Geo

metr

ical

op

tics

Geo

met

rica

l opti

cs is

bas

ed o

n r

ay-t

raci

ng. T

he

appro

ach

assu

mes

that

the

dim

ensi

on o

f opti

cal el

emen

ts is

much

la

rger

than

lig

ht

wav

elen

gth

, i.e.

, .

Voca

bula

ry-

Conju

gat

e poin

ts:

obje

ct s

pac

e an

d im

age

spac

e

-C

onve

x (

conve

rgin

g,

posi

tive

) le

ns

vs. co

nca

ve (

div

ergin

g,

neg

ativ

e) len

s

-Fo

cal poin

t an

d f

oca

l le

ngth

/ c

onju

gat

e fo

ci (

front

FL,

bac

k FL

)

-Rea

l im

age

vs.

virt

ual

im

age

-A

ber

rati

ons

(longit

udin

al/l

ater

al o

r tr

aver

se s

pher

ical

, ax

ial/

longit

udin

al

chro

mat

ic,

com

a, a

stig

mat

ism

)

-D

iffr

acti

on-lim

ited

sys

tem

s

-A

chro

mat

s(d

ouble

ts,

trip

lets

)

-Fr

aunhofe

r-ce

men

ted a

chro

mat

ic d

ouble

t (c

hro

mat

ic,

spher

ical

aber

rati

ons,

com

a)

Geo

metr

ical

op

tics

Geo

metr

ical

op

tics

: len

s

Lense

s fo

rm im

ages

by

refr

acti

ng lig

ht.

Lense

s ar

e tr

ansp

aren

t an

d a

llow

lig

ht

thro

ugh it

unlik

e m

irro

rs t

hat

are

hig

hly

ref

lect

ive.

Lense

s usu

ally

hav

e tw

o “

face

s”.

Each

fac

e ca

n b

e co

nca

ve,

conve

x,

or

pla

no

For

lense

s, t

her

e ar

e 2

foca

l poin

ts –

in f

ront

of

and b

ehin

d

the

lens.

Rea

l im

ages

are

beh

ind t

he

lens.

(Im

ages

are

opposi

te o

f th

e obje

ct)

Vir

tual

im

ages

are

on t

he

sam

e si

de

the

obje

ct is

loca

ted.

htt

p:/

/mrs

cgri

ffin

.word

pre

ss.c

om

/ap-p

hys

ics/

light-

geo

met

ric-

opti

cs/

Geo

metr

ical

op

tics

: len

s

Lens

form

ula

(Le

nsm

aker

’s f

orm

ula

)

RR

nf

fos

isan

d

Opti

cal ce

nte

r: r

ays

pas

sing t

hro

ugh c

an b

e dra

wn a

s st

raig

ht

lines

.

f o

f i

Mag

nif

icat

ion

oi ssM

Geo

metr

ical

op

tics

: len

s

Sign c

onve

nti

on

F-num

ber

:

typic

al c

amer

a le

nse

s hav

e m

arki

ngs

of

1,

1.4

, 2

, 2

.8,

4,

5.6

, 8

, 1

1,

16

, 2

2 …

(fa

st t

o s

low

)

Posi

tive

(+

)N

egat

ive

(–)

R1

Conve

x len

sC

onca

ve len

s

R2

Conca

ve

lens

Conve

x len

s

s oRea

l obje

ctV

irtu

al o

bje

ct

s iRea

l im

age

Vir

tual

im

age

fC

onve

x len

sC

onca

ve len

s

MEr

ect

imag

eIn

vert

ed

imag

e

Dff

Geo

metr

ical

op

tics

: mir

ror

an

d p

rism

Mir

ror:

form

ula

for

spher

ical

mir

rors

Pris

m:

Rig

ht-

angle

pri

sm a

nd d

ove

pri

sm

Ros

is

Fiber

: num

eric

al a

per

ture

(N

A)

fn

NA

f

fGeo

metr

ical

op

tics

: im

ag

e f

orm

ati

on

(c

on

vex l

en

s)

Geo

metr

ical

op

tics

: im

ag

e f

orm

ati

on

(c

on

cave l

en

s)

