Opti GeometricalgeoOptics I

7/27/2019 Opti GeometricalgeoOptics I

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Geometrical Optics

Geometrical light rays

Ray matrices and ray vectors

Matrices for various optical

components

The Lens Makers Formula

Imaging and the Lens Law

Mapping angle to position

Prof. Rick Trebino, Georgia Tech

www.frog.gatech.edu


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Geometrical optics (ray optics) is the

simplest version of optics.

Ray

optics

Its the roughestapproximation of

optical reality.

Itll handle

reflection andrefraction, but

not interference

or diffraction.

However, when it

does work, it will

handle complex

problems much

more easily than,

say, wave optics.


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How will geometrical optics go wrong?

It neglects the phase.

As a result, the ray pictureimplies that a lens can focus a

beam to a point with zero

diameterand so obtain infinite

intensity and infinitely good

spatial resolution.

Not true. The smallest possible

focal spot is actually about the

wavelength, l. Same for the

best spatial resolution of an

image. This is fundamentallydue to the wave nature of

light, which is not included in

geometrical optics.

But it will be easy to use.

> ~l

~0

Ray optics

Reality


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Ray Optics

We'll define light rays as directions in space, corresponding,

roughly, to k-vectors of light waves.

Each optical system will have an optic axis, and all light rays

will be assumed to propagate at small angles to it. This is called

the Paraxial Approximation.

Optic axis

Light ray

Optical system


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The Optic Axis

A mirror deflects the optic axis into a new direction with theangle of reflection equal to the angle of incidence.

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.


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Ray

Vectors

At every position,z, along the optic axis, a light ray can be definedby two co-ordinates:

its position,x

its slope, q

Optic axis

x

q

These parameters define aray vector,

which will change with distance,z, as

the ray propagates through optics.

x

q

zz0

xin, qin

xout, qout

Optic

axisz0

0


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Ray Matrices

An optical elements effect on a ray is found by multiplying the rayvector by the elements ray matrix.

Ray vector

before lens

after lens before lens

after lens before lens

x xA B

C Dq q

Lens 2 x 2 ray matrix

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

Distance 2 x 2 ray matrix

Ray matrix

for lens

Ray vector

after lens

xin, qin

xout, qout

We can do thesame for theother lenses andthe distances.


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Ray Matrices

as Derivatives

We can write

these equationsin matrix form.

out in

out inD

B x

C

x A

q q

out

in

q

q

out

in

x

x

out

inx

q

out

in

x

q

Angular

magnification

Spatial

magnification

out i

outout

i n

i

n i

n nx x

xx

xq

q

out in i

outut

nin i

o

nx

xq

q

qq

q

Since the displacements,xin andxout,

and angles, qin and qout, are all

assumed to be small, we can think in

terms of partial derivatives.


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For cascaded elements, we simply multiply

together all the individual ray matrices.

3 2 1 3 2 1

out in in

out in in

x x xO O O O O O

q q q

Notice that the order looks opposite to what it should be,

but it makes sense when you think about it.

O1 O3O2in

in

x

q

out

out

x

q

Component #1 Component #2 Component #3


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Ray Matrix for Free Space or a Medium

Ifxin and qin are the position and slope upon entering, letxoutand qout

be the position and slope after propagating an arbitrary distance,z.

out in in

out in

x x zq

q q

xin, qin

0

xout, qout

z

10 1

out in

out in

x xz

q q

Rewriting these expressions

in matrix notation:

1=

0 1space

zO


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Ray Matrix for

an Interface

At the interface:

qout= [n1 / n2]qin

qin

n1

qout

n2

xin xout

1 2

1 0

0 /interfaceO

n n

which, for small angles, becomes: n1q

in = n

2q

out

Snell's Law says: n1sin(qin) = n2sin(qout)

Now calculate qout:

xout=xin


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A thin lens is just two curved interfaces.

1 2 1 2

1 0

( / 1) / /curvedinterface

On n R n n

Well neglect the glass in between (its a

really thin lens!), and well 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

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-Makers Formulawhere:

n1= 1

R1 R2

n2 = n 1

n1= 1


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Ray Matrix for a Lens

The quantity,f, is the focal lengthof the lens. Its the single most

important parameter of a lens. It can be positive or negative.

1 0=

-1/ 1lensO

f

Iff > 0, the lens deflects

rays toward the axis.

f > 0

Iff < 0, the lens deflects

rays away from the axis.

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

R1 > 0

R2 < 0f < 0

R1 < 0

R2 > 0

Its easy to extend the Lens Makers Formula to real lenses of

greater thickness.

Sign convention:R > 0 if the sphere

center is to the

right (z> 0), and

R < 0 if the sphere

center is to the

left (z< 0).


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Types of Lenses

Lens nomenclature

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

aberrations and application.


