Astronom

ische Waarneem

technieken(Astronom

ical Observing T

echniques)

6thLecture 19

Octob

er 2011

1

Geom

etric

al O

ptic

s

bullD

efin

itions

bullA

berra

tions

Based

on ldquoAstronom

ical Opticsrdquo b

y Daniel J

Schroed

er ldquoPrinciples of Opticsrdquo b

y Max Born amp

Emil W

olf the ldquoO

ptical Engineerrsquos D

esk Referencerdquo b

y William

L Wolfe Lena b

ook and W

ikipedia

bullA

berra

tions

2

Diffra

ctio

n O

ptic

s

bullF

raunhofe

rD

iff

bullP

SF

bullS

R amp

EE

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Consid

er two points a and

b and

various paths b

etween th

em T

he

travel time b

etween th

em is τ

Cond

ition τwill h

ave a stationary value

for the actual path

Preface Fermatrsquos Principle

0=

part part=

part part

yx

ττ

a

b

Equivalently travel tim

e optical path

length(OPL)

where v is the speed

of light in the medium

of index n

Ferm

atrsquos principle states that th

e OPL is th

e shortest d

istance a b

int=

=sdot

=sdot

==b

ds

n

ds

nd

td

sn

dt

cd

tc

d

a

O

PL

dt

vv

)

OP

L(

Aperture and Field S

tops

Aperture stop

determ

ines the d

iameter of th

e light cone from

an axial

point on the ob

jectField

stop determ

ines the field

of view of th

e system

Entrance pupil

image of th

e aperture stop in the ob

ject spaceExit pupil

image of th

e aperture stop in the im

age spaceMarginal ray

ray from an ob

ject point on the optical ax

is that passes at

the ed

ge of the entrance pupil

Chief ray

ray from an ob

ject point at the ed

ge of the field

passing through

the center

of the aperture stop

cam

era

tele

scope

The S

peed of the Systemf

DΘ

The speed

of an optical system is d

escribed by th

e numerical

aperture NA and

the F

number w

here

NA

)(

2

1

an

d

sinN

A=

equivsdot

=D f

Fn

θ

Generally fast optics (large N

A) h

as a high

light pow

er is compact

has low

tolerances and is d

ifficult to manufacture S

low optics

(small N

A) is just th

e opposite

The M

aximum

Field of View of a Telescope

f D

f dD

d

ta

n

2

2ta

nm

ax

small

=

rArr=

le

ωω

ωD d

f

ωωω ω2

The Real

Field of View of a Telescope

However th

e practically useable F

OV of a telescope is

much

smaller and

limited

by ab

errations

bullfield

curvature

bulldifference [parab

ola ndashsph

ere]

bullcom

a astigmatism

Tele

scop

e (fo

cus)

fM

ax

FO

VR

ea

l FO

V

Osch

inS

chm

idt (p

rime

)2

52

2d

eg

24

0 a

rcmin

5m

Ha

le (p

rime

focu

s)5

51

0 d

eg

30

arcm

in

35

mN

TT

(Na

smy

th)

11

05

de

g3

0 a

rcmin

40

m E

-ELT

(Na

smu

th)

17

43

de

g5

arcm

in

Eacutetendue AxΩ

and Throughput

The geom

etrical eacutetendue

(frz `extentrsquo) is th

e product of area A

of the source tim

es the solid

angle Ω(of th

e systems entrance pupil

as seen from th

e source)

The eacutetend

ueis th

e maximum

beam

size the instrum

ent can accept

Hence th

e eacutetendue

is also called acceptance th

roughput or A

timesΩ

product

The eacutetend

uenever increases in any optical system

A

perfect optical system prod

uces an image w

ith th

e same

eacutetendue

as the source

Area A

=h2π

Solid

angle Ωgiven

by th

e marginal ray

Aberrations

Generally ab

errations are departures of th

e performance

of an optical system from

the pred

ictions of paraxial

optics

Until ph

otographic plates b

ecame availab

le only on-axis

telescope performance w

as relevant

There are tw

o categories of aberrations

1On-ax

is aberrations

(defocus sph

erical aberration)

2Off-ax

is aberrations

a)Aberrations th

at degrad

e the im

age coma

astigmatism

b)Aberrations th

at alter the im

age position distortion field

curvature

Consid

er a reference sphere S

of curvature R

for a off-axis

point Prsquo and an ab

erratedwavefront

W

An ldquoab

erratedrdquo ray from

the

object intersects th

e image

plane at Prdquo

Relation between Wave and Ray A

berrations

The ray ab

erration is PrsquoPrdquo

The w

ave aberration is nQ

Q

()

r

rW

n Rr

i

ipart

part=

For sm

all FOVs and

a radially

symmetric ab

erratedwavefront

W(r) w

e

can approximate th

e intersection with

the im

age plane

Defocus m

eans ldquoout of focusrdquo

defocused

image

Defocus

focused

image

2

2

NA 1

22

==

λλ

δF

The d

epth of focus usually refers to an optical path

difference of λ4

The am

ount of defocus can b

e characterized

by th

e depth

of focus

Rays furth

er from th

e optical axis h

ave a different focal point th

an rays closer to th

e optical axis

Spherical A

berration

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

Optical prob

lemHST prim

ary mirror suffers

from sph

erical aberration

Reason

the null corrector used

to measure th

e mirror sh

ape had been incorrectly assem

bled

(one lens w

as misplaced

by 13

mm)