Geo

metr

ical

op

tics

: im

ag

e f

orm

ati

on

Geo

metr

ical

op

tics

: co

nca

ve m

irro

r

Geo

metr

ical

op

tics

: im

ag

e f

orm

ati

on

(c

on

cave m

irro

r)G

eo

metr

ical

op

tics

: im

ag

e f

orm

ati

on

(c

on

cave m

irro

r)

Geo

metr

ical

op

tics

: co

nca

ve m

irro

rG

eo

metr

ical

op

tics

: co

nvex m

irro

r

Geo

metr

ical

op

tics

: im

ag

e f

orm

ati

on

(c

on

vex m

irro

r)C

ata

log

ues

Cata

log

ues

Cata

log

ues

Cata

log

ues

Cata

log

ues

Cata

log

ues

Ab

err

ati

on

s:

Chro

mat

ic a

ber

rati

on:

-n is

a fu

nct

ion o

f fr

equen

cy.

-a

lens

hav

ing a

dif

fere

nt

refr

acti

ve index

for

dif

fere

nt

wav

elen

gth

s of

light

(the

dis

per

sion o

f th

e le

ns)

Ab

err

ati

on

s:A

berr

ati

on

s:

Spher

ical

aber

rati

on (

SA):

-Sp

her

ical

surf

aces

in g

ener

al y

ield

per

fect

im

ager

y only

in t

he

par

axia

l re

gio

n.

-Im

age

imper

fect

ion t

hat

occ

urs

due

to t

he

incr

ease

d r

efra

ctio

n

of

light

rays

that

occ

urs

when

ray

s st

rike

a len

s or

mir

ror

nea

r it

s ed

ge,

in c

om

par

ison w

ith t

hose

that

str

ike

nea

rer

the

cente

r.-

Longit

udin

al S

A-

Tra

vers

e/la

tera

l SA

-W

avef

ront

dis

tort

ion

Ab

err

ati

on

s:

Com

a:

-off

-axis

poin

t so

urc

es s

uch

as

star

s ap

pea

ring d

isto

rted

. -

def

ined

as

a va

riat

ion in m

agnif

icat

ion o

ver

the

entr

ance

pupil.

Ast

igm

atis

m:

an o

pti

cal sy

stem

has

dif

fere

nt

foci

for

rays

th

at p

ropag

ate

in t

wo p

erpen

dic

ula

r pla

nes

Geo

metr

ical

op

tics

: para

xia

l ap

pro

xim

ati

on

Para

xia

lap

pro

xim

atio

n:

-M

ore

gen

eral

appro

ach

for

ray-

trac

ing

thro

ugh

com

pound

lense

s

-A

ngula

rdev

iati

on

of

ara

yfr

om

the

cylin

dri

cal

axis

or

the

axis

of

pro

pag

atio

n(z

)is

smal

lsu

chth

atsi

nor

tan

can

be

appro

xim

ated

by

.Ba

sed

on

this

,th

edef

lect

ion

of

ara

yby

ath

inle

ns

can

be

des

crib

edas

Geo

metr

ical

op

tics

: para

xia

l ap

pro

xim

ati

on

Fromr i=r o

and

, fr

fr

ri

oi

ii

rr

oo

rr

ii

oo

rr

frr

Geo

metr

ical

op

tics

: para

xia

l ap

pro

xim

ati

on

Sim

ilarl

y,th

epro

pag

atio

nof

ara

yth

rough

ast

raig

ht

sect

ion

of

ahom

ogen

eous

med

ium

of

lengthd

can

be

trac

edw

ith

am

atri

x

.

Itca

nbe

easi

lysh

ow

nth

atth

em

atri

xfo

rth

epro

pag

atio

nof

ara

yth

rough

ast

raig

ht

sect

ion

follo

wed

by

ath

inle

ns

isgiv

enby,

.

d

d

ff

df

d

Geo

metr

ical

op

tics

: ABC

D (

or

ray t

ran

sfer)

matr

ix

The

ray

pro

pag

atio

nth

rough

ase

ries

of

opti

cal

com

ponen

tsca

nbe

des

crib

edw

ith

mat

rix

pro

duct

s.In

equat

ion

form

,

or

The

mat

rix

wit

hth

eel

emen

tsof

A,

B,C

,an

dD

isca

lled

ABC

Dm

atri

xor

ray-

tran

sfer

mat

rix.