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A lens focuses parallel rays to a point

one 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 fq

f

f

At the focal plane, all rays

converge to thezaxis (xout

= 0)

independent of input position.

Parallel rays at a different angle

focus at a differentxout.

A lens followed by propagation by one focal length:

Assume all input

rays have qin = 0

For all rays,

xout

= 0!

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


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f

f

f

f

And it maps input position

to angle:

Lenses can simultan-

eously map angle to

position and position

to angle.

From input to output, use:

1) A distance f

2) A lens of focal lengthf3) Another distancef

1 1 0 1

0 1 1/ 1 0 1

1 1

0 1 1/ 1

0

/1/ 0

out in

out in

in

in

in in

in in

x xf f

f

xf f

f

x ff

x ff

q q

q

q

q

out inx q

out inxq

f

So this arrangement maps

input angle to position:

independent of

input position

independent of

input angle


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Spectrometers

f

f

Entrance

slit

Diffraction

grating

ff

Camera

To best distinguish different wave-

lengths, a slit confines the beam to

the optic axis. A lens collimates the

beam, and a diffraction gratingdisperses the colors. A second

lens focuses the beam to a

point that depends on its

beam input angle (i.e.,

the wavelength).

q ll0 There aremany

types ofspectrom-

eters. But

most are

based on

this principle.


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Lenses and Phase Delay

Equal phasedelays

Focus

ff

Ordinarily phase isnt considered in geometrical optics, but its

worth computing the phase delay vs.x andy for a lens.

It turns out that all paths through a lens to its focus have the same

phase delay, and hence yield constructive interference there!


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Lenses and

Phase Delay

( , ) ( 1) ( , )lens x y n k x y

2 2 2

1( , ) ( 1) ( )lens x y n k R x y d

neglecting phase delays

independent ofx andy.2 2

1( , ) ( 1)( / 2 )( )lens x y n k R x y

2 22 2 2 2 2 2

1 1 1 1

1

1 ( ) /2

x yR x y R x y R R

R

2 2 2

1( , )x y R x y d

( , )x yd

First consider variation (the

x andy dependence) in the

path through the lens.

But:

Extra phase

delay due to

the glass


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Ray Matrix for a Curved Mirror

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

Note that a flat mirror hasR = and hence an identity ray matrix.

1 ( )2 /

out s in s s

in inx R

q q q q q q q

Consider a mirror with radius of curvature,R, with its optic axis

perpendicular to the mirror:

qin

qout

xin = xout

R

z

q1

q1

qs

1 /in s s inx Rq q q q

1 0

= 2 / 1mirrorO R


24/31

Two flat mirrors, the flat-flatlaser cavity, is difficult to align

and maintain aligned.

Two concave curved mirrors,

the usually stable laser cavity,

is generally easy to align and

maintain aligned.

Two convex mirrors, the

unstable laser cavity, is

impossible to align!

Mirror curvatures matter in lasers.Laser Cavities


25/31

But an unstable cavity (orunstable resonator) can be useful!

In fact, it produces a large beam, useful for high-power lasers, which

must have large beams.

The mirror curvatures

determine the beam size,

which, for a stable resonator,

is small (100 mm to 1 mm).

An unstable resonator can

have a very large beam. But

the gain must be high. And

the beam has a hole in it.

Unstable Resonators


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Consecutive Lenses

2 1 1 2

1 0 1 0 1 0= =

1/ 1 1/ 1 1/ 1/ 1tot

O

f f f f

f1 f2

Consider two lenses right next to

each other (with no space inbetween).

1 21/ = 1/ +1/totf f f

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

As a result, we define a measure ofinverse lens focal length, the

diopter:

1 diopter = 1 m-1


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A system images an object when B= 0.

WhenB = 0, all rays from a

point xin arrive at a point xout,

independent of angle.

xout= A xin WhenB = 0, A is the magnification.

0out in in

out in in in

x x A xAC x DC Dq q q

Lens

ImageObject

do di

f


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/

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 Law

From the object to the

image, we have:

1) A distance do2) A lens of focal lengthf

3) A distance di

1 1 1

o id d f

This is the Lens Law.

Lens

ImageObject

do di

f


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Imaging

Magnification

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:

Lens

ImageObject

do di

f

Angular

magnification


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Negative-flenses have virtual images, and positive-flenses do

also if the object is less than one focal length away.

Object

f > 0

Virtual

image

Virtual

Images

f < 0

Virtual

image

Simply looking at a flat mirror yields a virtual image.

A virtual image occurs when the outgoing rays

from a point on the object never actually intersect

at a point but can be traced backwards to one.

Object infinitely

far away


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The F-number, f / #, of a lens is the ratio of its focal length and its

diameter.

f / # = f / d

f

f

d1 f

f

d2

f / # = 1 f / # = 2

Large f-number lenses collect more light but are harder to engineer

F-Number

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