Physics 1230: Light and Color
Ray tracing for lenses
http://www.colorado.edu/physics/phys1230
simple lens applet
http://phet.colorado.edu/new/simulations/sims.php?sim=Geometric_Optics
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
incident rays
diagram works in
either direction
glass or water
refracted
rays
How do rays behave at spherical convex interfaces?
•  Convex glass surfaces are focusing
•  Concave glass surfaces are defocusing
C
F
glass
convex interface
Rays parallel to the axis are deflected through the focus
How do rays behave at spherical convex interfaces?
•  Convex glass surfaces are focusing
F
C
F
glass
convex interface
Rays parallel to the axis are deflected through the focus
How do rays behave at spherical convex interfaces?
•  Convex glass surfaces are focusing
C
F
glass
convex interface
Rays from the focus are made parallel to the axis
How do rays behave at spherical interfaces?
• Concave glass surfaces are defocusing
F
C
F
glass
concave interface
Rays parallel to the axis are deflected as if they emerged
from the focus
How do rays behave at spherical convex interfaces?
• Concave glass surfaces are defocusing
F
C
F
glass
concave interface
Rays parallel to the axis are deflected as if they came
from the focus
Thin Lenses
Ray Tracing for Thin Lenses
Unlike a mirror there are two
focal points of a lens, one on
either side
•  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
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?
Why not just take two prisms stacked one above the other?
•  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
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
To locate the image 1.  Draw at least 2 rays: here is one
object
F
F
Images
To locate the image 1.  Draw at least 2 rays: here is a second
object
F
F
Images
To locate the image 1.  Draw at least 2 rays: here is a third
Where is the image?
Is it upright or inverted?
object
F
F
Concept question
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
object
F
F
Concept question
Answer: Notice that with a convex lens, the image is flipped
object
F
F
image
height Si
Images in THIN Lenses
What rules do rays 1, 2 and 3 obey for thin lenses?
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
real
images
photography case:
real images
(can be projected on film)
real
images
real image
of tree
magnifying glass case:
virtual image
http://micro.magnet.fsu.edu/primer/java/lenses/
converginglenses/index.html
virtual image
of man
Virtual images in thin convex lenses magnifying glass
image
object
F
F
thin convex lens
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
Object
simple magnifying glass applet
http://micro.magnet.fsu.edu/primer/java/lenses/simplemagnification/index.html
More Ray Tracing in Lenses: Concave Lens
Virtual images in thin concave lenses –
”demagnifying glass”
object
F
image F
thin concave lens
Is there one more ray that you can draw here?
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
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
image
height Si
xo
xi
Magnification
Magnification = Size of image = Image distance
Size of object
Object distance
object
height So
image
height Si
F
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
Chromatic aberration can be
F
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
Aberrations - Chromatic
http://micro.magnet.fsu.edu/primer/java/aberrations/chromatic/index.html
Example of Chromatic Aberration
Adding lenses
Lens #1 Lens #2
F1
F2
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?
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
Aberrations - Spherical
http://micro.magnet.fsu.edu/primer/java/aberrations/spherical/index.html
Concept Question: Spherical Aberration
Hubble Space Telescope Main Mirror
When spherical aberration is
present, what part of the image is
blurred?
A.  Center
B.  Edges
C.  All
http://micro.magnet.fsu.edu/primer/java/aberrations/spherical/index.html
http://en.wikipedia.org/wiki/Hubble_Space_Telescope#Flawed_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 offaxis 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
Off-Axis Aberrations: Barrel and Pincushion Distortions
Barrel aberration
• DISTORTION: Barrel and pincushion
aberrations distort the image. These
aberrations can be limited by using stops
to eliminate the extreme rays.
Pincushion aberration