Note

that

the

above

dis

cuss

ion

inher

entl

ypre

sum

espla

ne

wav

e.

nn

n

nn

n

Dr

Cr

rBr

Arr

DC

qB

AqD

rC

rBr

Arrr

qnn

nn

nn

nnn

Geo

metr

ical

op

tics

:

Ath

ick

lens

=tw

osp

her

ical

refr

acti

ng

surf

aces

+die

lect

ric

separ

atio

n-

Lens

form

ula

still

hold

sw

ith

effe

ctiv

efo

calle

ngth

.

Anal

ytic

alra

y-tr

acin

g:use

Snel

l’s

law

.

RnR

dn

RR

nf

d n n RR

n n R ff

Cata

log

ues

Cata

log

ues

Geo

metr

ical

op

tics

: Gau

ssia

n b

eam

s

For

aG

auss

ian

bea

m,

star

ting

from

Max

wel

l’s

equat

ions

wit

ha

few

assu

mpti

ons,

itca

nbe

obta

ined

that

wher

e,

,an

d

zRik

zw

rz

kziz

wwE

zy

xE

zzw

zw

zzz

Rzz

z

nw

z

Geo

metr

ical

op

tics

: Gau

ssia

n b

eam

s

Als

o,

note

that

k=

2n/

.In

this

equat

ion,

w(z

)an

dR

resp

ecti

vely

den

ote

the

bea

msp

ot

size

and

the

radiu

sof

curv

ature

of

aG

auss

ian

bea

m.

Geo

metr

ical

op

tics

: Gau

ssia

n b

eam

s

The

div

ergen

cean

gle

Gis

giv

enby

whic

his

am

anif

esta

tion

of

wav

edif

frac

tion,

indic

atin

gtr

ansv

erse

spre

adin

gin

the

far

fiel

dfo

rz>

>z 0

.

nw

G

Geo

metr

ical

op

tics

: Gau

ssia

n b

eam

s &

ABC

D

The

use

of

ABC

Dm

atri

xis

equal

lyva

lidfo

rG

auss

ian

bea

ms,

exce

pt

that

qn

inth

epla

ne

wav

eis

repla

ced

wit

h

inth

eca

seof

Gau

ssia

nbea

ms.

Inoth

erw

ord

s,th

epro

pag

atio

nof

aG

auss

ian

bea

mth

rough

anar

bit

rary

num

ber

of

lensl

ike

elem

ents

can

be

trac

edby

mult

iply

ing

ABC

Dm

atri

ces

and

reco

veri

ng

the

bea

mra

diu

sR(z

)an

dsp

ot

size

w(z

).T

his

isca

lled

the

ABC

Dla

w.

znw

iz

Rz

qn

nn

Geo

metr

ical

op

tics

: eyes

-Ph

oto

rece

pto

rs (

rods;

sco

topic

, nig

ht,

cones

; photo

pic

, day

)-

Acc

om

modat

ion: m

yopia

(neg

ativ

e le

nse

s),

hyp

eropia

/hyp

erm

etro

pia

(posi

tive

len

ses)

, as

tigm

atis

m

(anam

orp

hic

len

ses)

, ca

tara

ct

Geo

metr

ical

op

tics

: refr

act

ing

tele

sco

pe

Geo

metr

ical

op

tics

: terr

est

rial

tele

sco

pe

Geo

metr

ical

op

tics

: Gali

lean

tele

sco

pe

Geo

metr

ical

op

tics

: refl

ect

ing

tele

sco

pe

Geo

metr

ical

op

tics

: N

ew

ton

ian

refl

ect

or

Geo

metr

ical

op

tics

: C

ass

eg

rain

refl

ect

or

Geo

metr

ical

op

tics

: Sch

mid

t-C

ass

eg

rain

refl

ect

or

CO

DE V

, ZEM

AX

, Tra

cePro

, OSLO

, BEA

M3

, …