31. Images & Optical Instruments

1. Images with Mirrors

2. Images with Lens

3. Refraction in Lenses: The Details

4. Optical Instruments

How does laser surgery provide permanent vision correction?

Ans: Laser light reshapes cornea to adjust the focal point.

Geometrical (or ray) optics: Light ray in homegeneous medium = straight line.

Valid when L >> .

Real image: image location is the point of convergence of actual light rays.Can be shown on screen.

Virtual image: some or all of the light rays that converge to form the image are virtual (straight line extension of the actual rays).

Can’t be shown on screen.

Light ray = line (or curve) wave front.

31.1. Images with Mirrors

Virtual

Front-to-back reversal.Right-to-left-handed coord.

Short-cut:OP = PO

GOT IT? 31.1.

You stand in front of a plane mirror whose top is at the same height as the top of your head.

Approximately how far down must the mirror extend for you to see your full image?

Ans. Half your height.

Curved Mirrors

Hubble telescope:Spherical mirror (upper)Parabolic mirror (lower)

Normal

equal angles

Parabolic mirror

Spherical aberration

Spherical aberration is small for paraxial rays.

conic sections: f = e d

Parabola: e = 1

f

d

Tactics 31.1: Paraxial Ray Tracing with Spherical Mirrors

1. Ray // axis reflected through focus.2. Ray through focus reflected // axis.3. Ray striking through mirror mid-point reflects symmetrically.4. Ray through mirror center reflects upon itself.

FC

Concave Spherical Mirror

Bear and image are both in front of the mirror.

Convex Mirrors

Convex mirror. Image always virtual, upright, and reduced in size. Wide-angle view.

The Mirror Equation

Shaded triangles are similar:

h s

h s

h < 0

M Magnification

h s f

h f

s

s

11fs

s f

1 1

f s

1 1 1

s s f

Mirror Equation

1

2f R R = radius of mirror

positive if

s always

s, f same side of outgoing ray

h, h

above axis

Table 31.1. Image Formation with Mirrors: Sign Conventions

Example 31.1. Hubble Space Telescope

A technician standing in front of the Hubble Space Telescope mirror.

During assembly, a technician stood 3.85 m in front of the concave mirror of the HST.Let the focal length of the telescope be 5.52 m. Find

(a) the location, and(b)the magnificationof the technician ‘s image.

1 1 1

3.85 5.52s

1

1 1

5.52 3.85s

12.7 m ( Virtual image; behind mirror )

12.7

3.85

sM

s

3.30 ( Upright; enlarged )

Example 31.2. Jurassic Park

Convex side-view mirror: Objects in mirror are closer than they seem.

If the curvature radius of the mirror is 12 m and the T. Rex is 9.0 m from the mirror,by what factor does the dinosaur appear reduced in size?

1 1 1

9 12 / 2s

11 1

9 6s

18

5

185

9

sM

s

20.4

5 (image is upright & smaller)

31.2. Images with Lens

Convex lens Concave lens

Thin lens: Light rays bend just once going through the lens.

Tactics 31.2. Ray Tracing with Thin Lens

entering // axis

passes thru focus

entering thru center

passes thru undeflected

Lens Images by Ray Tracing

Getting Quantitative: The Lens Equation

h sM

h s

Shaded triangles similar:h s f

h f

s

s

1 1 1

f s s

lens equation

2s f s f f

positive if

s always

s, f same side of outgoing ray

h, h

above axis

Table 31.2. Image Formation with Lens: Sign Convention

GOT IT?. 31.3.

You look through a lens at this page and see the words enlarged and right-side up.

Is the image you observe real or virtual?

Is the lens concave or convex?

Example 31.3. Fine Print

You ‘re using a magnifying glass (converging lens) with a 30-cm focal length to read a telephone book.

How far from the page should you hold the lens in order to see the print enlarged three times?

1 1 1

f s s

1 1 1

30 3 s s

3h s

Mh s

Image is upright enlarged so it must be virtual.

Hence, h > 0, s < 0.

230

3s 20 cm

31.3. Refraction in Lenses: The Details

Refraction at a Curved Surface in the Paraxial Approximation

Snell’s law: 1 1 2 2sin sinn n 1 1 2 2n n paraxial approx.

