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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

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


convex interface

C F

glass

F

Rays parallel to the axis are deflected through the focus



convex interface

C F

glass

Rays from the focus are made parallel to the axis


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


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

Thin Lenses

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 one

Images

F

object

F

To locate the image - 1.  Draw at least 2 rays: here is a second

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

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

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

More Ray Tracing in Lenses: Concave Lens

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

Magnification


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


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

Aberrations - Chromatic

http://micro.magnet.fsu.edu/primer/java/aberrations/chromatic/index.html

Example of Chromatic Aberration

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

Aberrations - Spherical

http://micro.magnet.fsu.edu/primer/java/aberrations/spherical/index.html

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

Off-Axis Aberrations: Barrel and Pincushion Distortions

• DISTORTION: Barrel and pincushion aberrations distort the image. These aberrations can be limited by using stops to eliminate the extreme rays.

Pincushion aberration

Barrel aberration

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