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
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
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
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
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
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
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
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
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