geometrical theory of aberration for off-axis reflecting telescope and its applications seunghyuk...
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Geometrical theory of aberration for off-axis reflecting telescope
and its applications
Seunghyuk Chang
2013.02.14.
SSG13
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On-Axis vs Off-Axis
On-Axis Off-Axis
Secondary mirror blocks incoming rays.
No obstruction.Clear aperture.
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On-Going Off-Axis Telescope ProjectAdvanced Technology Solar Telescope (ATST)
4-m aperture, largest solar telescope, off-axis Gregorian design
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On-Going Off-Axis Telescope Project
Wide Field Infrared Survey Telescope (WFIRST)
• Top-ranked large space mission in the New Worlds, New Horizon Decadal Survey of Astronomy and Astrophysics• Sky surveys, Exoplanet – Microlensing, Dark Energy• 1.3m aperture off-axis Three Mirror Anastigmat (TMA) design
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Basic Off-Axis Telescope
Eccentric section of an on-axis parent system
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Confocal Plane-Symmetric Off-Axis Two-Mirror System
The mirrors of a confocal system do not need to have a common axis for a perfect image at the system focus
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Vertex Equation for Off-Axis Portion of Conic Sections of Revolution
2 22 (1 ) 0Rz K z
2 20
0 20
2 2 20
(1 cos )
2sin 2 )
1 sin
(1 sin ) 0
K z
RxK z
K
K x y
• A localized coordinate system is convenient to describe a mirror near a point (x0’, z0’)
• Vertex equation of conic sections of revolution :
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Expansion of Vertex Equation
2 2 3 21 2 3 4 (4)z a x a y a x a xy O
32 2
01
(1 sin )
2
Ka
R
12 2
02
(1 sin )
2
Ka
R
2 20 0
3 2
sin 2 (1 sin )
4
K Ka
R
20 0
4 2
sin 2 (1 sin )
4
K Ka
R
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Optical Path Length (OPL)
2 2 3 21 1 2 2 (4)OPL s s A x A y A x A xy O
Astigmatism Coma
• To compute the aberrations, the OPL for an arbitrary reflection point on the mirror is necessary
• The OPL is constant in a perfect focusing mirror
• The variance of the OPL
yields aberrations
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Astigmatic Images
1 0A
TangentialAstigmatic Image:
32cos1 1
cos( )s
stR ss
1 0A
SagittalAstigmatic Image:
2cos cos( )1 1s s
sR ss
The second order terms yields the two astigmatic image points
2 2 3 21 1 2 2OPL s s A x A y A x A xy
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Tilted Astigmatic Image Planes
Tangential AstigmaticImage Plane
22cos1 1(1 tan )s
s
tR ss
Sagittal AstigmaticImage Plane
22cos1 1(1 tan )s
s
sR ss
2sin 2 s
t s
s
Rs s
Linear Astigmatism:
Expanding the two astigmatic image distances to the first order of yields the tangential and sagittal astigmatic image planes and linear astigmatism
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IMAGE PLANES OF PARABOLOID
On-Axis Off-Axis
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Coma and Third Order Astigmatism
32
20
cos 1 1 1 1sin (cos sin ) coss
s s sAR s s s R
22sin 2 2 1 1s
t st s
s
R R s ss s
• The A2 term yields tangential coma aberration
• Expanding the two astigmatic image points to second order on yields third order astigmatism
2 2 3 21 1 2 2 (4)OPL s s A x A y A x A xy O
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Aberrations of Classical Off-axis Two-mirror Telescopes
• Aberrations of classical off-axis two-mirror telescopes can be obtained by cascading the aberrations of each mirror
• Assume the aperture stop is located at the primary mirror
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Aperture Stop
When aperture stop is displaced from the mirror surface,the reflection point of the chief ray depends on the field angle.
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Aperture Stop
2
0 0 0
1 1 tan s
W W W
s s s
0 0
1 1
2s s
W
s s
• A displaced aperture stop yields a new field angle and a new chief ray incidence angle s for the mirror
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Aperture Stop
0
2 2
0 0 0 00
sin 22 1
2 1 1 21 1 cos 2 sin
s
t s
s s
s W
s Rs s
W W WW
R s s s ss
32 2
2 20 0 0
cos 1 1 1 1sin cos 1 cos sins
s s s s
W WA
R s s s s R s
• A displaced aperture stop yields new astigmatism and coma aberration coefficient.
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Aberrations of Classical Off-Axis Two-mirror Telescopes
Astigmatism
0
2
0
sin 2 sin 22
1 tan tan
m m s
s m s
m s
s
f R R
W f
Coma
0
cos s s
ATCf x
Rm
Rs
Rm (Rs) is the radius of curvature of the primary (secondary) parent mirror at its vertex.