Managem

ent problem

The m

irror manufacturer

had analyzed

its surface with

other null

Side note the H

ST mirror

had analyzed

its surface with

other null

correctors which

indicated

the prob

lem but th

e test results w

ere ignored because th

ey were

believed

to be less accurate

A null corrector

cancels the non-spherical portion of an aspheric m

irror figure When the correct

mirror is view

ed from

point A the com

bination

looks precisely spherical

Coma

Com

a appears as a variation in magnification across th

e entrance pupil Point sources w

ill show a com

etarytail C

oma is an inh

erent property of telescopes using parab

olic mirrors

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Consid

er an off-axis point A

The lens d

oes not appear symmetrical

from A but sh

ortened in th

e plane of incidence th

e tangential plane The em

ergent wave w

ill have a sm

aller radius of curvature for th

e tangential plane th

an for the plane norm

al to it (sagittalplane) and

form

an image closer to th

e lens Astigm

atism

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Field CurvatureOnly ob

jects close to the optical ax

is will b

e in focus on a flat image

plane Close-to-ax

is and far off-ax

is objects w

ill have d

ifferent focal points

due to th

e OPL d

ifference

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Straigh

t lines on the sky b

ecome curved

linesin th

e focal plane The

transversal magnification d

epends on th

e distance from

the optical ax

is

Distortion (1)

Ch

ief ra

y

un

dis

torte

d

Ch

ief ra

y

pin

cu

sh

ion

-d

isto

rted

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Generally th

ere are two cases

1Outer parts h

ave smaller m

agnification

barrel d

istortion

2Outer parts h

ave larger magnification

pincush

ion distortion

Distortion (2)

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Consid

er two points a and

b and

various paths b

etween th

em T

he

travel time b

etween th

em is τ

Cond

ition τwill h

ave a stationary value

for the actual path

Preface Fermatrsquos Principle

0=

part part=

part part

yx

ττ

a

b

Equivalently travel tim

e optical path

length(OPL)

where v is the speed

of light in the medium

of index n

Ferm

atrsquos principle states that th

e OPL is th

e shortest d

istance a b

int=

=sdot

=sdot

==b

ds

n

ds

nd

td

sn

dt

cd

tc

d

a

O

PL

dt

vv

)

OP

L(

Aperture and Field S

tops

Aperture stop

determ

ines the d

iameter of th

e light cone from

an axial

point on the ob

jectField

stop determ

ines the field

of view of th

e system

Entrance pupil

image of th

e aperture stop in the ob

ject spaceExit pupil

image of th

e aperture stop in the im

age spaceMarginal ray

ray from an ob

ject point on the optical ax

is that passes at

the ed

ge of the entrance pupil

Chief ray

ray from an ob

ject point at the ed

ge of the field

passing through

the center

of the aperture stop

cam

era

tele

scope

The S

peed of the Systemf

DΘ

The speed

of an optical system is d

escribed by th

e numerical

aperture NA and

the F

number w

here

NA

)(

2

1

an

d

sinN

A=

equivsdot

=D f

Fn

θ

Generally fast optics (large N

A) h

as a high

light pow

er is compact

has low

tolerances and is d

ifficult to manufacture S

low optics

(small N

A) is just th

e opposite

The M

aximum

Field of View of a Telescope

f D

f dD

d

ta

n

2

2ta

nm

ax

small

=

rArr=

le

ωω

ωD d

f

ωωω ω2

The Real

Field of View of a Telescope

However th

e practically useable F

OV of a telescope is

much

smaller and

limited

by ab

errations

bullfield

curvature

bulldifference [parab

ola ndashsph

ere]

bullcom

a astigmatism

Tele

scop

e (fo

cus)

fM

ax

FO

VR

ea

l FO

V

Osch

inS

chm

idt (p

rime

)2

52

2d

eg

24

0 a

rcmin

5m

Ha

le (p

rime

focu

s)5

51

0 d

eg

30

arcm

in

35

mN

TT

(Na

smy

th)

11

05

de

g3

0 a

rcmin

40

m E

-ELT

(Na

smu

th)

17

43

de

g5

arcm

in

Eacutetendue AxΩ

and Throughput

The geom

etrical eacutetendue

(frz `extentrsquo) is th

e product of area A

of the source tim

es the solid

angle Ω(of th

e systems entrance pupil

as seen from th

e source)

The eacutetend

ueis th

e maximum

beam

size the instrum

ent can accept

Hence th

e eacutetendue

is also called acceptance th

roughput or A

timesΩ

product

The eacutetend

uenever increases in any optical system

A

perfect optical system prod

uces an image w

ith th

e same

eacutetendue

as the source

Area A

=h2π

Solid

angle Ωgiven

by th

e marginal ray

Aberrations

Generally ab

errations are departures of th

e performance

of an optical system from

the pred

ictions of paraxial

optics

Until ph

otographic plates b

ecame availab

le only on-axis

telescope performance w

as relevant

There are tw

o categories of aberrations

1On-ax

is aberrations

(defocus sph

erical aberration)

2Off-ax

is aberrations

a)Aberrations th

at degrad

e the im

age coma

astigmatism

b)Aberrations th

at alter the im

age position distortion field

curvature

Consid

er a reference sphere S

of curvature R

for a off-axis

point Prsquo and an ab

erratedwavefront

W

An ldquoab

erratedrdquo ray from

the

object intersects th

e image

plane at Prdquo

Relation between Wave and Ray A

berrations

The ray ab

erration is PrsquoPrdquo

The w

ave aberration is nQ

Q

()

r

rW

n Rr

i

ipart

part=

For sm

all FOVs and

a radially

symmetric ab

erratedwavefront

W(r) w

e

can approximate th

e intersection with

the im

age plane

Defocus m

eans ldquoout of focusrdquo

defocused

image

Defocus

focused

image

2

2

NA 1

22

==

λλ

δF

The d

epth of focus usually refers to an optical path

difference of λ4

The am

ount of defocus can b

e characterized

by th

e depth

of focus

Rays furth

er from th

e optical axis h

ave a different focal point th

an rays closer to th

e optical axis

Spherical A

berration

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

Optical prob

lemHST prim

ary mirror suffers

from sph

erical aberration

Reason

the null corrector used

to measure th

e mirror sh

ape had been incorrectly assem

bled

(one lens w

as misplaced

by 13

mm)

