1. Geometrical theory of aberration for off-axis reflecting telescope and its applications Seunghyuk Chang 2013.02.14. SSG13

2. On-Axis vs Off-Axis On-AxisOff-Axis Secondary mirror blocks incoming rays. No obstruction. Clear aperture.

3. On-Going Off-Axis Telescope Project Advanced Technology Solar Telescope (ATST) 4-m aperture, largest solar telescope, off-axis Gregorian design

4. 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

5. Basic Off-Axis Telescope Eccentric section of an on-axis parent system

6. 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

7. Vertex Equation for Off-Axis Portion of Conic Sections of Revolution A localized coordinate system is convenient to describe a mirror near a point (x 0 ’, z 0 ’) Vertex equation of conic sections of revolution :

8. Expansion of Vertex Equation

9. Optical Path Length (OPL) AstigmatismComa 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

10. Astigmatic Images Tangential Astigmatic Image: Sagittal Astigmatic Image: The second order terms yields the two astigmatic image points

11. Tilted Astigmatic Image Planes Tangential Astigmatic Image Plane Sagittal Astigmatic Image Plane Linear Astigmatism: Expanding the two astigmatic image distances to the first order of  yields the tangential and sagittal astigmatic image planes and linear astigmatism

12. IMAGE PLANES OF PARABOLOID On-AxisOff-Axis

13. Coma and Third Order Astigmatism The A 2 term yields tangential coma aberration Expanding the two astigmatic image points to second order on  yields third order astigmatism

14. 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

15. Aperture Stop When aperture stop is displaced from the mirror surface, the reflection point of the chief ray depends on the field angle.

16. Aperture Stop A displaced aperture stop yields a new field angle  and a new chief ray incidence angle  s for the mirror

17. Aperture Stop A displaced aperture stop yields new astigmatism and coma aberration coefficient.

18. Aberrations of Classical Off-Axis Two-mirror Telescopes Astigmatism Coma Rm Rs Rm (Rs) is the radius of curvature of the primary (secondary) parent mirror at its vertex.

19. Linear Astigmatism of a Two-mirror Telescope

20. Elimination of Linear Astigmatism and Third Order Coma Linear astigmatism can be eliminated by enforcing Third order coma is identical to an on-axis paraboloid

21. Example D=1000mm, f=2000mm Satisfies zero-linear- astigmatism condition Astigmatism

22. Spot Diagram Comparison ExampleOn-Axis Paraboloid Spot diagrams of the two systems are identical as the presented theory predicted

23. Example 1m f/8 classical Cassegrain Off-axisOn-axis Side View Spot Diagrams

24. Example 1m f/20 classical Gregorian Off-axisOn-axis Side View Spot Diagrams

25. Example 2.4m f/24 aplanatic Cassegrain Off-axisOn-axis Side View Spot Diagrams

26. Example 10cm f/4 off-axis Schwarzschild flat-field anastigmat Side View Spot Diagrams

27. Off-axis Reflector Design for SPICA Channel 1 MIR Camera Collimator Camera Both the collimator and the camera are off-axis reflecting telescopes with zero linear astigmatism.

28. Off-axis Reflector Design for SPICA Channel 4 MIR Camera Collimator Camera Both the collimator and the camera are off-axis reflecting telescopes with zero linear astigmatism.

29. 6.5-m TAO Telescope Mid-infrared re-imaging optics of 6.5m-TAO telescope has been developed based on linear-astigmatism theory.

30. Off-axis Reflector Design for McDonald 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

31. Three-Mirror Off-Axis Telescope 3 rd order aberration Two MirrorThree Mirror CassegrainGregorianCouderSchwartzschild Three Mirror Anastismat (TMA) Spherical RRRRR Coma RRRRR Astigmatism XXRRR Field Curvature XXXRR Two Mirror vs. Three Mirror R: removable, X:not removable

32. Linear Astigmatism of Confocal Off-Axis N-Mirror System

33. Image Planes of K th mirror in Confocal Off-Axis N-Mirror System : Radius of curvature of the parent mirror at its vertex

34. Image Planes of Confocal Off-Axis N-Mirror System Tangential image plane: Sagittal image plane:

35. Elimination of Linear Astigmatism in Confocal Off-axis N-mirror System Two-mirror telescope : Three-mirror telescope :

36. 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 Linear astigmatism can be eliminated by adding M3

37. Advanced Technology Solar Telescope (ATST) ATSTATST + M3

38. WFIRST 1.3m-Aperture Off-Axis TMA Telescope

39. Linear-astigmatism-free modification

40. WFIRST 1.3m-Aperture Off-Axis TMA Telescope NASA Design Linear-astigmatism- free Design Aperture diameter1.3m Focal length20675mm l1~ 3330mm3330mm i1~ -12 deg.-12 deg. l2~ -800mm-800mm i2~ 12 deg.12 deg. m2~ -3.25-3.25 l3~ 2700mm2696mm 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

41. 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