Aoptics-Matrix Optics

8/14/2019 Aoptics-Matrix Optics

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Matrix Optics

Matrix O tics-Geometrical light rays

-Ray matrices and ray vectors

-Matrices for various optical

components

-The Lens Makers Formula

-Imaging and the Lens Law

-

-Cylindrical lenses

Is geometrical optics the whole story?

. .

Also, our ray pictures seem

to im l that if we could

~

just remove all aberrations,

we could focus a beam to a

good spatial resolution.

Not true. The smallest

possible focal spot is the

wavelength, . Same for thebest s atial resolution of an

>

image. This is due todiffraction, which has not

geometrical optics.

Geometrical optics (ray optics) is the

simplest version of optics.

Ray

optics


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Ra O tics

axis

' , ,

roughly, to k-vectors of light waves.

We wont worr about the hase.

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.

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 propagating

e e ne a rays re a ve o e re evan op c ax s.

The Ra xin, in

Vectorxout,out

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

its position, x

its slope, x

Optical axis

These parameters define a ray vector,

which will change with distance and asx

the ray propagates through optics.

Ray Matrices

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

An elements effect on a ray is found by multiplying its ray vector.

ay ma r ces

can describe

simple and com-Optical system 2 x 2 Ray matrix

plex systems.

inx A B outx

in

C D out

These matrices are often (uncreatively) called ABCD Matrices.


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Ray matrices as

derivatives out ioutout

i n

i

n i

n nx x

x

= +

out in ioututo

nxx

= +

Since the displacements and

angles are assumed to be small,nn

derivatives.

out

in

x

x

out

in

x

spatial

magnification

We can write

these equationsout inB xx A

=

.out in

outout angularin

inx magn ca on

For cascaded elements, we simply multiplyray matrices.

O1 O3O2in

in

out

out

x x x

3 2 1 3 2 1

out in in

O O O O O O

= =

be, but it makes sense when you think about it.

Ray matrix for free space or a medium

Ifxinand inare the position and slope upon entering, let xoutand outbe the osition and slo e after ro a atin from z =0 to z.

out in in

out in

x x z

= +=

x ,

xoutout

Rewriting these

expressions in matrix

z = 0 z

1

out inx xz =

out in1

=z

O

0 1space

Ray Matrix for an Interface

At the interface, clearly:

xout= xin.in

out

xin xout

Now calculate .n1 n2

Snell's Law says: n1 sin(in) = n2 sin(out)

which becomes for small angles: n1 in = n2 out

= n / n 1 0

O =

1 20 /n n


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Ray matrix for a curved interface

At the interface, again:

=R

ou n.

To calculate out

, we

must calculate 1 and 2. in

out

x

12

s

s=

Ifs is the surface slopeat the height xin, then

n1 n2z

z= 0 z

s in

1 =in+sand 2 =out+s1 =in+ xin/ R and 2 =out+ xin/ R

Snell's Law: n1 1 = n2 2

1 2( / ) ( / )in in out inn x R n x R + = +

1 2( / )( / ) /

out in in inn n x R x R = +

= 1 0

curvedO

=

1 2 1 2out in in1 2 1 2interface n n n n

A thin lens is just two curved interfaces.

Well neglect the glass in between

R1 R2

1 0 =

,

n1 = 1.

n1

1 2 1 2( / 1) / /

curvedinterface n n R n n

1 0 1 0

= =

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

thin lens curved curved interface interface n R n n R n

= =

1 0 1 0

2 1 2 1( 1) / (1/ 1) / (1/ ) ( 1) / (1 ) / 1n R n n R n n n R n R= =

+ +

2 1( 1)(1/ 1/ ) 1n R R= 1/ 1f

This can be written:

1 21/ ( 1)(1/ 1/ ) f n R R= The Lens-Makers Formulawhere:

=

-1/ 1lens

Of

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

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

im ortant arameter of a lens. It can be ositive or ne ative.

In a homework problem, youll extend the Lens Makers Formula

to lenses of greater thickness.

R1 > 0 R1 < 0

Iff >0, the lens Iff 0

deflects rays toward

the axis.

rays away from the axis.

T es of lenses

Lens nomenclature

aberrations and application.


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A lens focuses parallel rays to a point oneoca engt away.

A lens followed by propagation by one focal

len th:

For all rays

xout= 0!

01 1 0 0out in inx f x f x /0 1 1/ 1 0 1/ 1 0out inx ff f

= = =

input rays have

in= 0

f

At the focal plane, all raysconverge to the zaxis (xout= 0)

independent of input position.Parallel rays at a different anglefocus at a different xout.

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

Spectrometers -lengths, a slit confines the beam tothe optic axis. A lens collimates the

Camera,

disperses the colors. A second

lens focuses the beam to a

f

po nt t at epen s on tsbeam input angle (i.e.,

the wavelength).