Managem

ent problem

The m

irror manufacturer

had analyzed

its surface with

other null

Side note the H

ST mirror

had analyzed

its surface with

other null

correctors which

indicated

the prob

lem but th

e test results w

ere ignored because th

ey were

believed

to be less accurate

A null corrector

cancels the non-spherical portion of an aspheric m

irror figure When the correct

mirror is view

ed from

point A the com

bination

looks precisely spherical

Coma

Com

a appears as a variation in magnification across th

e entrance pupil Point sources w

ill show a com

etarytail C

oma is an inh

erent property of telescopes using parab

olic mirrors

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Consid

er an off-axis point A

The lens d

oes not appear symmetrical

from A but sh

ortened in th

e plane of incidence th

e tangential plane The em

ergent wave w

ill have a sm

aller radius of curvature for th

e tangential plane th

an for the plane norm

al to it (sagittalplane) and

form

an image closer to th

e lens Astigm

atism

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Field CurvatureOnly ob

jects close to the optical ax

is will b

e in focus on a flat image

plane Close-to-ax

is and far off-ax

is objects w

ill have d

ifferent focal points

due to th

e OPL d

ifference

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Straigh

t lines on the sky b

ecome curved

linesin th

e focal plane The

transversal magnification d

epends on th

e distance from

the optical ax

is

Distortion (1)

Ch

ief ra

y

un

dis

torte

d

Ch

ief ra

y

pin

cu

sh

ion

-d

isto

rted

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Generally th

ere are two cases

1Outer parts h

ave smaller m

agnification

barrel d

istortion

2Outer parts h

ave larger magnification

pincush

ion distortion

Distortion (2)

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Aperture and Field S

tops

Aperture stop

determ

ines the d

iameter of th

e light cone from

an axial

point on the ob

jectField

stop determ

ines the field

of view of th

e system

Entrance pupil

image of th

e aperture stop in the ob

ject spaceExit pupil

image of th

e aperture stop in the im

age spaceMarginal ray

ray from an ob

ject point on the optical ax

is that passes at

the ed

ge of the entrance pupil

Chief ray

ray from an ob

ject point at the ed

ge of the field

passing through

the center

of the aperture stop

cam

era

tele

scope

The S

peed of the Systemf

DΘ

The speed

of an optical system is d

escribed by th

e numerical

aperture NA and

the F

number w

here

NA

)(

2

1

an

d

sinN

A=

equivsdot

=D f

Fn

θ

Generally fast optics (large N

A) h

as a high

light pow

er is compact

has low

tolerances and is d

ifficult to manufacture S

low optics

(small N

A) is just th

e opposite

The M

aximum

Field of View of a Telescope

f D

f dD

d

ta

n

2

2ta

nm

ax

small

=

rArr=

le

ωω

ωD d

f

ωωω ω2

The Real

Field of View of a Telescope

However th

e practically useable F

OV of a telescope is

much

smaller and

limited

by ab

errations

bullfield

curvature

bulldifference [parab

ola ndashsph

ere]

bullcom

a astigmatism

Tele

scop

e (fo

cus)

fM

ax

FO

VR

ea

l FO

V

Osch

inS

chm

idt (p

rime

)2

52

2d

eg

24

0 a

rcmin

5m

Ha

le (p

rime

focu

s)5

51

0 d

eg

30

arcm

in

35

mN

TT

(Na

smy

th)

11

05

de

g3

0 a

rcmin

40

m E

-ELT

(Na

smu

th)

17

43

de

g5

arcm

in

Eacutetendue AxΩ

and Throughput

The geom

etrical eacutetendue

(frz `extentrsquo) is th

e product of area A

of the source tim

es the solid

angle Ω(of th

e systems entrance pupil

as seen from th

e source)

The eacutetend

ueis th

e maximum

beam

size the instrum

ent can accept

Hence th

e eacutetendue

is also called acceptance th

roughput or A

timesΩ

product

The eacutetend

uenever increases in any optical system

A

perfect optical system prod

uces an image w

ith th

e same

eacutetendue

as the source

Area A

=h2π

Solid

angle Ωgiven

by th

e marginal ray

Aberrations

Generally ab

errations are departures of th

e performance

of an optical system from

the pred

ictions of paraxial

optics

Until ph

otographic plates b

ecame availab

le only on-axis

telescope performance w

as relevant

There are tw

o categories of aberrations

1On-ax

is aberrations

(defocus sph

erical aberration)