Green line is tangent to lens surface at point A. It merges with segment AB in the paraxial approx.

1 T(BOC):

tan T(BOA):

tan T(BAC):

AB

s

AB

R

2 T(BCI): tan T(IBA):AB

s

Snell’s law: 1 2

AB AB AB ABn n

s R R s

1 2 2 1n n n n

s s R

Example 31.4. Cylindrical Aquarium

An aquarium consists of a thin-walled plastic tube 70 cm in diameter.For a cat looking directly into the aquarium,what is the apparent distance to a fish 15 cm from the aquarium wall?

Top view

1 2 2 1n n n n

s s R

Plastic wall thin negligible.nwater = 1.333

1.333 1. 1. 1.333

15 70 / 2s

10.333 1.333

35 15s

12.6 cm

positive if

s always

s, f, R

same side of outgoing ray

h, h above axis

Lenses, Thick & Thin

O1 I1 = O2 I2 1 2 2 1n n n n

s s R

positive if

s always

s, f, R

same side of outgoing ray

h, h above axis

LHS (O1 I1 ) :1 1 1

1 1n n

s s R

1 10 , 0s R

RHS (O2 I2 ) :1 2 2

1 1n n

s t s R

2 0R

Thin lens (t 0) : 1 2

1 1 1 11n

s s R R

1 2

1 1 11n

f R R

Lensmaker’s formula

Common Lens Types

Example 31.5. Plano-Convex Lens

positive if

s always

s, f, R

same side of outgoing ray

h, h above axis

Find an expression for the focal length of the plano-convex lens,given refractive index n and radius R for the curved surface.

1 2

1 1 11n

f R R

1 1 11n

f R 1

Rf

n

Object on left hand side:

1 1 11n

f R

Object on right hand side:

same result

Lens Aberrations

Spherical aberration

Stopped down better focus

Astigmatism: cause: different R in different direction.

Chromatic aberration: cause: n = n( ).Minimized by using composite lens.

31.4. Optical Instruments

The Eye

Myopic (nearsighted)Corrective: Divergent lens.

Hyperopic (farsighted)Corrective: Convergent lens.

Corrective power P = 1 / f.[P] = diopter = m1

Application: Laser Vision Correction

LASIK procedure

Conceptual Example 31.6. Contact Lens Mix-Up

You and your roommate have gotten your boxes of disposable contact lenses mixed up.

One box is marked “1.75 diopter”, the other “+2.5 diopter”.

You are farsighted and your roommate is nearsighted.

Which lenses are yours?

Converging lens correct far-sightedness.

f > 0

P = 1 / f > 0

Making the Connection

What’s the focal length of your contact?

12.5P

f

1

2.5f m 40 cm

Example 31.6. Lost Your Glasses!

You’ve lost your reading glasses; without them, your eyes can’t focus closer than 70 cm.

Nonprescription reading glasses come in 0.25-diopter increments.

Which glasses should you buy so you can focus at the standard 25-cm near point?

1P

f 1 1

s s

1 1

0.25 0.70m m

2.57 diopters

Aim: make object at 25 cm appears to be at 70 cm.i.e., s = 25 cm , s = 70 cm.

Ans: buy glasses with P = 2.5 diopters

Cameras

Works like the eye.

Magnifiers & Microscopes

Angular magnification :

Angle subtended by image

Angle subtended by object at near pointm

/

/ 25

h f

h cm

Closest distance eye can focus is 25 cm (near point).

25 cm

f Simple magnifier

max 4m

Compound microscope

1 os f

2 es f

For ,o ef f L

from eyepiece

1s L from objective0

o

LM

f

25e

e

cmm

f

For the eyepiece:

Overall magnification: o eM M m25

o e

L cm

f f Compound

telescope

Telescopes

Refracting telescope

m

1

1

/

/e

o

h f

h f o

e

f

f Object inverted

Extra diverging eyepiece or set of reflecting prisms needed to get upright image.

Reflecting Telescope

Cassegrain design (large telescopes)

Newtonian design (small telescopes)

Advantages:1.No chromatic aberration.2.Mirror (adjustable) can be much larger (~10m) than lens (~1m).

Only light gathering power is important for astronomical telescopes.

Giant Magellan Telescope (2016)