Abstract

We present an analysis of the accuracy and information content of three-dimensional reconstructions of the dielectric susceptibility of a sample from noisy, near-field holographic measurements, such as those made in scanning probe microscopy. Holographic measurements are related to the dielectric susceptibility via a linear operator within the accuracy of the first Born approximation. The maximum-likelihood reconstruction of the dielectric susceptibility is expressed as a linear combination of basis functions determined by singular value decomposition of the weighted measurement operator. Maximum a posteriori estimates based on prior information are also discussed. Semianalytical expressions are given for the likely error due to measurement noise in the basis function coefficients, resulting in effective resolution limits in all three dimensions. These results are illustrated by numerical examples.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. P. S. Carney and J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. D. G. Fischer, “The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures,” J. Mod. Opt. 47, 1359–1374 (2000).
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    [CrossRef]
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    [CrossRef]
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  31. C. W. Helstrom, Elements of Signal Detection & Estimation (Prentice Hall, 1995).
  32. G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
    [CrossRef]
  33. P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
    [CrossRef]
  34. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).
  35. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd revised ed. (McGraw-Hill, 1986).
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    [CrossRef]
  37. J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography based on volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
    [CrossRef]
  38. F. Natterer, The Mathematics of Computerized Tomography (SIAM, 2001).
    [CrossRef]

2009

J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography based on volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
[CrossRef]

2007

2006

G. Y. Panasyuk, V. A. Markel, P. S. Carney, and J. C. Schotland, “Nonlinear inverse scattering and three-dimensional near-field optical imaging,” Appl. Phys. Lett. 89, 221116 (2006).
[CrossRef]

A. Sentenac, P. C. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97, 243901 (2006).
[CrossRef]

2005

2004

2003

2002

P. S. Carney and J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A: Pure Appl. Opt. 4, S140–S144 (2002).
[CrossRef]

2001

2000

D. G. Fischer, “Sub-wavelength depth resolution in near-field microscopy,” Opt. Lett. 25, 1529–1531 (2000).
[CrossRef]

P. S. Carney and J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

D. G. Fischer, “The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures,” J. Mod. Opt. 47, 1359–1374 (2000).

1999

A. K. Louis, “A unified approach to regularization methods for linear ill-posed problems,” Inverse Probl. 15, 489–498 (1999).
[CrossRef]

1998

1997

J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

1995

1992

E. Betzig and J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
[CrossRef]

1989

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

1984

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Ä spatial resolution light microscope. I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

1972

E. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

1928

E. Synge, “A suggested method for extending microscopic resolution into the ultra-microscopic region,” Phil. Mag. 6, 356–362 (1928).

Ash, E.

E. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

Belkebir, K.

Bertaux, N.

Betzig, E.

E. Betzig and J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Boltasseva, A.

P. S. Carney, R. A. Frazin, S. Bozhevolnyi, V. S. Volkov, A. Boltasseva, and J. C. Schotland, “A computational lens for the near field,” Phys. Rev. Lett. 92, 163903 (2004).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

Bozhevolnyi, S.

P. S. Carney, R. A. Frazin, S. Bozhevolnyi, V. S. Volkov, A. Boltasseva, and J. C. Schotland, “A computational lens for the near field,” Phys. Rev. Lett. 92, 163903 (2004).
[CrossRef] [PubMed]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd revised ed. (McGraw-Hill, 1986).

Brancaccio, A.

Carminati, R.

J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Carney, P. S.

J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography based on volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
[CrossRef]

J. Sun, J. C. Schotland, and P. S. Carney, “Strong probe effects in near-field optics,” J. Appl. Phys. 102, 103103 (2007).
[CrossRef]

G. Y. Panasyuk, V. A. Markel, P. S. Carney, and J. C. Schotland, “Nonlinear inverse scattering and three-dimensional near-field optical imaging,” Appl. Phys. Lett. 89, 221116 (2006).
[CrossRef]

D. L. Marks and P. S. Carney, “Near-field diffractive elements,” Opt. Lett. 30, 1870–1872 (2005).
[CrossRef] [PubMed]

R. A. Frazin, D. G. Fischer, and P. S. Carney, “Information content of the near field: two-dimensional samples,” J. Opt. Soc. Am. A 21, 1050–1057 (2004).
[CrossRef]

P. S. Carney, R. A. Frazin, S. Bozhevolnyi, V. S. Volkov, A. Boltasseva, and J. C. Schotland, “A computational lens for the near field,” Phys. Rev. Lett. 92, 163903 (2004).
[CrossRef] [PubMed]

P. S. Carney and J. C. Schotland, “Theory of total-internal-reflection tomography,” J. Opt. Soc. Am. A 20, 542–547(2003).
[CrossRef]

P. S. Carney and J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A: Pure Appl. Opt. 4, S140–S144 (2002).
[CrossRef]

P. S. Carney and J. C. Schotland, “Three-dimensional total internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
[CrossRef]

P. S. Carney and J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

Chaumet, P. C.

