Abstract

The transport-of-intensity equation links the intensity and phase of an optical source to the longitudinal variation of its intensity in the presence of Fresnel diffraction. This equation can be used to provide a simple, accurate spatial-phase measurement for optical testing of flat surfaces. The properties of this approach are derived. The experimental demonstration is performed by quantifying the surface variations induced by the magnetorheological finishing process on laser rods.

© 2007 Optical Society of America

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    [CrossRef]
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2006

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

2005

2004

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

2003

2002

2000

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

1998

1997

T. E. Gureyev and K. A. Nugent, "Rapid quantitative phase imaging using the transport of intensity equation," Opt. Commun. 133, 339-346 (1997).
[CrossRef]

1996

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670−1682 (1996).
[CrossRef]

1995

1990

1985

1983

1982

Bagnoud, V.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

V. Bagnoud, M. J. Guardalben, J. Puth, J. D. Zuegel, T. Mooney, and P. Dumas, "High-energy, high-average-power laser with Nd:YLF rods corrected by magnetorheological finishing," Appl. Opt. 44, 282−288 (2005).
[CrossRef] [PubMed]

V. Bagnoud, I. A. Begishev, M. J. Guardalben, J. Puth, and J. D. Zuegel, "5-hz, >250-mJ optical parametric chirped-pulse amplifier at 1053 nm," Opt. Lett. 30, 1843−1845 (2005).
[CrossRef] [PubMed]

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

Begishev, I. A.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

V. Bagnoud, I. A. Begishev, M. J. Guardalben, J. Puth, and J. D. Zuegel, "5-hz, >250-mJ optical parametric chirped-pulse amplifier at 1053 nm," Opt. Lett. 30, 1843−1845 (2005).
[CrossRef] [PubMed]

Bennett, J. M.

Bromage, J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Dalton, S.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Dorrer, C.

Dumas, P.

Duparré, A.

Ferre-Borrull, J.

Fienup, J. R.

Folnsbee, L.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Gliech, S.

Greenaway, A. H.

Guardalben, M. J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

V. Bagnoud, M. J. Guardalben, J. Puth, J. D. Zuegel, T. Mooney, and P. Dumas, "High-energy, high-average-power laser with Nd:YLF rods corrected by magnetorheological finishing," Appl. Opt. 44, 282−288 (2005).
[CrossRef] [PubMed]

V. Bagnoud, I. A. Begishev, M. J. Guardalben, J. Puth, and J. D. Zuegel, "5-hz, >250-mJ optical parametric chirped-pulse amplifier at 1053 nm," Opt. Lett. 30, 1843−1845 (2005).
[CrossRef] [PubMed]

Gureyev, T. E.

Jungquist, R.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Kang, I.

Kelly, J. H.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Kessler, T. J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Kruschwitz, B. E.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Loucks, S. J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Maywar, D. N.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

McCartney, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

McCrory, R. L.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

Meyerhofer, D. D.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Mooney, T.

Morse, S. F. B.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Notni, G.

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

T. E. Gureyev and K. A. Nugent, "Rapid quantitative phase imaging using the transport of intensity equation," Opt. Commun. 133, 339-346 (1997).
[CrossRef]

T. E. Gureyev and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination," J. Opt. Soc. Am. A 13, 1670−1682 (1996).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: Matrix solution with use of zernike polynomials," J. Opt. Soc. Am. A 12, 1932−1941 (1995).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942−1946 (1995).
[CrossRef]

Oliver, J. B.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Puth, J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

V. Bagnoud, M. J. Guardalben, J. Puth, J. D. Zuegel, T. Mooney, and P. Dumas, "High-energy, high-average-power laser with Nd:YLF rods corrected by magnetorheological finishing," Appl. Opt. 44, 282−288 (2005).
[CrossRef] [PubMed]

V. Bagnoud, I. A. Begishev, M. J. Guardalben, J. Puth, and J. D. Zuegel, "5-hz, >250-mJ optical parametric chirped-pulse amplifier at 1053 nm," Opt. Lett. 30, 1843−1845 (2005).
[CrossRef] [PubMed]

Rigatti, A. L.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Roberts, A.

Roddier, F.

Schmid, A. W.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Shoup, M. J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Steinert, J.

Stoeckl, C.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Teague, M. R.

Wall, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

Waxer, L. J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Weiner, D.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Woods, S. C.

Zuegel, J. D.

Zuegel, J. D.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Appl. Opt.

J. Microsc.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. IV France

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C.  Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. ShoupIII, D. Weiner, and J. D. Zuegel, "OMEGA EP: High-energy petawatt capability for the OMEGA laser facility," J. Phys. IV France 133, 75−80 (2006).
[CrossRef]

Opt. Commun.

