ss’

h’

h

f

Refraction on Spherical Surface

ba θβφφαθ +=+= ;

bbaa

bbaann

nnθθ

θθ≅

= sinsin ( )φβα abba nnnn −=+⎫⎬⎭

δφ

δβ

δα

−=

−=

+=

Rh

sh

sh tan;

'tan;tan

Rh

sh

sh

≅≅≅ φβα ;'

; Rnn

sn

sn abba −

=+'

⇒

Refraction on Spherical SurfaceR

nnsn

sn abba −

=+'Magnification

θθ

θθ

θθ

sintan

sinsin

tantan

≅

=′′−

==

bbaa

ba

nnsy

sy

For small angles snsn

yym

syn

syn

b

a

ba

′−=

′=

′′

−=

R positive if C ontransmission side;negative otherwise

Refraction on Spherical SurfaceR

nnsn

sn abba −

=+'

snsn

yym

b

a ′−=

′=

Example: Fish bowl. A fish is 7.5cm from the front of the bowl.Find the location and magnification of thefish as seen by the cat. Ignore effect of bowl.Radius of bowl = 15 cm. na = 1.33

OI s’

scm

cmcmsn

Rnnn

s

sn

Rnn

sn

Rnn

sn

sn

aab

b

aabbabba

44.6

5.733.1

1533.0

1''

−=−

−−

=−

−=′

−−

=−

=+

14.15.7

)44.6(133.1

=−

∗−=′

−=cmcm

snsn

mb

a The fish appears closer and larger.

a

b

R negative

Lenses• A lens is a piece of transparent material shaped such that

parallel light rays are refracted towards a point, a focus:– Convergent Lens

» light moving from air into glasswill move toward the normal» light moving from glass back into air will move away from the normal» real focus

– Divergent Lens» light moving from air into glasswill move toward the normal» light moving from glass back into air will move away from the normal» virtual focus

Negative f

Positive f

Lens Equation – thin lens

222111 '' Rnn

sn

sn

Rnn

sn

sn baababba −

=+−

=+

For air, na=1 and glass, nb=n, and s2=-s1’. Neglects l.

221111

1'

1'

1'

1R

nss

nR

nsn

s−

=+−−

=+

( )fRR

nss

1111'

11

21

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−=+

Eliminate, s1’, s1 → s, s2’ → s’⇐ Lensmaker eqn.f > 0, convergingf < 0, diverging

Lensmaker’s FormulaThe thin lens equation for a lens with two curved sides:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

21

1111RR

nf

1R 2R

R1 is first surface; R2 is second.

R is positive if center of curvature on transmission side.

f positive if converging lens; negative if diverging.

Lensmaker’s example

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

21

1111RR

nf R2 = -15 cm

R1 = +10 cmIncident Light

cmff

12151

101)15.1(1

=

⎥⎦⎤

⎢⎣⎡

−−−=

n = 1.5

R1 is first surface; R2 is second.

R is positive if center of curvature on transmission side.

f positive if converging lens; negative if diverging.

Light coming from ∞,focuses at focal point.

Convergent Lens example

• Draw some ray diagrams:

s’

fh’

s

h

'

1 1 1s s f+ =

fsfss−

='

hh

ssM

''

=−=

F1

F2f

Parallel ray – goes through F2Focal ray (F1 ) – becomes parallelCentral ray - undeviated

Sign conventions for lenses(same as mirrors)

• IF object & incoming ray on same side, positive s, otherwise negative.• IF image & outgoing ray on same side, positive s′, otherwise negative.• If converging lens, then f is positive; for diverging lens, f negative.

'

1 1 1s s f+ =

fsfss−

='

hh

ssM

''

=−=

13. A parallel laser beam of width w1 is incident on a two lens system as shown below.

a) The beam is convergingb) The beam is divergingc) The beam is parallel to the axis with a width < w1d) The beam is parallel to the axis with a width = w1e) The beam is parallel to the axis with a width > w1

Each lens is converging. The second lens has a larger focal length than the first (f2> f1).

What does the beam look like when it emerges from the second lens?

w2

6. The lens in your eyea) is always a positive (i.e., converging) lensb) is always a negative (i.e., diverging) lensc) is sometimes positive, and sometimes negative, depending on

whether you are looking at an object near or far awayd) is positive if you are near-sighted and negative if you are far-

sighted7. The image on the back of your retina is

a) invertedb) noninverted

8. The image on the back of your retina isa) realb) virtual

9. The image on the back of your retina isa) enlargedb) reduced

Lecture 26, ACT 1• A lens is used to image an object on a

screen. The right half of the lens is covered. – What is the nature of the image on the

screen?

(a) left half of image disappears(b) right half of image disappears(c) entire image reduced in intensity

object

lens

screen

Lecture 26, ACT 1• A lens is used to image an object on a

screen. The right half of the lens is covered. – What is the nature of the image on the

screen?

