Abstract

We present a simple formalism that has proven useful in on-axis alignment of two-element telescopes when wavefront information is available from only a limited region (here two noncontiguous subapertures) of the pupil. Misalignments cause predictable full-aperture aberrations, which in turn cause predictable tip/tilt modes in the subapertures. For the most useful case in which secondary mirror tilts are independently constrained by optical monitoring, the four subaperture tip/tilt modes provide enough information to solve for the state of misalignment uniquely. A practically important and intuitively appealing simplification of this inversion occurs if the tip/tilts of the two subapertures are first transformed into a new basis consisting of differential and common-mode tilts in each of the x and y directions. Then the matrices interpreting subaperture modes as full-aperture aberrations and those in turn as mechanical misalignments become diagonal, so the mechanical adjustment required to align each degree of freedom is just a constant sensitivity multiplying one of the measured differential or common-mode tilt basis modes. Knowing that this simplification occurs allows rapid empirical calibration of sensitivities in the lab and then deterministic alignment, simply and transparently, with no need for ray tracing to model the optical effects of the adjustments at each step of the alignment.

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  1. R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).
  2. B. A. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996).
  3. H. N. Chapman and D. W. Sweeney, “A rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998).
  4. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II: Optical system alignment using differential wavefront sampling,” Opt. Express 15, 15424–15437 (2007).
  5. H. Lee, “Optimal collimation of misaligned optical systems by concentering primary field aberrations,” Opt. Express 18, 19249–19262 (2010).
  6. T. Schmid, K. P. Thompson, and J. P. Rolland, “Misalignment-induced nodal aberration fields in two-mirror astronomical telescopes,” Appl. Opt. 49, D131–D144 (2010).
  7. X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).
  8. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  9. N. Kaiser, J. Morgan, and B. Stalder, “Collimation and alignment of the Pan-STARRS PS1 telescope,” Proc. Advanced Maui Optical and Space Surveillance Technologies Conference 2009, September4, 2009, p. E41 (Curran Associates, Inc., 2010).
  10. H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).
  11. S. Kim, H.-S. Yang, Y.-W. Lee, and S.-W. Kim, “Merit function regression method for efficient alignment control of two-mirror optical systems,” Opt. Express 15, 5059–5068 (2007).
  12. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. R. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd Ed. (Cambridge University, 1992).

2010

H. Lee, “Optimal collimation of misaligned optical systems by concentering primary field aberrations,” Opt. Express 18, 19249–19262 (2010).

T. Schmid, K. P. Thompson, and J. P. Rolland, “Misalignment-induced nodal aberration fields in two-mirror astronomical telescopes,” Appl. Opt. 49, D131–D144 (2010).

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

2007

2006

H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).

1998

H. N. Chapman and D. W. Sweeney, “A rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998).

1996

B. A. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996).

1980

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

An, X.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Bloemhof, E.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Chang, Z.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Chapman, H. N.

H. N. Chapman and D. W. Sweeney, “A rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998).

Dalton, G. B.

Dekens, F.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Flannery, B. R.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. R. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd Ed. (Cambridge University, 1992).

Goullioud, R.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Jeganathan, M.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Kaiser, N.

N. Kaiser, J. Morgan, and B. Stalder, “Collimation and alignment of the Pan-STARRS PS1 telescope,” Proc. Advanced Maui Optical and Space Surveillance Technologies Conference 2009, September4, 2009, p. E41 (Curran Associates, Inc., 2010).

Kim, S.

Kim, S.-H.

H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).

Kim, S.-W.

Kuan, G.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Lee, H.

Lee, H.-Y.

H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).

Lee, I.-W.

H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).

Lee, J.-H.

H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).

Lee, Y.-W.

S. Kim, H.-S. Yang, Y.-W. Lee, and S.-W. Kim, “Merit function regression method for efficient alignment control of two-mirror optical systems,” Opt. Express 15, 5059–5068 (2007).

H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).

Lin, S.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

McLeod, B. A.

B. A. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996).

Moore, D.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Moore, J.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Morgan, J.

N. Kaiser, J. Morgan, and B. Stalder, “Collimation and alignment of the Pan-STARRS PS1 telescope,” Proc. Advanced Maui Optical and Space Surveillance Technologies Conference 2009, September4, 2009, p. E41 (Curran Associates, Inc., 2010).

Page, N.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. R. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd Ed. (Cambridge University, 1992).

Rhee, H.-G.

H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).

Rolland, J. P.

Schmid, T.

Schmitigal, W.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Shack, R. V.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

Song, J.-B.

H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).

Stalder, B.

N. Kaiser, J. Morgan, and B. Stalder, “Collimation and alignment of the Pan-STARRS PS1 telescope,” Proc. Advanced Maui Optical and Space Surveillance Technologies Conference 2009, September4, 2009, p. E41 (Curran Associates, Inc., 2010).

Sutherland, K.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Sweeney, D. W.

H. N. Chapman and D. W. Sweeney, “A rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998).

Tang, H.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. R. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd Ed. (Cambridge University, 1992).

Thompson, K.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

Thompson, K. P.

Tosh, I. A. J.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. R. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd Ed. (Cambridge University, 1992).

Weilert, M.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Yang, H.-S.

