Abstract

We present an analytical derivation of the coupling parameter relating the angle between two interfering beams in a heterodyne interferometer to the differential phase signals detected by a quadrant photodiode. This technique, also referred to as differential wavefront sensing, is commonly used in space-based gravitational wave detectors to determine the attitude of a test mass in one of the interferometer arms from the quadrant diode signals. Successive approximations to the analytical expression are made to simplify the investigation of parameter dependencies. Motivated by our findings, we propose what we believe to be a new measurement method to accurately determine the absolute wavefront curvature of a single measurement beam. We also investigate the change in the coupling parameter when the interferometer “test mirror” is moved from its nominal position, an effect which mediates the coupling of mirror displacement noise into differential phase measurements.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  7. C. M. Wu, S. T. Lin, and J. Fu, “Heterodyne interferometer with two spatially separated polarization beams for nanometrology,” Opt. Quantum Electron. 34, 1267–1276 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.
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  15. H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
    [CrossRef]

2008 (1)

D. A. Shaddock, “Space-based gravitational wave detection with LISA,” Class. Quantum Grav. 25, 114012 (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (3)

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern approaches in phase measuring metrology,” Proc. SPIE 5856, 14–22 (2005).
[CrossRef]

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

2004 (1)

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

2003 (1)

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

2002 (1)

C. M. Wu, S. T. Lin, and J. Fu, “Heterodyne interferometer with two spatially separated polarization beams for nanometrology,” Opt. Quantum Electron. 34, 1267–1276 (2002).
[CrossRef]

1988 (1)

1953 (1)

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Arai, K.

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

Audley, H.

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

Babcock, H. W.

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Barbero, S.

Berger, M.

D. Weise, P. Marenaci, P. Weimer, M. Berger, H. R. Schulte, P. Gath, and U. Johann, “Opto-mechanical architecture of the LISA instrument,” in Proceedings of the 7th International Conference on Space Optics (ICSO); www.icsoconference2008.com (2008).

Braxmaier, C.

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

Brock, N.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern approaches in phase measuring metrology,” Proc. SPIE 5856, 14–22 (2005).
[CrossRef]

Caldwell, M.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

Campbell, H. I.

H. I. Campbell and A. H. Greenaway, “Wavefront sensing: From historical roots to state-of-the-art,” in Astronomy with High Contrast Imaging III: Instrumental Techniques, Modelling and Data Processing, EAS Publications Series, M.Carbillet, ed. (EDP Sciences, 2006), Vol. 22, pp. 165–185.

Casey, M. M.

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

Danzmann, K.

F. Guzman Cervantes, G. Heinzel, A. F. García Marín, V. Wand, F. Steier, O. Jennrich, and K. Danzmann, “Real-time phase-front detector for heterodyne interferometers,” Appl. Opt. 46, 4541–4548 (2007).
[CrossRef]

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

Draaisma, F.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

Freise, A.

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

Fu, J.

C. M. Wu, S. T. Lin, and J. Fu, “Heterodyne interferometer with two spatially separated polarization beams for nanometrology,” Opt. Quantum Electron. 34, 1267–1276 (2002).
[CrossRef]

Garcia, A.

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

Garcia Marin, A. F.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

García Marín, A. F.

Garcia-Marin, A.

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

Gath, P.

D. Weise, P. Marenaci, P. Weimer, M. Berger, H. R. Schulte, P. Gath, and U. Johann, “Opto-mechanical architecture of the LISA instrument,” in Proceedings of the 7th International Conference on Space Optics (ICSO); www.icsoconference2008.com (2008).

Gerndt, R.

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

Gossler, S.

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

Greenaway, A. H.

H. I. Campbell and A. H. Greenaway, “Wavefront sensing: From historical roots to state-of-the-art,” in Astronomy with High Contrast Imaging III: Instrumental Techniques, Modelling and Data Processing, EAS Publications Series, M.Carbillet, ed. (EDP Sciences, 2006), Vol. 22, pp. 165–185.

Grote, H.

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

Guzman Cervantes, F.

Hayes, J.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern approaches in phase measuring metrology,” Proc. SPIE 5856, 14–22 (2005).
[CrossRef]

Hechenblaikner, G.

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

Heinzel, G.