(a) left half of image disappears(b) right half of image disappears(c) entire image reduced in intensity

object

lens

screen

• All rays from the object are brought to a focus at the screen by the lens.• The covering simply blocks half of the rays.• Therefore the intensity is reduced but the image is of the entire object!•( one more thing… In our examples of image formation, we only needed the top half of the lens to form the image)

s’

f h’s

h

Divergent lens example• Because the lens is divergent, f is negative:

s’f

h’

s

h

'

1 1 1s s f+ =

fsfss−

='

hh

ssM

''

=−= f is negative.

f

F1F2

Parallel ray – goes through F2Focal ray (F1 ) – becomes parallelCentral ray - undeviated

Sign conventions for lenses(same as mirrors)

• IF object & incoming ray on same side, positive s, otherwise negative.• IF image & outgoing ray on same side, positive s′, otherwise negative.• If converging lens, then f is positive; for diverging lens, f negative.

'

1 1 1s s f+ =

fsfss−

='

hh

ssM

''

=−=

Graphical images of Thin Lens

Compound Lens Example with ray tracing

Image of 1st lens becomes object of 2nd lens Resulting image is upright

Work on board

More generally…Lensmaker’s Formula

Two arbitrary indices of refraction

R > 0 if center on transmission side

R < 0 otherwise

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

21

11)1(1RR

nf

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

211

12 111RRn

nnf

The complete generalized case…

Note: for one surface Planar, ∞=R

Lecture 26, ACT 3• What happens to the focal length of a lens

when it is submerged in water? – In air, the focal length of a glass lens

(n=1.5) is determined to be fair. When the lens is submersed in water(n=1.33), itsfocal length is measured to be fwater. What is the relation between fair and fwater?

(a) fwater < fair (b) fwater = fair (c) fwater > fair

air airfair

water waterfwater

Lecture 26, ACT 3

(a) fwater < fair (b) fwater = fair (c) fwater > fair• The refraction is determined by the difference in the two indices of refraction

− The smaller the difference, the less the bend

the longer the focal length• Think of the case where it is glass “in glass” ! f → infinity !• Quantitatively, look at the lensmaker’s formula

air airfair

water waterfwater

• What happens to the focal length of a lens when it is submerged in water? – In air, the focal length of a glass lens

(n=1.5) is determined to be fair. When the lens is submersed in water(n=1.33), its focal length is measured to be fwater. What is the relation between fair and fwater?

glass glass

water air

1 1 1 11 1water air

n nf n R f n R

⎛ ⎞ ⎛ ⎞≈ − < ≈ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

12. If you are swimming underwater (without goggles), the focusing power of the lens in your eye will

a) increase b) decrease c) stay the same

and sometimes inverted.Note: The “power” of a lens is defined as 1/f. The unit “diopter” is used for the power of a 1-m focal length lens.(E.g., the eyeglass lens of a slightly near-sighted person might have a power of -0.5 diopter f = -2 m)

and sometimes inverted.

Question: How many diopters is a typical eye?

Answer: f ~ 1 inch ~ 2.5 cm = 0.025 m Power = 40 D

Summary of Lenses and Mirrors• We have derived, in the paraxial (and thin lens)

approximation, the same equations for mirrors and lenses:

when the following sign conventions are used:

Variable

f > 0f < 0

s > 0s < 0

s’ > 0s’ < 0

Mirror

concaveconvex

real (front)virtual (back)

real (front)virtual (back)

Lens

convergingdiverging

real (front)virtual (back)

real (back)virtual (front)

Principal rays “connect”object and image

one goes through the center of the lens or mirror

the other goes parallel to the optic axis and then is refracted or reflected through a focal point

the third one is like the second one….

ssM′

−=fss111

=′

+

Optical AberrationsWhy really good lenses cost a lot

Chromatic aberrationDue to dispersion (index of refraction depends on frequency), focal length can be different for different colors.

Spherical aberrationOutside the “paraxial limit”, optimal focusing occurs only for a parabolic lens. Spherical lenses look ~parabolic for narrow field of view.

(But spherical lenses are

much cheaper to grind & polish!)

AstigmatismCurvature of the lens not symmetric in transverse directions → slightly cylindrical → different focal lengths

Multiple lenses can be used to improve aberrations

www-optics.unine.ch/education/optics_tutorials/achromat.html

Spherical Aberration Chromatic Aberration

Summary

• Lens Equation– same as mirror equation

• Lensmaker’s Formula

• Aberrations– spherical – ideal lens shape is parabolic– chromatic – index of refraction depends on frequency– multiple lenses can reduce aberrations

Next time: Multiple lenses, Optical instruments, Lasers

fss111

=′

+

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

211

12 111RRn

nnf

The Lens Equation• We now derive the lens equation which determines the image

distance in terms of the object distance and the focal length.– Convergent Lens:

s’f

h’s

h

Ray Trace:• Ray through the center of the lens (light blue) passes through undeflected.

two sets of similar triangles:ss

hh ′

=′

fh

fsh

=−′′

eliminating h’/h:f

fsss −′=

′fss111

=′

+same as mirror eqn

if we defines’ > 0 f > 0

magnification: also same as mirror eqn!!M < 0 for inverted image. s

sM′

−=

• Ray parallel to axis (white) passes through focal point f. These are principal rays!!!