Introduction to aberrations...IHH u W HH2' Ж Alternatively Prof. Jose Sasian OPTI 518 10 Irradiance transport equation Prof. Jose Sasian OPTI 518 11 Inverse square law 22 2 2 21cos

Prof. Jose SasianOPTI 518

1

Introduction to aberrations

OPTI 518

Lecture 16

Irradiance function


2

Irradiance function


3

Conservation of flux at pupils


4

Jacobian determinant


5

Irradiance equations


6

Irradiance change

'020 020 3114I I WЖ

'111 111 220 2224 6I H I H W W HЖ Ж

'200 200 1314I H H I H H W H HЖ


7

Pinhole camara 2 40 2

' '' ' cos ''

ds dad Ls

4 '

2 20 00 02 22 2

0 0

' '' 2 2 ' sin cos ' sin '' ''

a UL LrI r dr L d n Us nr s

4cos 'RI

4 2 2, cos ' 1 2 ' 4 ' ' 2 ' ...RI H u u u H u H H

F/100, F/2, F/1


8

Irradiance at exit pupil

1'000 I

' 2020 31142I u WЖ

'111 220 2224 64I H uu W W HЖ Ж

' 2200 13142I H H u W H HЖ

4 2 2, cos 1 2 4 2 ...I H RI H u uu H u H H


9

Irradiance at exit pupil

1'000 I

' 2020 13142 'I u WЖ

'111 220 2224 64 ' 'I H u u W W HЖ Ж

' 2200 31142 'I H H u W H HЖ

Alternatively


10

Irradiance transport equation


11

Inverse square law

2 2

2

2

2 1 cos

a a a b b b

ab a

b

I r I r

rI Ir


12

Element of Throughput

2 2

2 22

coscos cosO P P O O P

n dS dSdT n dS d n dS d

l


13

Radiance times throughput gives the optical power (flux)

2

,L HT

n


14

Invariance upon refraction

sin 'sin 'n I n I

'cos 'cos 'M Mn I dI n I dI

''T TndI n dI


15


O M Td dI dI ' ' 'O M Td dI dI

2 2 'cos ' cos 'O On I d n I d

cos cosP SdS I dS

' 'cos ' cos 'P SdS I dS 'S SdS dS

'cos cos 'cos cos 'P P

dS dSI I


16


2 22

2

coscosO P P O

n dS dSdT n dS d

l

2 2 ' '2 ' '

2

2' 2

' cos '' ' cos '

'' cos cos '

coscos

O PP O

P O P O

n dS dSdT n dS d

ln I

dS d n dS d dTI

Thus the element of throughput is invariant upon light refraction.


17

Invariance upon transfer

2 2 ' '

2 ' ' 2 ' '2

' cos '' cos ' ' cos '

'O P

P O P O

n dS dSn dS d n d dS

l

Since the element of throughput after refraction is also equalto the element of throughput for the next surface,

We have that the element of throughput is invariant upon transfer.

The conclusion is that the element of throughput is an invariant in a lens system.


18

Radiance theorem

2

,L Hd dT

n

dL

d T

The element of optical flux

transmitted is given by the product the source radiance

in units of watts/m2-steradian,

in units of watt/m2-steradian,

It follows that the quantity 2/L nis invariant; this is known as the radiance theorem.

divided by the square of the index of refraction

and the element of throughput


19

Image and pupil aberrations relationships

2 2 2 2 ' '2 2

cos ' cos ''

O P O Pn dS dS n dS dSl l

0

cosll

'0'

cos 'll

2 4 2 4 ' '2 20 0

cos ' cos ''

O P O Pn dS dS n dS dSl l

2 21cos 1 22 u uu H u H H

2 21cos ' 1 ' 2 ' ' '2 u u u H u H H


20


2 2

2 22 22 20

2 2 '2 '20

1 1 1 2 ' 2 ' ' '

' 1 2 2'

H

pupil object

pupil image

H u u u H u H H

y yn l u uu H u H Hl n y y

2 2 22 2 22 20

2 2 '2 '2 2 '2 '2 20

' 1' '

pupil object object

pupil image image

y y yn l un Жn l y y n u y Ж

2 2

2 2

1 1 1 2 ' 2 ' ' '

1 2 2

H H u u u H u H H

u uu H u H H

'2'

2, 1pupilP

PP pupil

ydS J HdS y

'2'

2, 1imageO

O HO object

ydS J H HdS y


21


2 22 2H H u uu H u H H

1 ,HW HЖ 1 ,H W HЖ

131 222 220 3111 4 6 4 4W H H W H W H WЖ

131 222 220 3111 4 6 4 4H H W W H W H W H HЖ


22


131 311 222 222

220 220 311 131

2 2

32

1 12 2

W W H H W W H

W W H W W

Ж u uu H u H H

2131 311 2ЖW W u

222 222 2ЖW W uu

220 220 4ЖW W uu

2311 131 2ЖW W u

!