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Linear Astigmatism of a Two-mirror Telescope
2
2
21
1
1
2
2 2sin2sinarctan iR
iRt
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Elimination of Linear Astigmatism and Third Order Coma
• Linear astigmatism can be eliminated by enforcing
sin 2 sin 2m sm s
m sR R
• Third order coma is identical to an on-axis paraboloid
202
3
4
xATC
f
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Example
• D=1000mm, f=2000mm• Satisfies zero-linear-astigmatism condition
Astigmatism
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Spot Diagram Comparison
Example On-Axis Paraboloid
Spot diagrams of the two systems are identical as the presented theory predicted
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Example
1m f/8 classical Cassegrain
Off-axis On-axis
Side View
Spot Diagrams
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Example
1m f/20 classical Gregorian
Off-axis On-axis
Side View
Spot Diagrams
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Example
2.4m f/24 aplanatic Cassegrain
Off-axis On-axis
Side View
Spot Diagrams
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Example
10cm f/4 off-axis Schwarzschild flat-field anastigmat
Side View Spot Diagrams
M1
M2
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Off-axis Reflector Design forSPICA Channel 1 MIR Camera
22:53:54
SPICA ch1 MIR Dcl:70 centered field Scale: 0.04 11-Jul-08
595.24 MM
Collimator
Camera
• Both the collimator and the camera are off-axis reflecting telescopes with zero linear astigmatism.
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13:31:39
SPICA ch4 MIR Dcl:50 offset field Scale: 0.04 20-Jul-08
609.76 MM
Off-axis Reflector Design forSPICA Channel 4 MIR Camera
Collimator
Camera
• Both the collimator and the camera are off-axis reflecting telescopes with zero linear astigmatism.
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6.5-m TAO Telescope
• Mid-infrared re-imaging optics of 6.5m-TAO telescope has been developed based on linear-astigmatism theory.
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Off-axis Reflector Design forMcDonald 2.1-m Telescope Focal Reducer
• Both the collimator and the camera are off-axis reflecting telescopes with zero linear astigmatism.• Reduce the telescope focal ratio from f/13.6 to f/4.56
Camera
Collimator
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Three-Mirror Off-Axis Telescope
3rd order aberration
Two Mirror Three Mirror
Cassegrain Gregorian Couder SchwartzschildThree Mirror Anastismat
(TMA)
Spherical R R R R R
Coma R R R R R
Astigmatism X X R R R
Field Curvature X X X R R
Two Mirror vs. Three Mirror
R: removable, X:not removable
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Linear Astigmatism of Confocal Off-Axis N-Mirror System
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Image Planes of Kth mirror inConfocal Off-Axis N-Mirror System
KK
KTKK
TK i
Rm 2sintantan 1
KK
KSKK
SK i
Rm 2sintantan 1
K
KKm
KR : Radius of curvature of the parent mirror at its vertex
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Image Planes of Confocal Off-AxisN-Mirror System
NN
Np
N
p p
pN
pqq
TN
pp
TN i
Ri
Rmm 2sin2sintantan
1
1 10
1
Tangential image plane:
Sagittal image plane: NN
Np
N
p p
pN
pqq
SN
pp
TN i
Ri
Rmm 2sin2sintantan
1
1 10
1
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Elimination of Linear Astigmatism in Confocal Off-axis N-mirror System
SN
TN tantan
STN
ppN
N
Np
N
p p
pN
pqq mi
Ri
Rm 00
1
1
1 1
tantan2
12sin2sin
Two-mirror telescope : 02sin1
2sin 22
2
21
1
1
iRm
iR
Three-mirror telescope : 02sin1
2sin1
2sin 33
3
322
2
2
21
1
1
iRmm
iRm
iR
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Advanced Technology Solar Telescope (ATST)
• 4m-aperture off-axis Gregorian design• Off-axis section of an on-axis telescope• Gregorian focus does not satisfy linear-astigmatism-free condition
02sin1
2sin 22
2
21
1
1
iRm
iR
• Linear astigmatism can be eliminated by adding M3
02sin1
2sin1
2sin 33
3
322
2
2
21
1
1
iRmm
iRm
iR
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Advanced Technology Solar Telescope (ATST)
ATST ATST + M3
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WFIRST 1.3m-Aperture Off-Axis TMA Telescope
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WFIRST 1.3m-Aperture Off-Axis TMA Telescope
Linear-astigmatism-free modification
02sin1
2sin1
2sin 33
3
322
2
2
21
1
1
iRmm
iRm
iR
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WFIRST 1.3m-Aperture Off-Axis TMA Telescope
NASA DesignLinear-astigmatism-
free Design
Aperture diameter 1.3m
Focal length 20675mm
l1 ~ 3330mm 3330mm
i1 ~ -12 deg. -12 deg.
l2 ~ -800mm -800mm
i2 ~ 12 deg. 12 deg.
m2 ~ -3.25 -3.25
l3 ~ 2700mm 2696mm
i3 ? -7.9427239 deg.
m3 ? 1.910339
Residual RMS wave front error for 0.8 deg x 0.46 deg FOV
12 ~ 18 nm* 0.9 ~ 3.5 nm
* : “Wide Field Infrared Survey Telescope [WFIRST]: telescope design and simulated performance,” Proc. SPIE 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, 84421U (September 21, 2012); doi:10.1117/12.927808
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References
• S. Chang and A. Prata, Jr., "Geometrical theory of aberrations near the axis in classical off-axis reflecting telescopes," Journal of the Optical Society of America A 22, 2454-2464 (2005)
• S. Chang, J. H. Lee, S. P. Kim, H. Kim, W. J. Kim, I. Song, and Y. Park, "Linear astigmatism of confocal off-axis reflective imaging systems and its elimination," Applied Optics 45, 484-488 (2006)
• S. Chang, " Off-axis reflecting telescope with axially-symmetric optical property and its applications," Proc. SPIE, Vol. 6265, 626548 (2006)
• S. Chang, “Elimination of linear astigmatism in N-confocal off-axis conic mirror imaging system,” in preparation