2Off-ax

is aberrations

a)Aberrations th

at degrad

e the im

age coma

astigmatism

b)Aberrations th

at alter the im

age position distortion field

curvature

Consid

er a reference sphere S

of curvature R

for a off-axis

point Prsquo and an ab

erratedwavefront

W

An ldquoab

erratedrdquo ray from

the

object intersects th

e image

plane at Prdquo

Relation between Wave and Ray A

berrations

The ray ab

erration is PrsquoPrdquo

The w

ave aberration is nQ

Q

()

r

rW

n Rr

i

ipart

part=

For sm

all FOVs and

a radially

symmetric ab

erratedwavefront

W(r) w

e

can approximate th

e intersection with

the im

age plane

Defocus m

eans ldquoout of focusrdquo

defocused

image

Defocus

focused

image

2

2

NA 1

22

==

λλ

δF

The d

epth of focus usually refers to an optical path

difference of λ4

The am

ount of defocus can b

e characterized

by th

e depth

of focus

Rays furth

er from th

e optical axis h

ave a different focal point th

an rays closer to th

e optical axis

Spherical A

berration

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

Optical prob

lemHST prim

ary mirror suffers

from sph

erical aberration

Reason

the null corrector used

to measure th

e mirror sh

ape had been incorrectly assem

bled

(one lens w

as misplaced

by 13

mm)

Managem

ent problem

The m

irror manufacturer

had analyzed

its surface with

other null

Side note the H

ST mirror

had analyzed

its surface with

other null

correctors which

indicated

the prob

lem but th

e test results w

ere ignored because th

ey were

believed

to be less accurate

A null corrector

cancels the non-spherical portion of an aspheric m

irror figure When the correct

mirror is view

ed from

point A the com

bination

looks precisely spherical

Coma

Com

a appears as a variation in magnification across th

e entrance pupil Point sources w

ill show a com

etarytail C

oma is an inh

erent property of telescopes using parab

olic mirrors

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Consid

er an off-axis point A

The lens d

oes not appear symmetrical

from A but sh

ortened in th

e plane of incidence th

e tangential plane The em

ergent wave w

ill have a sm

aller radius of curvature for th

e tangential plane th

an for the plane norm

al to it (sagittalplane) and

form

an image closer to th

e lens Astigm

atism

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Field CurvatureOnly ob

jects close to the optical ax

is will b

e in focus on a flat image

plane Close-to-ax

is and far off-ax

is objects w

ill have d

ifferent focal points

due to th

e OPL d

ifference

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Straigh

t lines on the sky b

ecome curved

linesin th

e focal plane The

transversal magnification d

epends on th

e distance from

the optical ax

is

Distortion (1)

Ch

ief ra

y

un

dis

torte

d

Ch

ief ra

y

pin

cu

sh

ion

-d

isto

rted

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Generally th

ere are two cases

1Outer parts h

ave smaller m

agnification

barrel d

istortion

2Outer parts h

ave larger magnification

pincush

ion distortion

Distortion (2)

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

The M

aximum

Field of View of a Telescope

f D

f dD

d

ta

n

2

2ta

nm

ax

small

=

rArr=

le

ωω

ωD d

f

ωωω ω2

The Real

Field of View of a Telescope

However th

e practically useable F

OV of a telescope is

much

smaller and

limited

by ab

errations

bullfield

curvature

bulldifference [parab

ola ndashsph

ere]

bullcom

a astigmatism

Tele

scop

e (fo

cus)

fM

ax

FO

VR

ea

l FO

V

Osch

inS

chm

idt (p

rime

)2

52

2d

eg

24

0 a

rcmin

5m

Ha

le (p

rime

focu

s)5

51

0 d

eg

30

arcm

in

35

mN

TT

(Na

smy

th)

11

05

de

g3

0 a

rcmin

40

m E

-ELT

(Na

smu

th)

17

43

de

g5

arcm

in

Eacutetendue AxΩ

and Throughput

The geom

etrical eacutetendue

(frz `extentrsquo) is th

e product of area A

of the source tim

es the solid

angle Ω(of th

e systems entrance pupil

as seen from th

e source)

The eacutetend

ueis th

e maximum

beam

size the instrum

ent can accept

Hence th

e eacutetendue

is also called acceptance th

roughput or A

timesΩ

product

The eacutetend

uenever increases in any optical system

A

perfect optical system prod

uces an image w

ith th

e same

eacutetendue

as the source

Area A

=h2π

Solid

angle Ωgiven

by th

e marginal ray

Aberrations

Generally ab

errations are departures of th

e performance

of an optical system from

the pred

ictions of paraxial

optics

Until ph

otographic plates b

ecame availab

le only on-axis

telescope performance w

as relevant

There are tw

o categories of aberrations

1On-ax

is aberrations

(defocus sph

erical aberration)

2Off-ax

is aberrations

a)Aberrations th

at degrad

e the im

age coma

astigmatism

b)Aberrations th

at alter the im

age position distortion field

curvature

Consid

er a reference sphere S

of curvature R

for a off-axis

point Prsquo and an ab

erratedwavefront

W

An ldquoab

erratedrdquo ray from

the

object intersects th

e image

plane at Prdquo

Relation between Wave and Ray A

berrations

The ray ab

erration is PrsquoPrdquo

The w

ave aberration is nQ

Q

()

r

rW

n Rr

i

ipart

part=

For sm

all FOVs and

a radially

symmetric ab

erratedwavefront

W(r) w

e

can approximate th

e intersection with

the im

age plane

Defocus m

eans ldquoout of focusrdquo

defocused

image

Defocus

focused

image

2

2

NA 1

22

==

λλ

δF

The d

epth of focus usually refers to an optical path

difference of λ4

The am

ount of defocus can b

e characterized

by th

e depth

of focus

Rays furth

er from th

e optical axis h

ave a different focal point th

an rays closer to th

e optical axis

Spherical A

berration

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

Optical prob

lemHST prim

ary mirror suffers

from sph

erical aberration

Reason

the null corrector used

to measure th

e mirror sh

ape had been incorrectly assem

bled

(one lens w

as misplaced

by 13

mm)