F. Guzman Cervantes, G. Heinzel, A. F. García Marín, V. Wand, F. Steier, O. Jennrich, and K. Danzmann, “Real-time phase-front detector for heterodyne interferometers,” Appl. Opt. 46, 4541–4548 (2007).
[CrossRef]

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

Hough, J.

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

Jennrich, O.

F. Guzman Cervantes, G. Heinzel, A. F. García Marín, V. Wand, F. Steier, O. Jennrich, and K. Danzmann, “Real-time phase-front detector for heterodyne interferometers,” Appl. Opt. 46, 4541–4548 (2007).
[CrossRef]

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

Johann, U.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

D. Weise, P. Marenaci, P. Weimer, M. Berger, H. R. Schulte, P. Gath, and U. Johann, “Opto-mechanical architecture of the LISA instrument,” in Proceedings of the 7th International Conference on Space Optics (ICSO); www.icsoconference2008.com (2008).

Killow, C.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

Kimbrough, B.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern approaches in phase measuring metrology,” Proc. SPIE 5856, 14–22 (2005).
[CrossRef]

Lin, S. T.

C. M. Wu, S. T. Lin, and J. Fu, “Heterodyne interferometer with two spatially separated polarization beams for nanometrology,” Opt. Quantum Electron. 34, 1267–1276 (2002).
[CrossRef]

Lueck, H.

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

Luetzow-Wentzky, P.

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

Marenaci, P.

D. Weise, P. Marenaci, P. Weimer, M. Berger, H. R. Schulte, P. Gath, and U. Johann, “Opto-mechanical architecture of the LISA instrument,” in Proceedings of the 7th International Conference on Space Optics (ICSO); www.icsoconference2008.com (2008).

Middleton, K.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

Millerd, J.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern approaches in phase measuring metrology,” Proc. SPIE 5856, 14–22 (2005).
[CrossRef]

Nofrarias, M.

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

North-Morris, M.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern approaches in phase measuring metrology,” Proc. SPIE 5856, 14–22 (2005).
[CrossRef]

Novak, M.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern approaches in phase measuring metrology,” Proc. SPIE 5856, 14–22 (2005).
[CrossRef]

Robertson, D.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

Roddier, F.

Rubinstein, J.

Ruediger, A.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

Schilling, R.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

Schulte, H. R.

D. Weise, P. Marenaci, P. Weimer, M. Berger, H. R. Schulte, P. Gath, and U. Johann, “Opto-mechanical architecture of the LISA instrument,” in Proceedings of the 7th International Conference on Space Optics (ICSO); www.icsoconference2008.com (2008).

Shaddock, D. A.

D. A. Shaddock, “Space-based gravitational wave detection with LISA,” Class. Quantum Grav. 25, 114012 (2008).
[CrossRef]

Steier, F.

F. Guzman Cervantes, G. Heinzel, A. F. García Marín, V. Wand, F. Steier, O. Jennrich, and K. Danzmann, “Real-time phase-front detector for heterodyne interferometers,” Appl. Opt. 46, 4541–4548 (2007).
[CrossRef]

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

Strain, K. A.

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

te Plate, M.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

Thibos, L. N.

Wand, V.

F. Guzman Cervantes, G. Heinzel, A. F. García Marín, V. Wand, F. Steier, O. Jennrich, and K. Danzmann, “Real-time phase-front detector for heterodyne interferometers,” Appl. Opt. 46, 4541–4548 (2007).
[CrossRef]

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

Ward, H.

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

Weimer, P.

D. Weise, P. Marenaci, P. Weimer, M. Berger, H. R. Schulte, P. Gath, and U. Johann, “Opto-mechanical architecture of the LISA instrument,” in Proceedings of the 7th International Conference on Space Optics (ICSO); www.icsoconference2008.com (2008).

Weise, D.

D. Weise, P. Marenaci, P. Weimer, M. Berger, H. R. Schulte, P. Gath, and U. Johann, “Opto-mechanical architecture of the LISA instrument,” in Proceedings of the 7th International Conference on Space Optics (ICSO); www.icsoconference2008.com (2008).

Wilke, B.

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

Wu, C. M.