Entrance

f

0 There aremany

types of

-

Diffraction

eters. Buttheyre all

f

same principle.

Lenses and phase delay

Ordinarily phase isnt considered in geometrical optics, but its. .

All paths through a lens to its focus have the same phase delay,and hence yield constructive interference there.

Equal phase

Focus

f

Lenses and ( , )x y

phase delay

First consider variation (thex and y dependence) in the

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

2 2 2

1( , )x y R x y d = .

2 2 2

1( , ) ( 1) ( )

lensx y n k R x y d = +

2 22 2 2 2 2 2 x y+

1 1 1 1

12

x y x y

R

= u :

phase delays.

2 2

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


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Lenses and hase dela x,y

(x,y)Now compute the phase delay in the

2 2 2 x k x z = + +

0Focusair after the lens:

a r

2 22 2 2 x y+

z

>>

2 2

2z ,

neglecting constant phase delays.,

air

2 2 2 21, ,lens air

1 1= if

1

nz R

= that is, if z = f!

Ra Matrix for a Curved MirrorConsider a mirror with radius of curvature, R, with its optic axis

1 /in s s inx R =

1( )

out s in s s = = out

R

2 /in inx R

in xin= xout

1s

z 1 0=

mirror

O

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

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

Mirror curvatures matter in lasers.Laser Cavities

, -laser cavity, is difficult to alignand maintain aligned.

,the usually stable laser cavity,is generally easy to align andmaintain aligned.

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

Unstable Resonators

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.

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

An unstable resonator can

have a very large beam. Butthe gain must be high. And.


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Consecutive lenses

Suppose we have two lenses

f1

f2

right next to each other (withno space in between).

1 0 1 0 1 0= =

2 1 1 2

-1/ 1 -1/ 1 -1/ 1/ 1

tot f f f f

tot 1 2=

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, theopter.

1 diopter = 1 m-1

A system images an object when B= 0.

When B= 0, all rays from a 0out in in x x AxA po n xinarr ve a a po n xout,independent of angle. out in in inCx DC D

= =+

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

The Lens Law

From the object tothe ima e we have:,

1) A distance do

3) A distance di

/o i o i

B d d d d f = + =

=

1 1 0 1

0 1 1/ 1 0 1

i od dO

f

=

0

o i o i

if11

oidd

= 1 1 1

1 / /

o

i o i o id f d d d d f

+ =

o id d f

+ =

1/ 1 /

o f d f This is the Lens Law.

ImagingMagnification

1 1 1+ =

t e mag ng con t on,

o id d f

is satisfied, then:

1 / 1i io i

A d f d d d

= = +

1 / 0i

d fO

= i

o

Md

= o

So:

1 11 / 1o oo i

D d f d d d= = +

0MO =

1/o

i

Md

= =


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Ma nification Power

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

,

number of possible magnifications. However, when a viewer

adjusts the object distance so that the image appears to be

essen a y a n n y w c s a com or a e v ew ng s ance

for most individuals), the magnification is given by the

relationship:

Thus a 25-mm focal-len th ositive lens would be a 10x

magnifier.

Virtual A virtual image occurs when the outgoing rays

Imagesfrom a point on the object never actually intersect

at a point but can be traced backwards to one.

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

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

Virtualimage

Virtualimage

Ob ect

f >0f


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De th of field exam le

isnt always desirable.

f/32 (very small aperture;

f/5 (relatively large aperture;

desirable for portraits.

Bokeh is the rendition of out-of-focus points of light.

Something deliberately out of focus should notdistract.

Poor Bokeh. Edge is sharply defined.

Neutral Bokeh. Evenly illuminated blur circle.Still bad because the edge is still well defined.

oo o e . ge s comp e e y un e ne .

Bokeh is where art and engineering diverge, since better bokeh is dueto an imperfection (spherical aberration). Perfect (most appealing)o e s a auss an ur, u enses are usua y esgne or neu ra

bokeh!

The inholeIf all light rays are directed through

cameraa pinhole, it forms an image withan infinite depth of field.

PinholeThe concept of thefocal length is

pinhole lens. Themagnification is still

ImageThe first person to

i o.

ment on t s eawas Aristotle.

With their low cost, small sizeand huge depth of field,the re useful in securitapplications.

The Camera Obscura A dark room with a smallhole in a wall. The termcamera obscura means

.Renaissance painters used

them to paint realistic

pa n ngs. ermeer pa n eThe Girl with a PearlEarring (1665-7) using

one.

A nice view of a camera obscurais in the movie, Addicted to Love,starring Matthew Broderick (who

plays an astronomer) and MegR an who set one u to s on

their former lovers.


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Another measure of a lens size is the numerical aperture. Its the.

Why thisdefinition?

Because themagnificationcan be shown to

f be the ratio ofthe NA on the

two sides of thelens.

High-numerical-aperture lenses are bigger.

Lenses can also mapang e to pos t on.