Managem

ent problem

The m

irror manufacturer

had analyzed

its surface with

other null

Side note the H

ST mirror

had analyzed

its surface with

other null

correctors which

indicated

the prob

lem but th

e test results w

ere ignored because th

ey were

believed

to be less accurate

A null corrector

cancels the non-spherical portion of an aspheric m

irror figure When the correct

mirror is view

ed from

point A the com

bination

looks precisely spherical

Coma

Com

a appears as a variation in magnification across th

e entrance pupil Point sources w

ill show a com

etarytail C

oma is an inh

erent property of telescopes using parab

olic mirrors

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Consid

er an off-axis point A

The lens d

oes not appear symmetrical

from A but sh

ortened in th

e plane of incidence th

e tangential plane The em

ergent wave w

ill have a sm

aller radius of curvature for th

e tangential plane th

an for the plane norm

al to it (sagittalplane) and

form

an image closer to th

e lens Astigm

atism

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Field CurvatureOnly ob

jects close to the optical ax

is will b

e in focus on a flat image

plane Close-to-ax

is and far off-ax

is objects w

ill have d

ifferent focal points

due to th

e OPL d

ifference

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Straigh

t lines on the sky b

ecome curved

linesin th

e focal plane The

transversal magnification d

epends on th

e distance from

the optical ax

is

Distortion (1)

Ch

ief ra

y

un

dis

torte

d

Ch

ief ra

y

pin

cu

sh

ion

-d

isto

rted

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Generally th

ere are two cases

1Outer parts h

ave smaller m

agnification

barrel d

istortion

2Outer parts h

ave larger magnification

pincush

ion distortion

Distortion (2)

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Eacutetendue AxΩ

and Throughput

The geom

etrical eacutetendue

(frz `extentrsquo) is th

e product of area A

of the source tim

es the solid

angle Ω(of th

e systems entrance pupil

as seen from th

e source)

The eacutetend

ueis th

e maximum

beam

size the instrum

ent can accept

Hence th

e eacutetendue

is also called acceptance th

roughput or A

timesΩ

product

The eacutetend

uenever increases in any optical system

A

perfect optical system prod

uces an image w

ith th

e same

eacutetendue

as the source

Area A

=h2π

Solid

angle Ωgiven

by th

e marginal ray

Aberrations

Generally ab

errations are departures of th

e performance

of an optical system from

the pred

ictions of paraxial

optics

Until ph

otographic plates b

ecame availab

le only on-axis

telescope performance w

as relevant

There are tw

o categories of aberrations

1On-ax

is aberrations

(defocus sph

erical aberration)

2Off-ax

is aberrations

a)Aberrations th

at degrad

e the im

age coma

astigmatism

b)Aberrations th

at alter the im

age position distortion field

curvature

Consid

er a reference sphere S

of curvature R

for a off-axis

point Prsquo and an ab

erratedwavefront

W

An ldquoab

erratedrdquo ray from

the

object intersects th

e image

plane at Prdquo

Relation between Wave and Ray A

berrations

The ray ab

erration is PrsquoPrdquo

The w

ave aberration is nQ

Q

()

r

rW

n Rr

i

ipart

part=

For sm

all FOVs and

a radially

symmetric ab

erratedwavefront

W(r) w

e

can approximate th

e intersection with

the im

age plane

Defocus m

eans ldquoout of focusrdquo

defocused

image

Defocus

focused

image

2

2

NA 1

22

==

λλ

δF

The d

epth of focus usually refers to an optical path

difference of λ4

The am

ount of defocus can b

e characterized

by th

e depth

of focus

Rays furth

er from th

e optical axis h

ave a different focal point th

an rays closer to th

e optical axis

Spherical A

berration

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

Optical prob

lemHST prim

ary mirror suffers

from sph

erical aberration

Reason

the null corrector used

to measure th

e mirror sh

ape had been incorrectly assem

bled

(one lens w

as misplaced

by 13

mm)

Managem

ent problem

The m

irror manufacturer

had analyzed

its surface with

other null

Side note the H

ST mirror

had analyzed

its surface with

other null

correctors which

indicated

the prob

lem but th

e test results w

ere ignored because th

ey were

believed

to be less accurate

A null corrector

cancels the non-spherical portion of an aspheric m

irror figure When the correct

mirror is view

ed from

point A the com

bination

looks precisely spherical

Coma

Com

a appears as a variation in magnification across th

e entrance pupil Point sources w

ill show a com

etarytail C

oma is an inh

erent property of telescopes using parab

olic mirrors

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Consid

er an off-axis point A

The lens d

oes not appear symmetrical

from A but sh

ortened in th

e plane of incidence th

e tangential plane The em

ergent wave w

ill have a sm

aller radius of curvature for th

e tangential plane th

an for the plane norm

al to it (sagittalplane) and

form

an image closer to th

e lens Astigm

atism

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Field CurvatureOnly ob

jects close to the optical ax

is will b

e in focus on a flat image

plane Close-to-ax

is and far off-ax

is objects w

ill have d

ifferent focal points

due to th

e OPL d

ifference

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Straigh

t lines on the sky b

ecome curved

linesin th

e focal plane The

transversal magnification d

epends on th

e distance from

the optical ax

is

Distortion (1)

Ch

ief ra

y

un

dis

torte

d

Ch

ief ra

y

pin

cu

sh

ion

-d

isto

rted

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Generally th

ere are two cases

1Outer parts h

ave smaller m

agnification

barrel d

istortion

2Outer parts h

ave larger magnification

pincush

ion distortion

Distortion (2)

