Geometrical theory of aberration for
off-axis reflecting telescope
and
its applications
SSG13
Seunghyuk Chang
2013.02.14.
On-Axis vs Off-Axis
On-Axis
Secondary mirror
blocks incoming rays.
Off-Axis
No obstruction.
Clear aperture.
On-Going Off-Axis Telescope Project
Advanced Technology Solar Telescope (ATST)
4-m aperture, largest solar telescope, off-axis Gregorian design
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
Basic Off-Axis Telescope
Eccentric section of an on-axis parent system
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
Vertex Equation for Off-Axis Portion of
Conic Sections of Revolution
• Vertex equation of conic sections
of revolution :
 2  2Rz  (1  K ) z2  0
• A localized coordinate system is
convenient to describe a mirror
near a point (x0’, z0’)
(1  K cos 2  0 ) z 2 


2R
 xK sin 2 0 
) z 
2


1

K
sin

0


(1  K sin 2  0 ) x 2  y 2  0
Expansion of Vertex Equation
z  a1x2  a2 y2  a3 x3  a4 xy 2  O(4)
(1  K sin 2  0 )
a1 
2R
3
2
(1  K sin 2  0 )
a2 
2R
K sin 2 0 (1  K sin 2  0 ) 2
a3 
4R2
1
2
K sin 2 0 (1  K sin 2  0 )
a4 
4R2
Optical Path Length (OPL)
OPL     s  s  A1 x 2  A1 y 2  A2 x 3  A2 xy 2  O (4)
Astigmatism
• 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
Coma
Astigmatic Images
OPL     s  s  A1 x 2  A1 y 2  A2 x 3  A2 xy 2
The second order terms yields the two astigmatic
image points
A1  0
A1  0
Tangential
Astigmatic Image:
Sagittal
Astigmatic Image:
2 cos3  s
1
1


s  R cos( s   ) s
1 2cos s cos( s   ) 1


R
s
s
t
s
Tilted Astigmatic Image Planes
Expanding the two astigmatic image distances to the first
order of  yields the tangential and sagittal astigmatic
image planes and linear astigmatism
Tangential Astigmatic
Image Plane
Sagittal Astigmatic
Image Plane
1 2 cos 2  s
1

(1   tan  s ) 
R
s
s
1 2 cos 2  s
1

(1   tan  s ) 
R
s
s
t
s
s 2sin 2s
Linear Astigmatism:    R 
s s
t
s
IMAGE PLANES OF PARABOLOID
On-Axis
Off-Axis
Coma and Third Order Astigmatism
OPL     s  s  A1 x 2  A1 y 2  A2 x 3  A2 xy 2  O (4)
• The A2 term yields tangential coma aberration
cos3  s
A2 
R

1 1 
1 
21
sin

(cos



sin

)



cos



s
s
s 


s
s
s
R


0 


• Expanding the two astigmatic image points to second
order on  yields third order astigmatism
1 1
s 2sin 2 s
2

  2   
R
R
 st ss 
st ss
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
Aperture Stop
When aperture stop is displaced from the mirror surface,
the reflection point of the chief ray depends on the field angle.
Aperture Stop
• A displaced aperture stop yields a new field angle 
and a new chief ray incidence angle s for the mirror
 W
   1 
 s0

W
2W 
 
1 
s0  s0

W
 s   s 
2
1 1 
  
 s0 s  
0 


 tan s

Aperture Stop
• A displaced aperture stop yields new astigmatism
and coma aberration coefficient.
 W  sin 2 s
s
 2 1  

 s0  R
stss
2 W
 1 
R  s0
cos3  s
A2 
R
  W
  1 
  s0

1 1 

2W
2
sin  s  2
  W    cos 2 s 
s
s0



 0 s0 


 1 1   W

sin s cos s     1 
 s s0   s0


 
 2 1 1  W

2
cos



sin





s
s
 2
s
R
s

 0

 

