Geometrical theory of aberration for off-axis reflecting telescope

and its applications

Seunghyuk Chang

2013.02.14.

SSG13

On-Axis vs Off-Axis

On-Axis Off-Axis

Secondary mirror blocks incoming rays.

No obstruction.Clear aperture.

On-Going Off-Axis Telescope ProjectAdvanced 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

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 :

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

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

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

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

IMAGE PLANES OF PARABOLOID

On-Axis Off-Axis

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

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

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

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.

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.

Linear Astigmatism of a Two-mirror Telescope

2

2

21

1

1

2

2 2sin2sinarctan iR

iRt

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

Example

• D=1000mm, f=2000mm• Satisfies zero-linear-astigmatism 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 On-axis

Side View

Spot Diagrams

Example

1m f/20 classical Gregorian

Off-axis On-axis

Side View

Spot Diagrams

Example

2.4m f/24 aplanatic Cassegrain

Off-axis On-axis

Side View

Spot Diagrams

Example

10cm f/4 off-axis Schwarzschild flat-field anastigmat

Side View Spot Diagrams

M1

M2

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.

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.

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 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

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

Linear Astigmatism of Confocal Off-Axis N-Mirror System

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

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

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

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

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

02sin1

2sin1

2sin 33

3

322

2

2

21

1

1

iRmm

iRm

iR

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

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