Chapter 5

Geometrical OpticsMirrors and Prisms

Phys 322Lecture 14

Optical bench

http://webphysics.davidson.edu/Applets/optics4/default.html

Mirrors

Ancient bronze mirror

Liquid mercury mirror

Hubble telescope mirror

Planar mirror

i rsi = -so

Sign convention: s on the object side is positive, and negative on the opposite side

also called plane, or flat mirrors

Planar mirror

For a plane mirror, a point source and its image are at the same distance from the mirror on opposite sides; both lie on the same normal line.Image is virtual, up-right, and life-size (MT = +1)

o

i

o

iT s

syyM The equation for lens works:

si = -so

Sign convention: s on the object side is positive, and negative on the opposite side

Exercise: plane mirror height

How high should be the mirror for a person to see a full image ofhim/her-self?

Solution:A

B

C

D

ETriangle ABC is twice as small as ADE

BC is half DE (the height of the guy)

1. Mirror (BC) should be at least half of the guy’s height (DE) 2. Its bottom should 1/2 of the height of guy’s eyes from the ground

‘Mirror image’

Mirror image of left hand is a right hand

Inversion: converting right-handed coordinate system into left-handed one

Even number of mirrors can be used to avoid inversion

Applications: steering light

reflex camera (SLR)

Atomic force microscope

DLP projection TV

http://www.plus-america.com/papers.html

Parabolic aspherical mirror

Make a mirror that will converge plane waves into a pointFermat’s principle:

FAAWFAAWOPL 222111

22221111 DAAWDAAW

111 DAFA 222 DAFA

In general: ADAF

This is the surface of paraboloid: y2 = 4fx (origin at vertext V)

Application:headlights, flashlights,radars,dish antenna,….

V

Aspherical mirrors

Collects light from one point to another

divergingdiverging

convergingconverging

convergingdiverging

divergingconverging

off-axis parabolic

Spherical mirror

222 RRxy

fxy 42

Paraboloid and sphere are similar in paraxial approximation

2222 2 RRxRxy 2 2 2y x xR

small when close to axis x

Spherical mirror formula

SAP is bisected by AC:

PACP

SASC

R<0 in real object spaceso>0 in real object spacesi>0 in real image spacef >0 concave mirror

Sign convention:

RsSC o

isRCP

Paraxial approximation: osSA isPA

i

i

o

o

sRs

sRs

Rss io

211

Focal lengths:lim1/ 1/ 2 /i

o o osf s f R

lim 1/ 1/ 2 /o

i i isf s f R

Mirror Formula

Rfss io

2111

Spherical mirrors

Rfss io

2111

Note: Both mirror and lens equations are the same, except the real image is in front of mirror, but it is behind the lensMagnification equations are the same as well.

Concave mirror: principal axes and image

S

f

1) Parallel to principal axis reflects through F.2) Through F, reflects parallel to principal axis.

3) Through center.

C

#1

#2#3

NOTE: Any other ray from object tip which hits mirror will reflect through image tip

Image is:Real (light rays actually cross)Inverted (Arrow points in opposite direction)Diminished (smaller than object, only if object is further than C)

si

Principal rays for concave mirror:Rfss io

2111

Convex mirror: principal axes and image

Image is:Virtual (light rays don’t really cross)Upright (same direction as object)Diminished (smaller than object)

SP

1) Parallel to principal axis appear to originate from F after reflection.2) Through F, reflects parallel to principal axis.3) Through center.

C

#1

#2#3

**For a real object, image is always virtual, upright and diminished

f

Principal rays for convex mirror:

F

Exercise: can a concave mirror form a virtual image?

fss io

111

Concave mirror: so and f are always positive, want to get negative si

siso

0111

oi sfs

fso

An object must be between mirror and its focal plane

F

so si

virtualimage

Spherical mirrors

Examples

Dispersing prism

ttii nn sinsin

Bending depends on wavelength: dispersing prism, i.e. n=n()

Can we use optical flat for dispersing light?Rays emerge parallel to each other.Practically we don’t see them (focused by the eye at the same spot).

Example

Dispersing prism equation

i1

ttii nn sinsin

t2

nt=nni=1

cossinsinsinarcsin 1122

1 iti n

Total deviation is a function of refraction index:

Minimum deviation min occurs when i1 = t2

2/sin

2/sin min

n can use to determine n

And this arrangementmaps position to angle:

Spectral analyzer

out inx

Prism spectrometers

Drawbacks:() - nonlinear dependenceLow spectral resolutionSmall aperture

Constant-deviation dispersing prisms

min=90o always!

Pellin-Broca prism:

Pellin, Ph. and Broca, André (1899), "A Spectroscope of Fixed Deviation". Astrophysical Journal 10 337

Abbe prism:

Ernst Abbe1840-1905

min=60o always!

Fix input-output at 90o or 60o and rotate prism for different wavelengths

Reflecting prismReflect the beam with no dispersion using total internal reflection

If we make t1 = i2 - like in flat glass plate

= 2i1 + “achromatic” prism

Reflecting prisms

The right-angle prism The Porro prism

The Dove prism

The penta prism