C. M. Wu, S. T. Lin, and J. Fu, “Heterodyne interferometer with two spatially separated polarization beams for nanometrology,” Opt. Quantum Electron. 34, 1267–1276 (2002).
[CrossRef]

Wyant, J. C.

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern approaches in phase measuring metrology,” Proc. SPIE 5856, 14–22 (2005).
[CrossRef]

Zhao, F.

F. Zhao, “Picometer laser metrology for the Space Interferometry Mission,” in Conference on Lasers and Electro-Optics (CLEO), Vol. 96 of Trends in Optics and Photonics Series (Optical Society of America, 2004).

Appl. Opt. (2)

Class. Quantum Grav. (5)

D. A. Shaddock, “Space-based gravitational wave detection with LISA,” Class. Quantum Grav. 25, 114012 (2008).
[CrossRef]

H. Grote, G. Heinzel, A. Freise, S. Gossler, B. Wilke, H. Lueck, H. Ward, M. M. Casey, K. A. Strain, D. Robertson, J. Hough, and K. Danzmann, “Alignment control of GEO600,” Class. Quantum Grav. 21, S441–S449 (2004).
[CrossRef]

G. Heinzel, C. Braxmaier, R. Schilling, A. Ruediger, D. Robertson, M. te Plate, V. Wand, K. Arai, U. Johann, and K. Danzmann, “Interferometry for the LISA technology package (LTP) aboard SMART-2,” Class. Quantum Grav. 20, S153–S161 (2003).
[CrossRef]

D. Robertson, C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier, “LTP interferometer—noise sources and performance,” Class. Quantum Grav. 22, S155–S163 (2005).
[CrossRef]

G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. F. Garcia Marin, J. Hough, O. Jennrich, U. Johann, C. Killow, K. Middleton, M. te Plate, D. Robertson, A. Ruediger, R. Schilling, F. Steier, V. Wand, and H. Ward, “Successful testing of the LISA Technology Package (LTP) interferometer engineering model,” Class. Quantum Grav. 22, S149–S154 (2005).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

C. M. Wu, S. T. Lin, and J. Fu, “Heterodyne interferometer with two spatially separated polarization beams for nanometrology,” Opt. Quantum Electron. 34, 1267–1276 (2002).
[CrossRef]

Proc. SPIE (1)

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris, and J. C. Wyant, “Modern approaches in phase measuring metrology,” Proc. SPIE 5856, 14–22 (2005).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Other (4)

H. I. Campbell and A. H. Greenaway, “Wavefront sensing: From historical roots to state-of-the-art,” in Astronomy with High Contrast Imaging III: Instrumental Techniques, Modelling and Data Processing, EAS Publications Series, M.Carbillet, ed. (EDP Sciences, 2006), Vol. 22, pp. 165–185.

G. Hechenblaikner, R. Gerndt, U. Johann, P. Luetzow-Wentzky, V. Wand, H. Audley, K. Danzmann, A. Garcia-Marin, G. Heinzel, M. Nofrarias, and F. Steier, “Coupling characterization and noise studies of the Optical Metrology System on-board the LISA Pathfinder mission” (submitted to Appl. Opt.); preprint on arXiv:1006.2122.

F. Zhao, “Picometer laser metrology for the Space Interferometry Mission,” in Conference on Lasers and Electro-Optics (CLEO), Vol. 96 of Trends in Optics and Photonics Series (Optical Society of America, 2004).

D. Weise, P. Marenaci, P. Weimer, M. Berger, H. R. Schulte, P. Gath, and U. Johann, “Opto-mechanical architecture of the LISA instrument,” in Proceedings of the 7th International Conference on Space Optics (ICSO); www.icsoconference2008.com (2008).

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Figures (5)

Fig. 1
Fig. 1

Experimental setup to measure the absolute curvature of the measurement beam wavefront. Mirrors 1 and 2 are successively rotated in the opposite sense of rotation. A thick line is printed to indicate the lever arm from mirror 1 to the photodiode. The path to the spare photodiode is printed as a dotted line.

Fig. 2
Fig. 2

Reference frames of the measurement beam (axes with tilde) and the reference beam (axes without tilde). The two beams propagate along z ̃ and z, respectively, and interfere on the plane described by z = 0 .