S. Kim, H.-S. Yang, Y.-W. Lee, and S.-W. Kim, “Merit function regression method for efficient alignment control of two-mirror optical systems,” Opt. Express 15, 5059–5068 (2007).

H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).

Zimmer, R.

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Appl. Opt.

Opt. Express

Proc. SPIE

H. N. Chapman and D. W. Sweeney, “A rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998).

H.-S. Yang, S.-H. Kim, Y.-W. Lee, J.-B. Song, H.-G. Rhee, H.-Y. Lee, J.-H. Lee, I.-W. Lee, and S.-W. Kim, “Computer aided alignment using Zernike coefficients,” Proc. SPIE 6293, 629301 (2006).

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

X. An, E. Bloemhof, H. Tang, G. Kuan, M. Weilert, D. Moore, N. Page, J. Moore, S. Lin, Z. Chang, K. Sutherland, R. Zimmer, M. Jeganathan, W. Schmitigal, F. Dekens, and R. Goullioud, “SIM brassboard Astrometric Beam Combiner (ABC) integration and performance testing,” Proc. SPIE 7734, 773423 (2010).

Publ. Astron. Soc. Pac.

B. A. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996).

Other

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

N. Kaiser, J. Morgan, and B. Stalder, “Collimation and alignment of the Pan-STARRS PS1 telescope,” Proc. Advanced Maui Optical and Space Surveillance Technologies Conference 2009, September4, 2009, p. E41 (Curran Associates, Inc., 2010).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. R. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd Ed. (Cambridge University, 1992).

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Figures (7)

Fig. 1.
Fig. 1.

Ray trace and physical layout of the two-element ATA telescope, whose clear aperture spans only a small subregion of the full-aperture pupil, the two subapertures shown. In the analysis here, pupil coordinates will be denoted x (along the line joining the two subapertures) and y (perpendicular to the x axis and passing through the center of the full aperture or the centers of the two subapertures).

Fig. 2.
Fig. 2.

Experimental layout for measuring wavefront quality in the two subapertures of ATA. A Fizeau test interferometer is at right; a high-quality retroreflecting flat mirror is at left. A reticle positionable at the focus of the interferometer allows setting focus and tip/tilt of the entering beam. An autocollimator (not shown) allows setting secondary tip/tilt with respect to those of both primary and Fizeau.

Fig. 3.
Fig. 3.

Subaperture appearance of full-aperture defocus. (Top) One unit of defocus across the full aperture. (Middle) The corresponding incoherent subaperture pattern, with pistons removed, primarily a mode of “differential” x-tilt in which the subaperture tilts are oppositely directed, with no y-tilt. (Bottom) Profile plot through the surviving subaperture wavefront.

Fig. 4.
Fig. 4.

Subaperture appearance of full-aperture x-coma. (Top) One unit of x-coma across the full aperture. (Middle) The corresponding incoherent subaperture pattern, with pistons removed, primarily a “common-mode” pattern of subaperture x-tilts. At a finer level of detail, each subaperture has a small component of local defocus as well and that defocus is of opposite sign in the two subapertures. (Bottom) Profile plot through the surviving subaperture wavefront.

Fig. 5.
Fig. 5.

Subaperture appearance of full-aperture y-coma. (Top) One unit of y-coma across the full aperture. (Middle) The corresponding incoherent subaperture pattern, with pistons removed, a common-mode pattern of subaperture y-tilts oppositely directed from the parent full-aperture aberration (the middle display has been rotated 180° for clarity). (Bottom) Profile plot through the surviving subaperture wavefront.

Fig. 6.
Fig. 6.

Fizeau interferometer screen capture for the ATA two-subaperture telescope system in a relatively early state of alignment, with large full-aperture defocus seen as differential x-tilt in the subapertures (see text). Local aberration diagnostics (not shown) are conveniently available, in particular x- and y- tilt in either subaperture, precisely what is needed for the technique presented in this paper.

Fig. 7.
Fig. 7.

Fizeau interferometer screen captures at good alignment showing beams from the two subapertures for the ATA (left) and FTA (right) two-element telescopes. (Left) The ATA subaperture beams are angular regions of spherical wavefronts sent out by the Fizeau, as indicated in Fig. 1. The beams shown are highly parallel and very flat, with surface 20nm rms. (Right) The FTA subaperture beams are transverse regions of a Fizeau plane wave, as the FTA is essentially a beam compressor. A common tilt has been imposed here to flatten the wavefront on the fiducial flat reference ring visible at the edge of the picture, but FTA surfaces are also flat to 20nm rms, within the design target.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

f=A·m,
m=A1·f.
A=[0.130000.0160000.016].
s=B·f,
B=[0.530.390000.130.530.390000.13].
f=B1·s.
B1=[0.9400.9401.2801.28003.8503.85].
B=[0.710.7100000.710.710.710.7100000.710.71][0.750000.550000.18000][100010001].
B6×5=[0.530.3900.200000.1300.200.530.3900.200000.1300.200.040.130000.04+0.13000].
B6×51=[0.9400.9400.070.0700003.853.8501.1401.14002.5002.507.57.501.7501.7600];
m=A1·f=A1B1·s=[(A11B11)1000(A22B22)1000(A33B33)1]·s.

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