Lecture 18 Ray Optics-4 Chapter 23 PHYSICS 270 Dennis Papadopoulos March 24, 2010.

Lecture 18Ray Optics-4 Chapter 23

PHYSICS 270

Dennis Papadopoulos

March 24, 2010

Convex Mirror

Virtual Image – Rays Diverge

Tactics: Ray tracing for a spherical mirror

Tactics: Ray tracing for a spherical mirror

1. A parallel ray reflects through focal point2. A ray through the focal point reflects parallel to the axis3. A ray striking the mirror center reflects at an equal angle

on the opposite side of the axis

Optical axis

Tactics: Ray tracing for a spherical mirror

2 2 2

2

2 2

( )( )

2

h d

s d

s d h

s d s d h

h h

s d s

A Little Math

sd

2 2

2

h h

s d s

Take rays near axis, paraxial rays

The excess time along OP vs. OQ is h2/2s and the excess time on the PO’ route vs. QO’ is n(h2/2s’). This is compensated by the delay time from V to Q (Notice that we took the VQ distance as traveled at c). This delay time is (n-1)VQ. For a spherical surface with radius R, VQ=h2/2R. We thus have

2 2 2( / 2 ) ( / 2 ') ( 1) / 2

(1/ ) ( / ') ( 1) /

h s nh s n h R

s n s n R

See equ. 23.21 of the book with n1=1, n2=n

s /c s' /v consts ns'const.

21

22

( / 2 )(1/ )

' '( / 2 ')(1/ )

OP OVt h s c

cPO QO

t h s vv

h

These are the excess time along OP and along PO’ if we assume that the distance VQ is covered at speed at the vacuum speed c and not v. In other words the time to go from V to Q is not as it was straight in vacuum but slower by the factor v/c. So the excess delay is then

t3 VQ

vVQ

cVQ

c(n 1)

For light to arrive at the same time travelling the path OVO’ and the path OPO’ we must have

t1 t2 t3(h2 /2s)(1/c) (h2 /2s')(1/v) (h2 /2R)(1/c)(n 1)

(h2 /2s) (h2 /2s)(c /v) (h2 /2R)(n 1)

1

s ns'

n 1

R

Where VQ= h2/2R

Details of mathematical proof

Q

A paraxial ray

A. moves in a parabolic path.B. is a ray that has been

reflected from parabolic mirror.

C. is a ray that moves nearly parallel to the optical axis.

D. is a ray that moves exactly parallel to the optical axis.

A paraxial ray

A. moves in a parabolic path.B. is a ray that has been

reflected from parabolic mirror.

C. is a ray that moves nearly parallel to the optical axis.

D. is a ray that moves exactly parallel to the optical axis.

The Mirror EquationFor a spherical mirror with negligible thickness, the object and image distances are related by

where the focal length f is related to the mirror’s radius of curvature by

What is specular reflection?

A. The image of a specimen.B. A reflection that separates different colors.

C. Reflection by a flat smooth object.D. When the image is virtual and special.

What is specular reflection?

A. The image of a specimen.B. A reflection that separates different colors.

C. Reflection by a flat smooth object.D. When the image is virtual and special.

A virtual image is

A. the cause of optical illusions.B. a point from which rays appear to

diverge.C.an image that only seems to exist.D. the image that is left in space after

you remove a viewing screen.

A virtual image is

A. the cause of optical illusions.B. a point from which rays appear to

diverge.C.an image that only seems to exist.D. the image that is left in space after

you remove a viewing screen.