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Aberrations

Generally ab

errations are departures of th

e performance

of an optical system from

the pred

ictions of paraxial

optics

Until ph

otographic plates b

ecame availab

le only on-axis

telescope performance w

as relevant

There are tw

o categories of aberrations

1On-ax

is aberrations

(defocus sph

erical aberration)

2Off-ax

is aberrations

a)Aberrations th

at degrad

e the im

age coma

astigmatism

b)Aberrations th

at alter the im

age position distortion field

curvature

Consid

er a reference sphere S

of curvature R

for a off-axis

point Prsquo and an ab

erratedwavefront

W

An ldquoab

erratedrdquo ray from

the

object intersects th

e image

plane at Prdquo

Relation between Wave and Ray A

berrations

The ray ab

erration is PrsquoPrdquo

The w

ave aberration is nQ

Q

()

r

rW

n Rr

i

ipart

part=

For sm

all FOVs and

a radially

symmetric ab

erratedwavefront

W(r) w

e

can approximate th

e intersection with

the im

age plane

Defocus m

eans ldquoout of focusrdquo

defocused

image

Defocus

focused

image

2

2

NA 1

22

==

λλ

δF

The d

epth of focus usually refers to an optical path

difference of λ4

The am

ount of defocus can b

e characterized

by th

e depth

of focus

Rays furth

er from th

e optical axis h

ave a different focal point th

an rays closer to th

e optical axis

Spherical A

berration

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

Optical prob

lemHST prim

ary mirror suffers

from sph

erical aberration

Reason

the null corrector used

to measure th

e mirror sh

ape had been incorrectly assem

bled

(one lens w

as misplaced

by 13

mm)

Managem

ent problem

The m

irror manufacturer

had analyzed

its surface with

other null

Side note the H

ST mirror

had analyzed

its surface with

other null

correctors which

indicated

the prob

lem but th

e test results w

ere ignored because th

ey were

believed

to be less accurate

A null corrector

cancels the non-spherical portion of an aspheric m

irror figure When the correct

mirror is view

ed from

point A the com

bination

looks precisely spherical

Coma

Com

a appears as a variation in magnification across th

e entrance pupil Point sources w

ill show a com

etarytail C

oma is an inh

erent property of telescopes using parab

olic mirrors

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Consid

er an off-axis point A

The lens d

oes not appear symmetrical

from A but sh

ortened in th

e plane of incidence th

e tangential plane The em

ergent wave w

ill have a sm

aller radius of curvature for th

e tangential plane th

an for the plane norm

al to it (sagittalplane) and

form

an image closer to th

e lens Astigm

atism

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Field CurvatureOnly ob

jects close to the optical ax

is will b

e in focus on a flat image

plane Close-to-ax

is and far off-ax

is objects w

ill have d

ifferent focal points

due to th

e OPL d

ifference

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Straigh

t lines on the sky b

ecome curved

linesin th

e focal plane The

transversal magnification d

epends on th

e distance from

the optical ax

is

Distortion (1)

Ch

ief ra

y

un

dis

torte

d

Ch

ief ra

y

pin

cu

sh

ion

-d

isto

rted

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Generally th

ere are two cases

1Outer parts h

ave smaller m

agnification

barrel d

istortion

2Outer parts h

ave larger magnification

pincush

ion distortion

Distortion (2)

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Defocus m

eans ldquoout of focusrdquo

defocused

image

Defocus

focused

image

2

2

NA 1

22

==

λλ

δF

The d

epth of focus usually refers to an optical path

difference of λ4

The am

ount of defocus can b

e characterized

by th

e depth

of focus

Rays furth

er from th

e optical axis h

ave a different focal point th

an rays closer to th

e optical axis

Spherical A

berration

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

Optical prob

lemHST prim

ary mirror suffers

from sph

erical aberration

Reason

the null corrector used

to measure th

e mirror sh

ape had been incorrectly assem

bled

(one lens w

as misplaced

by 13

mm)

Managem

ent problem

The m

irror manufacturer

had analyzed

its surface with

other null

Side note the H

ST mirror

had analyzed

its surface with

other null

correctors which

indicated

the prob

lem but th

e test results w

ere ignored because th

ey were

believed

to be less accurate

A null corrector

cancels the non-spherical portion of an aspheric m

irror figure When the correct

mirror is view

ed from

point A the com

bination

looks precisely spherical

Coma

Com

a appears as a variation in magnification across th

e entrance pupil Point sources w

ill show a com

etarytail C

oma is an inh

erent property of telescopes using parab

olic mirrors

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Consid

er an off-axis point A

The lens d

oes not appear symmetrical

from A but sh

ortened in th

e plane of incidence th

e tangential plane The em

ergent wave w

ill have a sm

aller radius of curvature for th

e tangential plane th

an for the plane norm

al to it (sagittalplane) and

form

an image closer to th

e lens Astigm

atism

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Field CurvatureOnly ob

jects close to the optical ax

is will b

e in focus on a flat image

plane Close-to-ax

is and far off-ax

is objects w

ill have d

ifferent focal points

due to th

e OPL d

ifference

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Straigh

t lines on the sky b

ecome curved

linesin th

e focal plane The

transversal magnification d

epends on th

e distance from

the optical ax

is

Distortion (1)

Ch

ief ra

y

un

dis

torte

d

Ch

ief ra

y

pin

cu

sh

ion

-d

isto

rted

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Generally th

ere are two cases

1Outer parts h

ave smaller m

agnification

barrel d

istortion

2Outer parts h

ave larger magnification

pincush

ion distortion

Distortion (2)