Fig. 3
Fig. 3

DWS signals, normalized by their value for σ 2 = 0 , are plotted against σ 2 while the value of k w eff φ is kept constant. The exact result is given by the solid line, the approximate results are given by the dotted lines, and the linearized approximate result is given by the dashed line.

Fig. 4
Fig. 4

Schematic of the interferometer described in [9]. The measurement beam is reflected from the test mirror at an angle α (solid line). When the mirror is moved the beam is displaced (dashed line). Its center on the photodiode can be aligned with its original position when the mirror is slightly tilted (dotted line).

Fig. 5
Fig. 5

Relative change in coupling coefficients D ϕ when the test mirror in the setup of Fig. 4 is moved. The change in coefficients for increasing beam displacement is given by the dashed line. The dotted line is given for the case where the beam displacement is compensated by adjusting the test mirror and increasing the angle between reference and measurement beams. The solid line is given for the case of α = 0 (no beam displacement).

Equations (24)

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A = 1 w ( z ) 2 π e ( x 2 + y 2 ) / w 2 ( z ) e i k ( x 2 + y 2 ) / ( 2 R ) e i k z ,
w 2 ( z ) = w 0 2 ( 1 + z 2 z R 2 ) ,
R ( z ) = z ( 1 + z R 2 z 2 ) .
A r = C r e ( x 2 + y 2 ) / w r 2 ( z ) e i k ( x 2 + y 2 ) / 2 R r ( z ) e i k z ,
A m = C m e ( x ̃ 2 + y ̃ 2 ) / w m 2 ( z ̃ ) e i k ( x ̃ 2 + y ̃ 2 ) / 2 R m ( z ̃ ) e i k z ̃ ,
( x ̃ y ̃ z ̃ ) = ( cos   φ 0 sin   φ 0 1 0 sin   φ 0 cos   φ ) ( x y z ) ( x 0 0 0 ) .
A m = e [ ( x x o ) 2 + y 2 ] / w m 2 ( z ) e i k [ ( x x o ) 2 + y 2 ] / 2 R m ( z ) e i k z e i k x φ .
F left = d y 0 d x A r A m = d y { } 0 d x e 2 x 2 / w eff 2 e i k ( x 2 / 2 R rel ) e i k x φ ( 1 z t m / R m ) e i k x ( x 0 s / R m ) ,
2 w eff 2 = 1 w m 2 + 1 w r 2 ,
1 R rel = 1 R r 1 R m ,
x 0 = x 0 s + φ z t m .
D W S φ = arg ( F left F right ) = arg ( d y 0 d x A r A m d y 0 d x A r A m ) = arg ( 0 d x e 2 x 2 / w eff 2 e i k ( x 2 / 2 R rel ) e i k x φ ( 1 z t m / R m ) e i k x ( x 0 s / R m ) 0 d x e 2 x 2 / w eff 2 e i k ( x 2 / 2 R rel ) e i k x φ ( 1 z t m / R m ) e i k x ( x 0 s / R m ) ) .
D W S φ = arg ( 1 + i α 1 i α ) ,
α = erfi ( w eff 2 3 / 2 ( k φ ( 1 z t m R m ) k x 0 s R m ) 1 + i σ 1 + σ 2 ) ,
σ = k w eff 2 4 R rel ,
D W S φ = 2 π ( k w eff φ ( 1 z t m R m ) k w eff x 0 s R m ) F ( σ ) + O ( φ 2 , x 0 s 2 ) ,
F ( σ ) = 1 2 1 + 1 + σ 2 1 + σ 2 ,
D ϕ m 1 = d ( D W S ϕ ) d ϕ = 2 π 4 w eff λ ( 1 z t m R m ) F ( σ ) .
β = D ϕ m 1 D ϕ m 2 = 1 z m 1 / R m 1 z m 2 / R m .
R m = z m 1 β z m 2 1 β .
Δ β = D ϕ m 1 D ϕ m 2 ( Δ D ϕ m 1 D ϕ m 1 ) 2 + ( Δ D ϕ m 2 D ϕ m 2 ) 2 .
Δ R m = Δ β 1 β ( R m z m 2 ) = 0.08   m .
φ ( w ) = k w 2 2 R m 2 π 3 λ 3 ,
Δ φ ( w ) = k w 2 2 R m Δ R m R m λ 40 .

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