23

The sine condition

sin ''sinUu

u U

'000 020 311

2 2

4' 1

41 1 2 '2

I I I WЖ

Ж u uЖ


24

Herschel’s condition

2 2 'sin ' 'sin2 2U Un l n l

' 2000 020 3114 3' 1 1 '

2I I I W u

Ж


25


26

Beam apodization in microscopes

2311 131 12W W Ж u 131 0W

2311 12W Ж u


27

Beam apodization

'020 3114I WЖ

' 2 2020 3114 4 4 '2 2ЖI W u u

Ж Ж

Exit pupil irradiance


28

‘Exact’

Exitpupil

4' cosds ds

ds

'ds


29

Relative illumination across the field of view

• RI is the ratio of the irradiance at a given field point to the on-axis irradiance.

4cosRI


30

Relative illumination across the field of view

Stop at entrance pupil


31

Pinhole camera and lens

Stop at entrance pupil 4cosRI


32

Pupil coma vs Image distortion


33


34


35

Aplanatic telecentric lens

Produces uniform illumination from Lambertian source

2020 3114[ 2 ' ]I H H u W H HЖ

111 220 2224 6[ 4 ' ' ]I H u u W W HЖ Ж

2200 1314[ 2 ' ]I u WЖ


36

Wave and irradiance functions

, ,, ,

000 200 111 020

22040 131 222

2

220 311 400

( , ) ( ) ( ) ( )

,

j m nk l m

j mlWH W H H H

W W H H W H W

W W H W H

W H H W H H H W H H

, ,, ,

000 200 111 020

2 2

040 131 222

2220 311 400

( , ) ( ) ( ) ( )

,

j m nk l m

j ml

WH W H H H

W W W H W H H

W H H W H H H W H

W H H W H W

, ,

000 200 111 020

22040 131 222

2

220 311 400

( , )

,

j m n

j mnI H I H H H

I I H H I H I

I I H I H

I H H I H H H I H H

, ,, ,

000 200 111 020

2 2

040 131 222

2220 311 400

( , ) ( ) ( ) ( )

,

j m nl k m

j ml

I H I H H H

I I I H I H H

I H H I H H H I H

I H H I H I


37

Basic irradiance changesor

Irradiance aberrations


38

2020 2 'I u

111 4 ' 'I H u u H

2200 2 'I H H u H H

2 24040 04016[3 ' ' ']I u W u uЖ

3 2131 040 13116 12[12 ' ' ' ' ']I H u u W u W u u HЖ Ж

2 22 2 2222 131 2228 8[12 ' ' ' ' ']I H u u W u W u u HЖ Ж

2 2 2220 131 2204 8[6 ' ' ' ' ']I H H u u W u W u u H HЖ Ж

3 2 2311 222 220 3118 8 4[12 ' ' ' ' ' ']I H H H u u W u W u W u u H H HЖ Ж Ж

2 24 2400 3114[3 ' ' ]I H H u W u H HЖ

Irradiance coefficients at image plane:second and fourth order


39

2020 3114[ 2 ' ]I H H u W H HЖ

111 220 2224 6[ 4 ' ' ]I H u u W W HЖ Ж

2200 1314[ 2 ' ]I u WЖ

Irradiance coefficients at exit pupil plane:second and fourth order

2 24 2040 511 311 311 31126 3 3[3 ' ' ]I H H u W W u W W H HЖ Ж Ж

3 2 2131 422 420 222 220 31110 8 5 2 6[12 ' ' ' ' ' 'I H H H u u W W W u W u W u uЖ Ж Ж Ж Ж

220 311 222 3112 24 10 ]W W W W H H HЖ Ж

2 2 2 2 2222 333 331 131 222 220 31112 4 2 12 8 2[12 ' ' ' ' ' ' ' 'I H u u W W W u W u u W u u W uЖ Ж Ж Ж Ж Ж

2222 222 222 220 311 1312 2 28 16 4 ]W W W W W W HЖ Ж Ж

2 2 2 2220 331 131 222 220 3116 5 2 4 5[6 ' ' ' ' ' ' ' 'I H H u u W W u W u u W u u W uЖ Ж Ж Ж Ж

311 131 222 2202 210 8 ]W W W W H HЖ Ж

3 2 2311 242 240 222 220 13110 8 5 2 6[12 ' ' ' ' ' 'I H u u W W W u W u W u uЖ Ж Ж Ж Ж

220 131 222 1312 24 10 ]W W W W HЖ Ж

2 24 2400 151 131 131 13126 3 3[3 ' ' ]I u W W u W WЖ Ж Ж


40

Summary

• It is important to understand the radiometry in an optical system

• Irradiance function• The element of throughput• Image-pupil relationships• Sine condition• Herschel’s condition

Documents

Introduction to aberrations...IHH u W HH2' Ж Alternatively Prof. Jose Sasian OPTI 518 10 Irradiance transport equation Prof. Jose Sasian OPTI 518 11 Inverse square law 22 2 2 21cos