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Optical prob

lemHST prim

ary mirror suffers

from sph

erical aberration

Reason

the null corrector used

to measure th

e mirror sh

ape had been incorrectly assem

bled

(one lens w

as misplaced

by 13

mm)

Managem

ent problem

The m

irror manufacturer

had analyzed

its surface with

other null

Side note the H

ST mirror

had analyzed

its surface with

other null

correctors which

indicated

the prob

lem but th

e test results w

ere ignored because th

ey were

believed

to be less accurate

A null corrector

cancels the non-spherical portion of an aspheric m

irror figure When the correct

mirror is view

ed from

point A the com

bination

looks precisely spherical

Coma

Com

a appears as a variation in magnification across th

e entrance pupil Point sources w

ill show a com

etarytail C

oma is an inh

erent property of telescopes using parab

olic mirrors

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Consid

er an off-axis point A

The lens d

oes not appear symmetrical

from A but sh

ortened in th

e plane of incidence th

e tangential plane The em

ergent wave w

ill have a sm

aller radius of curvature for th

e tangential plane th

an for the plane norm

al to it (sagittalplane) and

form

an image closer to th

e lens Astigm

atism

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Field CurvatureOnly ob

jects close to the optical ax

is will b

e in focus on a flat image

plane Close-to-ax

is and far off-ax

is objects w

ill have d

ifferent focal points

due to th

e OPL d

ifference

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Straigh

t lines on the sky b

ecome curved

linesin th

e focal plane The

transversal magnification d

epends on th

e distance from

the optical ax

is

Distortion (1)

Ch

ief ra

y

un

dis

torte

d

Ch

ief ra

y

pin

cu

sh

ion

-d

isto

rted

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Generally th

ere are two cases

1Outer parts h

ave smaller m

agnification

barrel d

istortion

2Outer parts h

ave larger magnification

pincush

ion distortion

Distortion (2)

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Consid

er an off-axis point A

The lens d

oes not appear symmetrical

from A but sh

ortened in th

e plane of incidence th

e tangential plane The em

ergent wave w

ill have a sm

aller radius of curvature for th

e tangential plane th

an for the plane norm

al to it (sagittalplane) and

form

an image closer to th

e lens Astigm

atism

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Field CurvatureOnly ob

jects close to the optical ax

is will b

e in focus on a flat image

plane Close-to-ax

is and far off-ax

is objects w

ill have d

ifferent focal points

due to th

e OPL d

ifference

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Straigh

t lines on the sky b

ecome curved

linesin th

e focal plane The

transversal magnification d

epends on th

e distance from

the optical ax

is

Distortion (1)

Ch

ief ra

y

un

dis

torte

d

Ch

ief ra

y

pin

cu

sh

ion

-d

isto

rted

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Generally th

ere are two cases

1Outer parts h

ave smaller m

agnification

barrel d

istortion

2Outer parts h

ave larger magnification

pincush

ion distortion

Distortion (2)

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Straigh

t lines on the sky b

ecome curved

linesin th

e focal plane The

transversal magnification d

epends on th

e distance from

the optical ax

is

Distortion (1)

Ch

ief ra

y

un

dis

torte

d

Ch

ief ra

y

pin

cu

sh

ion

-d

isto

rted

φis

the w

ave a

berra

tion

Θis

the a

ngle

in th

e p

upil p

lane

ρis

the ra

diu

s in

the p

upil p

lane

ξ=

ρsinΘ

η=

ρcosΘ

y0

= p

ositio

n o

f the o

bje

ct in

the fie

ld

Generally th

ere are two cases

1Outer parts h

ave smaller m

agnification

barrel d

istortion

2Outer parts h

ave larger magnification

pincush

ion distortion

Distortion (2)

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Summary Prim

ary Wave A

berrations

Spheric

al

aberra

tion

Com

a

Dependence

on p

upil size

Dependence

on im

age size

~ρ

4

~ρ

3

co

nst

~y

Astig

matis

m

Fie

ld

curv

atu

re

Dis

tortio

n

~ρ

2

~ρ

2

~ρ

~ρ

2D

efo

cus

co

nst

~y

2

~y

2

~y

3

Chromatic A

berrationSince th

e refractive index n = f(λ) th

e focal length of a lens = f(λ) and

different w

avelengths h

ave different foci (M

irrors are usually ach

romatic)

Mitigation use tw

o lenses of different

material w

ith different d

ispersion ach

romatic d

oublet

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Part IGeom

etrical Optics

Part IIDiffraction O

ptics

Ferm

atrsquos view ldquoA

wavefront

is a surface on which every point has

the same O

PDrdquo

Huygensrsquo view

ldquoAt a given tim

e each point on primary w

avefrontacts as a source of second

ary spherical wavelets T

hese propagate with the sam

e speed and

frequency as the primary w

averdquo

Huygens-Fresnel Principle

The H

uygens-Fresnel principle w

as theoretically d

emonstrated

by

Kirch

hoff (

Fresnel-K

irchhoff d

iffraction integral)

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Fresnel d

iffraction = near-field diffraction

When a w

ave passes through

an aperture and diffracts in th

e near field

it causes the ob

served diffraction pattern to d

iffer in size and

shape for d

ifferent distances

For F

raunhofer d

iffractionat infinity (far-field

) the w

ave becom

es planar Fresnel and Fraunhofer

Diffraction

1

1

2 2

ltlt

sdot=

gesdot

=

λ λ

d rF

d rF

Fresnel

Fraunh

ofer

(where F

= Fresnel num

ber r =

aperture size and d = d

istance to screen)

