Geometrical OpticsGeometrical light rays

Ray matrices and ray vectors

Matrices for various opticalcomponents

The Lens Maker’s Formula

Imaging and the Lens Law

Slides taken from R. Trebino’s online lectures

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 beassumed to propagate at small angles to it. This is called theParaxial Approximation.

axis

The Optic Axis

A mirror deflects the optic axis into a new direction.

This ring laser has an optic axis that scans out a rectangle.

Optic axis A ray propagatingthrough this system

We define all rays relative to the relevant optic axis.

The RayVector

A light ray can be defined by two co-ordinates:

xin, qin

xout, qout

its position, x

its slope, q

Optical axis

optical ray

x

q

These parameters define a ray vector,which will change with distance and asthe ray propagates through optics.

x

!

" #$ %& '

Ray MatricesFor many optical components, we can define 2 x 2 ray matrices.

An element’s effect on a ray is found by multiplying its ray vector.

Ray matricescan describesimple and com-plex systems.

These matrices are often called ABCD Matrices.

in

in

x

!

" #$ %& '

A B

C D

! "# $% &

Optical system ↔ 2 x 2 Ray matrix

out

out

x

!

" #$ %& '

Ray matrices asderivatives

We can writethese equationsin matrix form.

out in

out inD

B x

C

x A

! !

" # " #" #=$ % $ %$ %& '& ' & '

out

in

!

!

"

"

out

in

x

x

!

!

out

inx

!"

"

out

in

x

!

"

"

angularmagnification

spatialmagnification

out i

outout

i n

i

n i

n nx x

xx

x!

!= +

"

"

"

"

out in i

outut

nin i

o

nx

x!

!

!!

!= +

"

"

"

"

Since the displacements and anglesare assumed to be small, we can

think in terms of partial derivatives.

For cascaded elements, we simplymultiply ray matrices.

3 2 1 3 2 1

out in in

out in in

x x xO O O O O O

! ! !

" #$ %& ' & ' & '( (= =) *+ ,- . - . - .

/ 0 / 0 / 0( (1 23 4

Notice that the order looks opposite to what it should be,but it makes sense when you think about it.

O1 O3O2in

in

x

!

" #$ %& '

out

out

x

!

" #$ %& '

Ray matrix for free space or a medium

If xin and qin are the position and slope upon entering, let xout andqout be the position and slope after propagating from z = 0 to z.

out in in

out in

x x z !

! !

= +

=xin, qin

z = 0

xout qout

z 1

0 1

out in

out in

x xz

! !

" # " #" #=$ % $ %$ %

& '& ' & '

Rewriting these expressionsin matrix notation:

1=

0 1space

zO

! "# $% &

Ray Matrix for an Interface

At the interface, clearly:

xout = xin.

Now calculate qout.

Snell's Law says: n1 sin(qin) = n2 sin(qout)

which becomes for small angles: n1 qin = n2 qout

=> qout = [n1 / n2] qin

qin

n1

qout

n2

xin xout

1 2

1 0

0 /interfaceO

n n

! "= # $% &

Ray matrix for a curved interface

At the interface, again:

xout = xin.

To calculate qout, we mustcalculate q1 and q2.

If qs is the surface slope atthe height xin, then

q1 = qin+ qs and q2 = qout+qs

If R is the surface radius of curvature, the surface z coordinate will be:

2 2 2 212

212

1 ( / ) 1 ( / )

( / )

in in in

in

z R R x R R x R R R x R

x R

! "= # # = # # $ # #% &

=/

s in

in

dzx R

dx!" # #

qin

n1

qout

n2

xin = xout

q1q2

qs

R

z

qs

z = 0

Ray matrix for a curved interface (cont’d)

1 2 1 2

1 0

( / 1) / /curvedinterface

On n R n n

! "= # $%& '

1 2( / )( / ) /out in in in

n n x R x R! !" # + $

1 2 1 2( / ) ( / 1) /out in in

n n n n x R! !" # + $

Now the output angle depends on the input position, too.

Snell's Law: n1 q1 = n2 q2

1 2( / ) ( / )in in out in

n x R n x R! !" + # +

q1 = qin+ xin / R and q2 = qout+ xin / R

n1 n2

q1q2

A thin lens is just two curved interfaces.

1 2 1 2

1 0

( / 1) / /curvedinterface

On n R n n

! "= # $%& '

We’ll neglect the glass in between (it’s areally thin lens!), and we’ll take n1 = 1.

2 1 2 1

1 0 1 0

( 1) / (1/ 1) / 1/thin lens curved curved

interface interface

O O On R n n R n

! " ! "= = # $ # $% %& ' & '

2 1 2 1

1 0 1 0

( 1) / (1/ 1) / (1/ ) ( 1) / (1 ) / 1n R n n R n n n R n R

! " ! "= =# $ # $% + % % + %& ' & '

n=1

R1 R2

n≠1n=1

2 1

1 0

( 1)(1/ 1/ ) 1n R R

! "= # $% %& '

1 0

1/ 1f

! "# $%& '

This can be written:

1 21/ ( 1)(1/ 1/ )f n R R= ! ! The Lens-Maker’s Formulawhere:

Ray matrix for a lens

The quantity, f, is the focal length of the lens. It’s the single mostimportant parameter of a lens. It can be positive or negative.In a homework problem, you’ll extend the Lens Maker’s Formula tolenses of greater thickness.