T. E. Gureyev and K. A. Nugent, "Rapid quantitative phase imaging using the transport of intensity equation," Opt. Commun. 133, 339-346 (1997).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Ultramicroscopy

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, "Quantitative phase-sensitive imaging in a transmission electron microscope," Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

Other

D. Malacara, "Optical Shop Testing," 2nd ed., Wiley series in Pure and Applied Optics (Wiley, New York, 1992).

D. Golini, S. Jacobs, W. Kordonski, and P. Dumas, "Precision optics fabrication using magnetorheological finishing," in Advanced Materials for Optics and Precision Structures, M. A. Ealey, R. A. Paquin, and T. B. Parsonage, eds., Critical Reviews of Optical Science and Technology (SPIE, Bellingham, WA, 1997), Vol. CR67, pp. 251−274.

Supplementary Material (1)

» Media 1: AVI (782 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

(a). Notations for the TIE. The intensity of the wave under test propagating along the z axis is measured as a function of x and y in two closely spaced planes at z 0 and z 0 + dz. (b) Algorithm for phase reconstruction from the measured intensity.

Fig. 2.
Fig. 2.

(a). Phase reconstructed for a 0.2-mm period and 0.1-rad amplitude for propagation distances of 1 mm and 1 cm; (b) phase reconstructed for a 0.2-mm period and 1-rad amplitude for propagation distances of 1 mm and 1 cm; (c) normalized intensity [I(x,dz) -I 0]/(I 0 dz) calculated for a 0.2-mm period and 1-rad amplitude for propagation distances of 1 mm and 1 cm. For short propagation distance, the sinusoidal phase leads to a sinusoidal intensity, while for longer propagation distances, higher harmonics are present in the intensity.

Fig. 3.
Fig. 3.

(a). rms error obtained for a propagation distance dz = 1 mm; (b) rms error obtained for a propagation distance dz = 1 cm; (c) rms error obtained for a propagation distance dz = 10 cm. In the absence of noise, the reconstruction accuracy decreases for increasing frequency of the phase under test. Longer propagation distances lead to lower reconstruction accuracy.

Fig. 4.
Fig. 4.

rms error obtained (a) for a propagation distance dz = 1 cm for additive noise of standard deviation 10−4, (b) for a propagation distance dz = 1 cm for additive noise of standard deviation 10−3, and (c) for a propagation distance dz = 10 cm for additive noise of standard deviation 10−3. For the sake of clarity, the rms error was limited to the maximal rms error in the absence of noise for each distance. In the presence of noise, the reconstruction accuracy decreases for decreasing amplitude of the phase under test.

Fig. 5.
Fig. 5.

(a). Layout for the characterization of the surface modulation of a laser rod using the TIE. The virtual distance between the image of the surface and the detector is changed by longitudinally translating the CCD camera used to spatially resolve the intensity. (b) Peak-to-valley of the reconstructed phase as a function of the distance dz.

Fig. 6.
Fig. 6.

Animation of the spatially resolved intensity as a function of the distance from the surface under test to the camera (0.78 MB). Intensity modulations increase when the propagation distance is increased. [Media 1]

Fig. 7.
Fig. 7.

The three columns represent the measured intensity (first column), calculated phase (second column), and calculated intensity (third column) for propagation distances equal to 12 mm (first line), dz = 24 mm (second line), and dz = 36 mm (third line).

Fig. 8.
Fig. 8.

(a). Two-dimensional PSD of the intensity measured for dz = 24 mm in the case of the reference face; (b) two-dimensional PSD of the intensity measured for dz = 24 mm in the case of the face with MRF processing; (c) PSD of the intensity integrated over kx measured for dz = 12 mm, dz = 24 mm, and dz = 36 mm.

Equations (8)

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i E z + 1 2 k 0 Δ E + k 0 E = 0 .
[ I φ ] = 2 π λ 0 I z ,
[ I x y z 0 x y z 0 ] = 2 π λ 0 I x y z 0 + dz I x y z 0 dz ,
Δ φ x y z 0 = 2 π λ 0 dz [ I x y z 0 + dz I 0 1 ] .
( k x 2 + k y 2 ) FT [ φ ] k x k y z 0 = 2 π λ 0 dz [ FT [ I ] k x k y z 0 + dz I 0 δ k x k y ] ,
φ = 2 π λ 0 dz IFT { 1 k x 2 + k y 2 [ FT [ I ] k x k y z 0 + dz I 0 δ k x k y ] } ,
FT [ φ ] k x k y z 0 2 = ( 2 π I 0 λ 0 dz ) 2 1 ( k x 2 + k y 2 ) 2 FT [ I ] k x k y z 0 + dz 2 .
ε rms = x 1 x 2 [ φ reconstructed ( x ) φ test ( x ) ] 2 dx 2 x 1 x 2 [ φ test ( x ) ] 2 dx .

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