F. OPTICS

22. Geometrical optics

Outline 22.1 Spherical mirrors 22.2 Refraction at spherical surfaces 22.3 Thin lenses

Objectives (a) use the relationship f = r/2 for spherical mirrors (b) draw ray diagrams to show the formation of

images by concave mirrors and convex mirrors (c) use the formula 1/f = 1/u + 1/v for spherical mirrors (d) use the formula n1/u + n2/v = (n2-n2)/r for

refraction at spherical surface (e) use the formula n1/u + n2/v = (n2-n2)/r to derive

thin lens formula 1/u + 1/v = 1/f and lens formula 1/f = (n-1)(1/r1 - 1/r2)

(f) use the thin lens formula and lens equation.

Introduction

Geometrical Optics

In describing the propagation of light as a wave we need to understand:

wavefronts: a surface passing through points of a wave that have the same phase.

rays: a ray describes the direction of wave propagation. A ray is a vector perpendicular to the wavefront.

Wavefronts We can chose to associate the wavefronts with the instantaneous surfaces where the wave is at its maximum. Wavefronts travel outward from the source at the speed of light: c. Wavefronts propagate perpendicular to the local wavefront surface.

Light Rays The propagation of the wavefronts can be described by light rays. In free space, the light rays travel in straight lines, perpendicular to the wavefronts.

Reflection and Refraction When a light ray travels from one medium to

another, part of the incident light is reflected and part of the light is transmitted at the boundary between the two media.

The transmitted part is said to be refracted in the second medium.

incident ray reflected ray

refracted ray

Reflection by plane surfaces

r1 = (x,y,z)

x

y

r2 = (x,-y,z)

Law of Reflection

r1 = (x,y,z) 2 = (x,-y,z) Reflecting through (x,z) plane

x

y

z r2= (-x,y,z)

r3=(-x,-y,z)

r4=(-x-y,-z)

r1 = (x,y,z) n2

Refraction by plane interface & Total internal reflection

n1

n1 > n2

C

P

1 1

1 1

2 2

1sin 1=n2sin 2

Examples of prisms and total internal reflection

45o

45o

45o

45o

Totally reflecting prism

Porro Prism

Types of Reflection

If the surface off which the light is reflected is smooth, then the light undergoes specular reflection (parallel rays will all be reflected in the same directions).

If, on the other hand, the surface is rough, then the light will undergo diffuse reflection (parallel rays will be reflected in a variety of directions)

The Law of Reflection For specular reflection the incident angle i

equals the reflected angle r

i

r

The angles are measured relative to the normal, shown here as a dotted line.

22.1 Spherical Mirrors

Spherical Mirrors A spherical mirror is a mirror whose surface shape is spherical with radius of curvature R. There are two types of spherical mirrors: concave and convex.

concave

Spherical Mirrors We will always orient the mirrors so that the

reflecting surface is on the left. The object will be on the left.

convex

Focal Point When parallel rays (e.g. rays from a distance

source) are incident upon a spherical mirror, the reflected rays intersect at the focal point F, a distance R/2 from the mirror.

Focal Point Locally, the mirror is a flat surface,

perpendicular to the radius drawn from C, at an angle from the axis of symmetry of the mirror.

Focal Point For a concave mirror, the focal point is in front

of the mirror (real).

Focal Point For a convex mirror, the focal point is behind

the mirror (virtual).

The incident rays diverge from the convex mirror, but they trace back to a virtual focal point F.

Focal Length The focal length f is the distance from the surface of the mirror to the focal point. CF = FA = FM = ½ radius

Focal Length The focal length FM is half the radius of curvature of a spherical mirror. Sign Convention: the focal length is negative if the focal point is behind the mirror. For a concave mirror, f = ½R For a convex mirror, f = ½R (R is always positive)

22.2 Refraction at spherical surfaces

Ray Diagram It is sufficient to use two of four principal rays

to determine where an image will be located.

M ray

The parallel ray (P ray) reflects through the focal point.

The focal ray (F ray) reflects parallel to the axis, and

The center-of-curvature ray (C ray) reflects back along its incoming path.

The Mid ray (M ray) reflects with equal angles at the axis of symmetry of the mirror.

Ray Diagram The parallel ray (P ray) reflects through the focal point. The focal ray (F ray) reflects parallel to the axis The center-of-curvature ray (C ray) reflects back along its

incoming path. The Mid ray (M ray) reflects with equal angles at the axis of

symmetry of the mirror.

Ray Diagram Examples: concave

Real image Put film here for Sharp Image.

Ray Diagram Examples: concave

Real image

Ray Diagram Examples: convex

Virtual image

Ray Diagram Examples: convex

Virtual image

The Mirror Equation The ray tracing technique

shows qualitatively where the image will be located. The distance from the mirror to the image, di, can be found from the mirror equation:

fdd io

111

do = distance from object to mirror

di = distance from image to mirror

f = focal length

m = magnification

Sign Conventions: do is positive if the object is in front of the

mirror (real object)

do is negative if the object is in back of the mirror (virtual object)

di is positive if the image is in front of the mirror (real image)

di is negative if the image is behind the mirror (virtual image)

f is positive for concave mirrors

f is negative for convex mirrors

m is positive for upright images

m is negative for inverted images o

iddm

Example 1 An object is placed 30 cm in front of a concave mirror of radius 10

cm. Where is the image located? Is it real or virtual? Is it upright or inverted? What is the magnification of the image?

cmdcmcmcmcmd

cmcmdfd

cmdfdd

cmRf

i

i

i

o

i

66

130

530

130

6130

15

111130

11152/

0

0

di>0 Real Image m = di / do = 1/5

Example 2 An object is placed 3 cm in front of a concave mirror of radius 20

cm. Where is the image located? Is it real or virtual? Is it upright or inverted? What is the magnification of the image?

