Prof. Jose Sasian

Coma aberration

Lens Design OPTI 517

Prof. Jose Sasian

Coma0.25 wave 1.0 wave 2.0 waves 4.0 waves Spot diagram

...cos

coscos

cos,,

4400

3311

22220

222222

3131

4040

1112

0202

200

HWHWHW

HWHWW

HWWHWHW

Prof. Jose Sasian

Coma though focus

Prof. Jose Sasian

Cases of zero coma

13112

uW AA yn

•At y=0, surface is at an image•A=0, On axis beam concentric with center of curvature•A-bar=0, Off-axis beam concentric, chief ray goes through the center of curvature•Aplanatic points

Prof. Jose Sasian

Cases of zero coma

Prof. Jose Sasian

Aplanatic-concentric

Aplanatic surface

Concentric surface A=0

0un

Prof. Jose Sasian

Coma as a variation of magnification with aperture I

1311,2

uW H AA y Hn

Prof. Jose Sasian

Coma as a variation of magnification with aperture II

m=s’/s

S

S’

Prof. Jose Sasian

Sine conditionComa aberration can be considered as a variation of magnification

with respect to the aperture. If the paraxial magnificationis equal to the real ray marginal magnification, then an optical system

would be free of coma.Spherical aberration can be considered as a variation

of the focal length with the aperture.

uu

UU'

sinsin '

U U’

Prof. Jose Sasian

Sine condition

'L L

' 'sin( ')L h U

On-axis

P P’

U U’

Y Y’

O

O’

Optical path length between O and O’ is Laxis anddoes not depend on Y or Y’

Sine condition

P P’

h

h’

Y Y’

Optical path length between y and y’ is

Loff-axis = Laxis + L’ - L

Loff-axis = Laxis + h’ n’ sin(U’) - h n sin(U)

O’

' 'sin( ') sin( )h n U hn U

'L

sin( )L h U

L

uu

UU'

sinsin '

That is: OPD has no linear phase errors as a functionof field of view!

Prof. Jose Sasian

Imaging a grating

dmU

)sin(

)'sin(')sin( UdmUd

Prof. Jose Sasian

Contribution from an aspheric surface

Wyy

A y n131 441

28

y-bar chief ray

y marginal ray

Prof. Jose Sasian

Aberrations and symmetry

...cos

coscos

cos,,

4400

3311

22220

222222

3131

4040

1112

0202

200

HWHWHW

HWHWW

HWWHWHW

•Coma is an odd aberration with respect to the stop•Natural stop position to cancel coma by symmetry

Prof. Jose Sasian

Structural coefficients: Thin lens (stop at lens)

DCYBXYAXySI 2234

41

2 212IIS Ж y EX FY

2IIIS Ж

2 1IVS Ж

n

0VS

12yCL

0TC

212

nnnA

2

2

1nnD

1

14

nnnB

11

nnnE

nnC 23

nnF 12

12

12

21

21

rrrr

cccc

X

uuuu

mmY

''

11

))(1( 1 xccncn

Prof. Jose Sasian

Coma vs Bending

Shape factor X

Coma

FYEXII

Prof. Jose Sasian

Principal surface

In an aplanat working at m=0the equivalent

refracting surface is a hemisphere

Prof. Jose Sasian

Cassegrain’s principal surface

Since the equivalent refracting surface in a Cassegrain telescope is a paraboloid then the coma of that Cassegrain is the same of a paraboloid mirror with the same focal length.

Prof. Jose Sasian

Aplanat doublets

Prof. Jose Sasian

Kingslake’s cemented aplanatChromatic correctionSpherical aberration correctionComa correctionStill cemented

Prof. Jose Sasian

Control of coma in the presence of an aspheric mirror near a pupil

Wyy

A y n131 441

28

In the presence of a strong asphericsurface near the stop or pupil, coma

aberration can be corrected by moving the surface

Prof. Jose Sasian

Coma correction by phantom stop position

Optical system

Object plane Image plane

Originalstop position

In the presence of spherical aberration there is a stop positionfor which coma is zero. At that stop position spherical aberrationmight be corrected. Then the system becomes aplanatic and the stopcan be shifted back to its original position.

III SyyS

Naturalstop position

Prof. Jose Sasian

The aplanatic member(s) in a family of solutions

Ritchey-ChretienDoublet

Prof. Jose Sasian

Camera Schmidt

Aspheric plate at mirror center of curvature A-bar=0Stop aperture at aspheric plateNote symmetry about mirror CCNo spherical aberrationNo comaNo astigmatism.Anastigmatic over a wide field of view!Satisfies Conrady’s D-d sum

Prof. Jose Sasian

Notes

• Need to make aplanatic zero-field systems (that are fast). The alignmentbecomes easier.

• Lenses for lasers diodes/optical fibers• Microscope objectives

Prof. Jose Sasian

Sine condition from optical flux conservation and radiance

theorem

Etendu considerations

Prof. Jose Sasian

Sine condition

A

sin ''sinUu

u U

Prof. Jose Sasian

Optical flux from a Lambertian source

20 0

0

2 cos sin sinAL d AL

0 0 00

2 sin 2 1 cosAL d AL AL

Compare with homogeneous source

Prof. Jose Sasian

Optical flux=radiance x throughput

2 2

0 0

' ''

n n TL L

02

L Tn

02

'''L Tn

2 2

0 0

' ''

n nU UL L

U

2 2

0 0

''

n nL L

Radiance theorem

Prof. Jose Sasian

Sine condition from optical flux conservation

sin ''sinUu

u U

sin ' 'sin 'nh U n h U

2 2

0 0

' ''

n nU UL L

2 2 2 2 2 2sin ' ' sin 'n h U h n U

2 2 2 2 2 2sin ' ' sin 'n h U h n U

2 2 2 2sin ' 'sin 'n A U n A U

Sine condition

Prof. Jose Sasian

Throughput Etendu

(area-omega product)

2 2' ' 'n A n A 2T Ж

“Capacity to transfer optical flux”

A 2 1 cos

Prof. Jose Sasian

Even better

2 2' ' 'n A n A

2 2 2sinA NA n A

Homogeneoussource

Lambertiansource

Prof. Jose Sasian

Etendu considerations are key to design an optical system

12 3 4

5

5 4 3 2 1

Start with the sensor at the end 25 5A NA

Prof. Jose Sasian

Summary

• Coma aberration• Coma as an odd aberration• Sine condition• Natural stop position• Aplanatic doublets• Zero-field systems