1 0=

-1/ 1lensO

f

! "# $% &

If f > 0, the lens deflectsrays toward the axis.

f > 0

If f < 0, the lens deflectsrays away from the axis.

f < 0

1 21/ ( 1)(1/ 1/ )f n R R= ! !

R1 > 0R2 < 0

R1 < 0R2 > 0

Types of lenses

Lens nomenclature

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

A lens focuses parallel rays to a pointone focal length away.

01 1 0 0

/0 1 1/ 1 0 1/ 1 0

out in in

out in

x f x f x

x ff f!

" # " #" # " # " # " # " #= = =$ % $ %$ % $ % $ % $ % $ % && &' ( ' ( ' ( ' ( ' (' ( ' (

f

f

At the focal plane, all raysconverge to the z axis (xout = 0)independent of input position.

Parallel rays at a different anglefocus at a different xout.

A lens followed by propagation by one focal length:

Assume all inputrays have qin = 0

For all raysxout = 0!

Looking from right to left, rays diverging from a point are made parallel.

Spectrometers

f

f

Entrance slit

Diffractiongrating

ff

Camera

To best distinguish different wave-lengths, a slit confines the beam tothe optic axis. A lens collimates the beam, and a diffraction grating disperses the colors. A second lens focuses the beam to a point that depends on its beam input angle (i.e., the wavelength).

q µ l-l0 There aremany

types ofspectrom-eters. Butthey’re all

based on thesame principle.

Ray Matrix for a Curved Mirror

Like a lens, a curved mirror will focus a beam. Its focal length is R/2.

Note that a flat mirror has R = ∞ and hence an identity ray matrix.

1 ( )

2 /

out s in s s

in inx R

! ! ! ! ! !

!

= " = " "

# "

Consider a mirror with radius of curvature, R, with its optic axisperpendicular to the mirror:

qin

qout

xin = xout

q1qs

R

z

q1

1/

in s s inx R! ! ! != " #

1 0 =

2 / 1mirrorO

R

! "# $%& '

!

Laser Cavities

Two flat mirrors, the flat-flatlaser cavity, is difficult to alignand maintain aligned.

Two concave curved mirrors,the usually stable laser cavity,is generally easy to align andmaintain aligned.

Two convex mirrors, theunstable laser cavity, isimpossible to align!

Mirror curvatures matter in lasers.

An unstable cavity (or unstable resonator) can work if you do itproperly!In fact, it produces a large beam, useful for high-power lasers, whichmust have large beams.

Unstable Resonators

The mirror curvaturesdetermine the beam size,which, for a stable resonator,is small (100 mm to 1 mm).

An unstable resonator canhave a very large beam. Butthe gain must be high. Andthe beam has a hole in it.

Consecutive lenses

2 1 1 2

1 0 1 0 1 0= =

-1/ 1 -1/ 1 -1/ 1/ 1totO

f f f f

! " ! " ! "# $ # $ # $%& ' & ' & '

f1 f2

Suppose we have two lensesright next to each other (withno space in between).

tot 1 21/ =1/ +1/f f f

So two consecutive lenses act as one whose focal length iscomputed by the resistive sum.

As a result, we define a measure of inverse lens focal length, thediopter. 1 diopter = 1 m-1

A system images an object when B = 0.

When B = 0, all rays from apoint xin arrive at a point xout,independent of angle.

xout = A xin When B = 0, A is the magnification.

0out in in

out in in in

x x A xA

C x DC D! ! !

" # " # " #" #= =$ % $ % $ %$ %

+& '& ' & ' & '

[ ]

/

1/ 1/ 1/

0

o i o i

o i o i

B d d d d f

d d d d f

= + ! =

+ ! =

if

1 1 0 1

0 1 1/ 1 0 1

11

1/ 1 /0 1

1 / /

1/ 1 /

i o

oi

o

i o i o i

o

d dO

f

dd

f d f

d f d d d d f

f d f

! " ! " ! "= # $ # $ # $%& ' & ' & '

! "! "= # $# $ % %& ' & '

% + %! "= # $% %& '

The Lens LawFrom the object tothe image, we have:

1) A distance do2) A lens of focal length f3) A distance di

1 1 1

o id d f+ =

This is the Lens Law.

ImagingMagnification

1 1 1

o id d f+ =

1 11 / 1i i

o i

A d f dd d

! "= # = # +$ %

& '

( i

o

dM

d= !

If the imaging condition,

is satisfied, then:

1 / 0

1/ 1 /

i

o

d fO

f d f

!" #= $ %! !& '

1 11 / 1o o

o i

D d f dd d

! "= # = # +$ %

& '1/

o

i

dM

d= ! =

0

1/ 1/

MO

f M

! "= # $%& '

So:

Magnification Power

Often, positive lenses are rated with asingle magnification, such as 4x.

In principle, any positive lens can be used at an infinite number ofpossible magnifications. However, when a viewer adjusts theobject distance so that the image appears to be essentially atinfinity (which is a comfortable viewing distance for mostindividuals), the magnification is given by the relationship:

Magnification = 250 mm / f

Thus, a 25-mm focal-length positive lens would be a 10x magnifier.

Object under observation

Virtual ImagesWhen the object is less than one focal length away from a lens,no image occurs, but a virtual image is said to occur if you lookback through the lens.

Object

f > 0

Virtualimage