43.1/29.4

307

3010

3031

31

101111

3

111102/

0

0

oi

i

i

i

o

i

ddmcmd

cmcmcmd

cmcmdfd

cmdfdd

cmRf

Virtual image, di <0 Magnified, |m| > 1, not inverted. m > 0

Example 3 An object is placed 5 cm in front of a convex mirror of focal length

10 cm. Where is the image located? Is it real or virtual? Is it upright or inverted? What is the magnification of the image?

66.0/33.3

103

102

1011

51

101111

5

111102/

0

0

oi

i

i

i

o

i

ddmcmd

cmcmcmd

cmcmdfd

cmdfdd

cmRf

Virtual image, di <0 De-Magnified, |m| < 1, not inverted. m > 0

22.3 Thin lenses

Positive Lenses Thicker in middle Bend rays toward axis Form real focus

Negative Lenses Thinner in middle Bend rays away from the axis Form virtual focus

Types of Lenses Lenses are used to focus light and form images. There are a variety of possible types; we will consider only the symmetric ones, the double concave and the double convex.

Types of lenses

Lens nomenclature

Which type of lens to use (and how to orient it) depends on the aberrations and application.

Raytracing made easier In principle, to trace a ray, one must calculate the intersection of each ray with the complex lens surface, compute the surface normal here, then propagate to the next surface

computationally very cumbersome We can make things easy on ourselves by making the following assumptions:

all rays are in the plane (2-d) each lens is thin: height does not change across lens each lens has a focal length (real or virtual) that is the same in both directions

Thin Lens Benefits If the lens is thin, we can say that a ray through the lens center is undeflected

real story not far from this, in fact: direction almost identical, just a jog the jog gets smaller as the lens gets thinner

Using the focus condition real foci virtual foci

s = f

f

s = f

f

Tracing an arbitrary ray (positive lens)

1. draw an arbitrary ray toward lens 2. stop ray at middle of lens 3. note intersection of ray with focal plane 4. from intersection, draw guiding (helper) ray

straight through center of lens (thus undeflected)

Tracing an arbitrary ray (positive lens)

Original ray leaves lens parallel to helper why? because parallel rays on one side of lens

meet each other at the focal plane on the other side

Tracing an arbitrary ray (negative lens)

1. draw an arbitrary ray toward lens 2. stop ray at middle of lens 3. draw helper ray through lens center (thus

undeflected) parallel to the incident ray 4. note intersection of helper with focal plane

Tracing an arbitrary ray (negative lens)

Emerging ray will appear to come from this (virtual) focal point why? parallel rays into a negative lens appear to

diverge from the same virtual focus on the input side

Image Formation

Place arrow (object) on left, trace through image:

1) along optical axis (no defl.); 2) parallel to axis, goes through far focus with optical axis ray; 3) through lens center; 4) through near-side focus, emerges parallel to optical axis; 5) arbitrary ray with helper

Image Formation

Note convergence at image position (smaller arrow)

could run backwards just as well

Notes on Image Formation

Note the following: image is inverted image size proportional to the associated s-value: ray 3 proves it both s and s f (s = 120; s80; f = 48)

Notes on Image Formation

Gaussian lens formula (simple form):

Virtual Images

If the object is inside the focal length (s < f): a virtual (larger) image is formed non-inverted

Ray numbers are same procedure as previous

Virtual Images

This time s s = 40; f = 60; s 120 negative image distances indicate virtual images

The lens- We saw the Gaussian lens formula before:

f is positive for positive lenses, negative for negative lenses s is positive on left, s

But in terms of the surface properties:

The lens- R1 is for the left surface (pos. if center of curvature to right)

R2 is for right surface (pos. if center of curvature to right) bi-convex (as in prev. examples) has R1 > 0; R2 < 0 n is the refractive index of the material (assume in air/vac)

1 / f = (n 1) (1 /R1 + 1 /R2)

© 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

Principal-ray diagrams showing the graphical method of locating an image

formed by a thin lens (converging and diverging).

© 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

Formation of images by a thin converging lens for various object distances.

© 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

Principal-ray diagram for an image formed by a thin diverging lens.

© 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

The real image of the first lens acts as the object for the second lens.

© 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

Telescope

eyepiece sharing a focal plane; giving the eye the parallel light it wants

Everything goes as ratio of focal lengths: f1/f2

magnification is just M = 2/ 1 = f1/f2

Telescope

magnification is just M = 2/ 1 = f1/f2 after all: magnification is how much bigger things look displacement at focal plane, = f1 1 = f2 2 relation above

ratio of collimated beam (pupil) sizes: P1/P2 = f1/f2 = M

Summary

Spherical Mirrors (i) Concave mirrors

Spherical Mirrors (ii) Convex mirrors

spherical Mirrors

Image Forming by Spherical Mirrors Image Forming by Spherical Mirrors

Image Forming by spherical Mirrors Image Forming by spherical Mirrors

Image Forming by spherical Mirrors Refraction on Curved Surfaces

Refraction on Curved Surfaces Thin Lens

Thin Lens Thin Lens

Thin Lens Telescopes and Microscopes

Example

(a) Determine the position of the final image (b) If the plane mirror is removed and the distance of separation between both lenses is 35 cm, determine the new position of the final image.

22.5 cm

Object f1 = 15cm f2 = 30cm

15cm 40cm Mirror