Lecture12 Ch5 geometrical optics

Chapter 5

Geometrical Optics

Phys 322Lecture 12

Geometrical optics (ray optics) is the simplest version of optics.

Rayoptics

Ray Optics

We'll define light rays as directions in space, corresponding, roughly, to k-vectors of light waves.

We won’t worry about the phase.

Each optical system will have an axis, and all light rays will be assumed to propagate at small angles to it. This is called the Paraxial Approximation.

axis

RaysRays are orthogonal trajectories of the wavefronts.

Normal congruence: group of rays for which you can find a surface that is orthogonal to all of them.

Theorem of Malus and Dupin:A group of rays will preserve its normal congruence after any

number of reflections and refractions.

Is geometrical optics the whole story?

No. We neglect the phase.

Also, our ray pictures seem to imply that, if we could just remove all aberrations, we could focus a beam to a point and obtain infinitely good spatial resolution.

Not true. The smallest possible focal spot is the wavelength, . Same for the best spatial resolution of an image. This is due to diffraction, which has not been included in geometrical optics.

>

~0

Focus

Each point on an illuminated or a self-illuminating surface is a source of spherical waves:Rays diverge from that point

Spherical wave can also converge to a point

A point from (to) which a portion of spherical wave diverges (converges) is a focus of the bundle of rays

Stigmatic optical system - perfect image

Reversibility: SPP and S are conjugate points

object space image space

Lens- refractive device that changes the wavefront curvature

The wavefront changed from convex to concave

OPL should be the same for red and blue rays

Qualitatively:Insert a transparent object with n>1 that is thicker in center and thinner at the edges

Aspherical surfacesThe shape of the interface:

constant1 ADnAFn ti

Time to ravel from S to DD’:

ti

ADAFvv

1

Time of travel from S to DD’ must be the same for any point in plane DD’:

or: cADnAFn ti

c1

constant1 ADnnAF

i

t

nti nt/ni > 1 - hyperbola nti nt/ni < 1 - ellipsoid

rays can be reversed

Change spherical wave to plane wave

Aspherical surfaces

Convex lensesConverging lenses

Concave lensDiverging lens

F: Focal points

When a bundle of parallel rays passes through a lens, the point to which they converge (converging lens) or the point from which they appear to diverge (diverging lens) is called focal point

real image

virtual image

object distance

image distance

Vertex

Opticalaxis

Spherical lensUse Fermat’s Principle

OPL=n1lo+ n2li

Law of cosines for SAC and ACP:

2/1222 cos2 RsRRsRl ooo

2/1222 cos2 RsRRsRl iii cos180cos

02

sin2

sin 21

i

i

o

o

lRsRn

lRsRn

dOPLd

0

122

0

1 1lsn

lsn

Rln

ln o

i

i

iFor different P will be different

object distance

image distance

Vertex

Opticalaxis

Spherical lens

0

122

0

1 1lsn

lsn

Rln

ln o

i

i

i

Approximate: for small :

ii

o

slsl

0

sin1cos

Rnn

sn

sn

i

122

0

1

The position of P is independent of the location of A over small area close to optical axis.Paraxial rays: rays that form small angles with respect to optical axisParaxial approximation: consider paraxial rays only

Spherical lens: focal length

Rnn

sn

sn

io

1221

Focal point F0 : si =

Rnn

fn

o

121 0

Rnn

nfo12

1

First focal length:

(object focal length)

Reverse:

Rnn

nfi12

2

Second focal length:

(image focal length)

object(or first)focus

image(or second)focus

R > 0, n2 > n1 f > 0 - converging lens

Spherical lens: focal lengthWhat if R is negative?

Rnn

nfo12

1

an image is virtual:it appears on object side

Rnn

nfi12

1

an object is virtual:it appears to be in the image side

object distance

image distance

Vertex

Opticalaxis

Sign convention

so, fo + left of Vsi, fi + right of VR + if C is right of V

Assume light entering from left:

Rnn

sn

sn

io

1221

Lens classification

Thicker inthe middle

Thinner inthe middle

R1<0 R2<0R1<0 R2=

R1>0 R2<0

(negative)(positive)

Thin lens equationR

nnsn

sn

i

122

0

1

For the first surface:

111 Rnn

sn

sn ml

i

l

o

m

221 Rnn

sn

dsn lm

i

m

i

l

Second surface:

Add two eq-ns and simplify using nm=1 (air) and d0:

Thin-lens equation(Lensmaker’s formula)

21

11111RR

nss l

io

Gaussian lens formula

21

11111RR

nss l

io

Find focal lengths (so, or si)fo = fi f

21

1111RR

nf l

Gaussian lens formula:

fss io

111

This is one of the most widely used equations.All one needs to know about the lens is its focal length.

s0 si

f fxo xi

Newtonian form: 2fxx io

Example

Plano-convex spherical lens

R = 50 mmn = 1.5

What is a focal length of this lens?

Solution

mm 100/1mm 50

1115.11111

21

RRn

f l

mm 100f

Example

Object is placed at 600 mm, 200 mm, 150 mm, 100 mm, 50 mm.Where would be the image?

Solutionfss io

111 so si

600 120200 200150 300100 80 -400

f = 100 mm

fsfss

o

oi

Focal plane

All bundles of parallel rays converge to focal points that lay on one plane: second, or back focal plane

Thin lens + paraxial approximation:All rays that go through center O do not bend

second, or backfocal plane

Fo- lies on first, or front focal plane.

Imaging with a lens

Each point in object plane is a point source of spherical waves and the lens will image them to respective points in the image plane.

Converging lens: principal rays

1) Rays parallel to principal axis pass through focal point Fi.2) Rays through center of lens are not refracted.3) Rays through Fo emerge parallel to principal axis.

Fo

Fi

Object

ImageOptical axis

In this case image is real, inverted and enlargedAssumptions:

• monochromatic light

• thin lens.• rays are all “near” the principal axis

(paraxial).

Since n is function of , in reality each color has different focal point: chromatic aberration. Contrast to mirrors: angle of incidence/reflection not a function of

Principal rays:

Diverging lens: forming image

1) Rays parallel to principal appear to come from focal point Fi.2) Rays through center of lens are not refracted.3) Rays toward Fo emerge parallel to principal axis.

Fi

Fo

Object Image

O.A.

Image is virtual, upright and reduced.

Principal rays:

Assumptions:• paraxial monochromatic rays• thin lens