From the object tothe image, we have:

1 A distance f2) A lens of focal length f3) A distance f

So1 1 0 1out inx xf f = out inx

And this arrangement1 1

out in

inxf f

= 0 1 1/ 1

0

in

in in

f

x ff

out inx

/1/ 0 in inx ff

Telescope(matrix)

Telesco es Image Image

M1 M2

Keplerian telescope

A telescope should image an object, but, because the object willhave a ver small solid an le it should also increase its solid an le

significantly, so it looks bigger. So wed like Dto be large. And usetwo lenses to square the effect.

0

1/ 1/imaging

MO

M

=

where M = - d i/ do

2 10 0M MO

=

Note that this iseasy for the firstlens, as the object

2 2 1 11 1 1 1 f M f M

1 2 0M M = So use di


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Telescopes (contd)

f1 < 0 f2 > 0

The analysis of this telescope is a homework problem!

The Cassegrain Telescope

Telescopes must collect as much light as possible from the generally- .

Its easier to create large mirrors than large lenses (only the surfaceneeds to be very precise).

Object

It may seem like the

image will have ahole in it, but onl ifits out of focus.

The Cassegrain Telescope

If a 45-mirrorreflects thebeam to the sidebefore thesmaller mirror,

Newtoniantelescope.


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No discussion oftelescopes would becom lete without afew pretty pictures.

Galaxy Messier 81

Uranus is surrounded by its four major ringsand by 10 of its 17 known satellites

- at s ye e u a-one o t emost complex planetary nebulae ever seen

ImageImage Eye-Objective-

M1 M2

scopes

Microscopes work on the same principle as telescopes, except thatthe ob ect is reall close and we wish to ma nif it.

When two lenses are used, its called a compound microscope.

= =for the objective, where sis the image distance beyond one focallength. In terms of s, the magnification of each lens is given by:

|M| = di/ do= (f + s) [1/f 1/(f+s)] = (f + s) / f 1 = s/ f

Many creative designs existfor microscope objectives.Exam le: the Burch reflectin

Object To eyepiece

microscope objective:

terminology

If an optical system lacks cylindricalsymmetry, we must ana yze ts x- an y-directions separately: Cylindrical lenses

A "spherical lens" focuses in both transverse directions.

A "c lindrical lens" focuses in onl one transverse direction. .

en us ng cy n r ca enses, we must per orm two separateray-matrix analyses, one for each transverse direction.


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Large-angle reflection off a curved mirroralso destroys cylindrical symmetry.

The optic axis makes a large angle with the mirror normal,and rays make an angle with respect to it.

Optic axisbefore reflection

tangentialray

Optic axis afterreflection

called "tangential.

ays a ev a e rom e op c ax s o e p ane o nc ence arecalled "sagittal. (We need a 3D display to show one of these.)

Ray Matrix for Off-Axis Reflection from aCurved Mirror

If the beam is incident at a large angle, , on a mirror with radius ofcurvature, R:

tangential ray

1 0

2 / 1eR R

pt c ax s

where Re= Rcos for tangential rays

and Re= R /cos for sagittal rays

Photo ra h lenses

.

Double Gauss Petzval

ese are o er esgns.

Camera


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hy lensesModern lenses canhave up to 20elements!

Canon 17-85mmf/3.5-4.5 zoom

Canon EF 600mm f/4L ISUSM Super Telephoto Lens17 elements in 13 rou s$12,000

Geometrical Opticsterms

Optical instrument(eyes)

Anatomy of theye

Incoming

light

Eye slides courtesy of Prasad Krishna, Optics I student 2003.


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The cornea iris and lensThe cornea is a thin membrane that has

. .It protects the eye and refracts light

(more than the lens does!) as it enterse eye. ome g ea s roug e

cornea, especially when its blue.

The iris controls the size of the pupil, an opening that allows lightto enter through.

The lens is jelly-like lens with an index of refraction of about 1.44.

This lens bends so that the vision process can be fine tuned.When you squint, you are bending this lens and changing itsproperties so that your vision is clearer.

The ciliary muscles bend and adjust the lens.

Near-sightednessmyop a

n nears g e ness, a person cansee nearby objects well, but has

difficulty seeing distant objects. Near-si htednessObjects focus before the retina.This is usually caused by an eyethat is too long or a lens systemthat has too much power to focus.

M o ia is corrected with anegative-focal-length lens. Thislens causes the light to diverge

.

Far-sightedness(hyperopia)

Far-si htedness h ero iaoccurs when the focal point isbeyond the retina. Such a

ar-s g e ness

well, but has difficulty seeingnearby objects. This is causedy an eye a s oo s or , or a

lens system that has too littlefocusing power. Hyperopia iscorrected with a positive-focal-length lens. The lens slightlyconverges the light before itenters the eye.

, ,likely to experience far-sightedness. Hence bifocals.

Astigmatism is a commonproblem in the eye.

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