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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-AxisOff-Axis Secondary mirror blocks incoming rays. No obstruction. Clear aperture.

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On-Going Off-Axis Telescope Project Advanced 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 A localized coordinate system is convenient to describe a mirror near a point (x 0 ’, z 0 ’) Vertex equation of conic sections of revolution :

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Expansion of Vertex Equation

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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

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

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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

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IMAGE PLANES OF PARABOLOID On-AxisOff-Axis

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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

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

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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.

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Linear Astigmatism of a Two-mirror Telescope

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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

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

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Spot Diagram Comparison ExampleOn-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-axisOn-axis Side View Spot Diagrams

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

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

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

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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.

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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.

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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 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

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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

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Linear Astigmatism of Confocal Off-Axis N-Mirror System

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

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

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

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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 Linear astigmatism can be eliminated by adding M3

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Advanced Technology Solar Telescope (ATST) ATSTATST + M3

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WFIRST 1.3m-Aperture Off-Axis TMA Telescope

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Linear-astigmatism-free modification

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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

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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

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