail (like a comet, from which this aberration is named!)
Coma aberration