An

exa

mp

le o

f an

op

tica

l se

tup

tha

t dis

pla

ys F

resn

el d

iffractio

n o

ccu

rring

in th

e

ne

ar-fie

ld O

n th

is d

iag

ram

a w

ave

is d

iffracte

d a

nd

ob

se

rve

d a

t po

int σ

As

this

po

int is

mo

ve

d fu

rthe

r ba

ck b

eyo

nd

the

Fre

sn

el th

resh

old

or in

the

far-

field

Fra

un

ho

fer d

iffractio

n o

ccu

rs

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

FraunhoferDiffraction at a Pupil

Consid

er a circular pupil function G(r) of unity w

ithin A

and zero outsid

e

()

()

2

2

11

01

λλ

λθ

λθ

θπ

dr

er

GA E

tV

ri

screenA

sdotminus

minus

intint

=

Theorem

When a screen is illum

inated by a source at

infinity the amplitud

e of the field diffracted

in any direction is the F

ourier transform of the pupil function

characterizing the screen A

Math

ematically th

e amplitud

e of the d

iffracted field

can be ex

pressed as

(see Lena b

ook pp 120ff for d

etails)

V(Θ

0 ) V(Θ

1 ) complex

field am

plitudes of points in ob

ject and im

age planeK(Θ

0 Θ1 )

ldquotransmissionrdquo of th

e system

Imaging and Filtering

()

()

()

()

()

2

2

00

10

01

wh

ere

λ

θθ

θθ

θθ

λπ

θd

re

rG

Kd

KV

V

ri

object

intintintint

minus

=minus

=

Then th

e image of an ex

tended

object can b

e describ

ed by

In Fourier space

()

()

()

()

()r

GV

FT

KF

TV

FT

VF

Tsdot

=sdot

=0

00

00

1θ

θθ

θ

The F

ourier transform of the im

age equals the product of F

ourier transform

of the object and

the pupil function G w

hich acts as alinear spatial filter

co

nvo

lutio

n

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

All ph

ysical pupils have finite sizes

cut-off frequencies must ex

ist The pupil function G

(r) acts as a low-pass filter on th

e spatial frequencies equivalent to a convolution

Accord

ing to the N

yquist-Shannon sam

pling theorem

I(Θ) sh

all be

sampled

with

a rate of at least

Point Spread Function (1)

()

21

22

cc

cv

u+

=ω

cω

θ2

1=

∆

()

()

()

()

()

()

θθ

θθ

θθ

PS

FI

IK

FT

VF

TV

FT

lowast=

hArrsdot

=0

10

00

1

Now consid

er circular telescope aperture with

Graph

ically

pupil function G(r)

autocorrelation G

(r)G(r)

and its M

TF

()

Π=

02

r rr

G

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

When th

e circular pupil is illuminated

by a point source [I

0 (Θ) = δ(Θ

)] then th

e resulting PSF is d

escribed by a 1

stord

er Bessel function

This is also called

the A

iry functionThe rad

ius of the first d

ark ring (minim

um) is at

Point Spread Function (2)

()

()

2

0

01

1

2

22

=λ

θπ

λθ

πθ

r

rJ

I

Rem

ember th

e Rayleigh

criterion states that

two sources can b

e resolved if th

e peak of the second

source is no closer than th

e 1st

dark A

iry ring of the first source

Df r

Fr

λα

λ2

2

1o

r

22

1

1

11

==

=

The PS

F is often ch

aracterized by th

e half pow

er beam

width

(HPBW)

in angular units to indicate th

e angular resolution of the im

age

Most ldquorealrdquo telescopes h

ave a central obscuration w

hich

modifies our

simplistic pupil function

The resulting PS

F can b

e describ

ed by a m

odified

function

where ε

is the fraction of central ob

scuration

Point Spread Function (3)

()

()

()

()

2

0

01

2

0

01

22

1

2

22

2

22

1

1

minus

minus=

λεθ

π

λεθ

πε

λθ

π

λθ

π

εθ

r

rJ

r

rJ

I

()

()0

2

rr

rG

Π=

where ε

is the fraction of central ob

scuration to total pupil area

Astronom

ical instruments som

etimes use a

phase mask to red

uce the secondary lob

es of the PS

F (from

diffraction at ldquohard

edgesrdquo)

Phase masks introd

uce a position depend

ent phase change T

his is called apod

isation

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

A convenient m

easure to assess the quality of an optical system

is the

Streh

l ratio

The S

trehl ratio (S

R) is th

e ratio of the ob

served peak intensity of

the PS

F com

pared to th

e theoretical m

aximum

peak intensity of a point source seen w

ith a perfect im

aging system working at th

e diffraction lim

it

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one

Strehl

Ratio

Using th

e wave num

ber k=2

πλ

and th

e RMS wavefront error ω

one can calculate th

at

Exam

ples

bullA SR gt 8

0 is consid

ered diffraction-lim

ited

averageWFE ~ λ14

bullA typical ad

aptive optics system delivers S

R ~ 10

-50 (d

epends on λ)

bullA seeing-lim

itedPSF on an 8

m telescope h

as a SR ~ 01-0

01

22

12

2

ωω

ke

SR

kminus

asymp=

minus

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber

Q W

hat is th

e maximum

concentration of light w

ithin a sm

all area The fraction of th

e total PSF intensity w

ithin a certain rad

ius is given b

y the encircled

energy(EE)

Encircled Energy

()

minus

minus

=F r

JF r

Jr

EE

λ π

λ π21

201

Note th

at the E

E depend

s strongly on the central ob

scuration εεε εof th

e telescope

Fis th

e f num

ber