Aberrations of Classical Off-Axis
Two-mirror Telescopes
Astigmatism

s
 2 
f


2
m
s
sin 2 m sin 2 s 


Rm
Rs 
W  f 1  tan  m tan  s  
0
Coma
Rm
Rs
ATC 

f cos  s xs
0
Rm (Rs) is the radius of curvature of the primary
(secondary) parent mirror at its vertex.
0
Linear Astigmatism of a Two-mirror Telescope
 2   1


 sin 2i1  2 sin 2i2 
R2

  2  R1
 t  arctan
Elimination of Linear Astigmatism
and Third Order Coma
• Linear astigmatism can be eliminated by enforcing
m
Rm
sin 2 m 
s
Rs
sin 2 s
• Third order coma is identical to an on-axis paraboloid
3x02
ATC  
4f 2
Example
• D=1000mm, f=2000mm
• Satisfies zero-linearastigmatism condition
Astigmatism
Spot Diagram Comparison
Example
On-Axis Paraboloid
Spot diagrams of the two systems are identical as the
presented theory predicted
Example
1m f/8 classical Cassegrain
Off-axis
Side View
Spot Diagrams
On-axis
Example
1m f/20 classical Gregorian
Off-axis
Side View
Spot Diagrams
On-axis
Example
2.4m f/24 aplanatic Cassegrain
Off-axis
Side View
Spot Diagrams
On-axis
Example
10cm f/4 off-axis Schwarzschild flat-field anastigmat
Spot Diagrams
Side View
M1
M2
22:53:54
Off-axis Reflector Design for
SPICA Channel 1 MIR Camera
Camera
Collimator
• Both the collimator and the camera are off-axis reflecting telescopes with
zero linear astigmatism.
13:31:39
Off-axis Reflector Design for
SPICA Channel 4 MIR Camera
Camera
Collimator
• Both the collimator and the camera are off-axis reflecting telescopes with
zero linear astigmatism.
6.5-m TAO Telescope
• Mid-infrared re-imaging optics of 6.5m-TAO telescope has been developed
based on linear-astigmatism theory.
Off-axis Reflector Design for
McDonald 2.1-m Telescope Focal Reducer
Camera
Collimator
• 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
Three-Mirror Off-Axis Telescope
Two Mirror vs. Three Mirror
Two Mirror
Three Mirror
3rd order
aberration
Cassegrain
Gregorian
Couder
Schwartzschild
Three 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
R: removable, X:not removable
Linear Astigmatism of
Confocal Off-Axis N-Mirror System
Image Planes of Kth mirror in
Confocal Off-Axis N-Mirror System
tan KT  mK tan KT 1 
tan  mK tan
S
K
mK 
RK
S
K 1
K
sin 2iK
RK
K

sin 2iK
RK
K
K
: Radius of curvature of the parent
mirror at its vertex
Image Planes of Confocal Off-Axis
N-Mirror System
N 1  N
 N

 p
N
T




sin 2i p 
sin 2iN
Tangential image plane: tan   m p tan 0    mq




R
R
p 1  q  p 1
N
 p 1

 p
T
N
Sagittal image plane:
N 1  N
 N

 p

S



tan    m p  tan 0     mq 
sin 2i p  N sin 2iN
RN
p 1  q  p 1
 p 1

 Rp
T
N
Elimination of Linear Astigmatism in
Confocal Off-axis N-mirror System
tan NT  tan NS
 N
 p

N
1 N



 tan 0T  tan 0S
m
sin
2
i

sin
2
i

m



q
p
N
p

R

RN
2  p 1
p 1  q  p 1
 p


N 1
Two-mirror telescope :
1
1 2
sin 2i1 
sin 2i2  0
R1
m2 R2
1
1 2
1 3
sin 2i1 
sin 2i2 
sin 2i3  0
Three-mirror telescope :
R1
m2 R2
m2 m3 R3

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
1
1 2
sin 2i1 
sin 2i2  0
R1
m2 R2
• Linear astigmatism can be
eliminated by adding M3
1
1 2
1 3
sin 2i1 
sin 2i2 
sin 2i3  0
R1
m2 R2
m2 m3 R3
Advanced Technology Solar Telescope (ATST)
ATST
ATST + M3
WFIRST 1.3m-Aperture Off-Axis TMA Telescope
WFIRST 1.3m-Aperture Off-Axis TMA Telescope
Linear-astigmatism-free modification
1
1 2
1 3
sin 2i1 
sin 2i2 
sin 2i3  0
R1
m2 R2
m2 m3 R3
WFIRST 1.3m-Aperture Off-Axis TMA Telescope
Linear-astigmatismfree Design
NASA 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
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