Courjon, D.

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

Demoment, G.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

Devaney, A. J.

Drsek, F.

Fessler, J. A.

Fischer, D. G.

Frazin, R. A.

P. S. Carney, R. A. Frazin, S. Bozhevolnyi, V. S. Volkov, A. Boltasseva, and J. C. Schotland, “A computational lens for the near field,” Phys. Rev. Lett. 92, 163903 (2004).
[CrossRef] [PubMed]

R. A. Frazin, D. G. Fischer, and P. S. Carney, “Information content of the near field: two-dimensional samples,” J. Opt. Soc. Am. A 21, 1050–1057 (2004).
[CrossRef]

Garcia, N.

Giovannini, H.

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Greffet, J.-J.

J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Gúerin, C.-A.

Guo, P.

Hansen, P. C.

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
[CrossRef]

Harootunian, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Ä spatial resolution light microscope. I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Helstrom, C. W.

C. W. Helstrom, Elements of Signal Detection & Estimation (Prentice Hall, 1995).

Hillenbrand, R.

J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography based on volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
[CrossRef]

Isaacson, M.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Ä spatial resolution light microscope. I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Leone, G.

R. Solimene, G. Leone, and R. Pierri, “Resolution in two-dimensional tomographic reconstructions in the Fresnel zone from Born scattered fields,” J. Opt. A: Pure Appl. Opt. 6, 529–536 (2004).
[CrossRef]

A. Brancaccio, G. Leone, and R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[CrossRef]

Lewis, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Ä spatial resolution light microscope. I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Liseno, A.

Louis, A. K.

A. K. Louis, “A unified approach to regularization methods for linear ill-posed problems,” Inverse Probl. 15, 489–498 (1999).
[CrossRef]

Markel, V. A.

G. Y. Panasyuk, V. A. Markel, P. S. Carney, and J. C. Schotland, “Nonlinear inverse scattering and three-dimensional near-field optical imaging,” Appl. Phys. Lett. 89, 221116 (2006).
[CrossRef]

Marks, D. L.

Moon, T. K.

T. K. Moon and W. C. Sterling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, 2000).

Muray, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Ä spatial resolution light microscope. I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (SIAM, 2001).
[CrossRef]

Nicholls, G.

E. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

Nieto-Vesperinas, M.

Panasyuk, G. Y.

G. Y. Panasyuk, V. A. Markel, P. S. Carney, and J. C. Schotland, “Nonlinear inverse scattering and three-dimensional near-field optical imaging,” Appl. Phys. Lett. 89, 221116 (2006).
[CrossRef]

Pierri, R.

Sarayeddine, K.

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

Schotland, J. C.

J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography based on volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
[CrossRef]

J. Sun, J. C. Schotland, and P. S. Carney, “Strong probe effects in near-field optics,” J. Appl. Phys. 102, 103103 (2007).
[CrossRef]

G. Y. Panasyuk, V. A. Markel, P. S. Carney, and J. C. Schotland, “Nonlinear inverse scattering and three-dimensional near-field optical imaging,” Appl. Phys. Lett. 89, 221116 (2006).
[CrossRef]

P. S. Carney, R. A. Frazin, S. Bozhevolnyi, V. S. Volkov, A. Boltasseva, and J. C. Schotland, “A computational lens for the near field,” Phys. Rev. Lett. 92, 163903 (2004).
[CrossRef] [PubMed]

P. S. Carney and J. C. Schotland, “Theory of total-internal-reflection tomography,” J. Opt. Soc. Am. A 20, 542–547(2003).
[CrossRef]

P. S. Carney and J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A: Pure Appl. Opt. 4, S140–S144 (2002).
[CrossRef]

P. S. Carney and J. C. Schotland, “Three-dimensional total internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
[CrossRef]

P. S. Carney and J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

Sentenac, A.

Soldovieri, F.

Solimene, R.

R. Solimene, G. Leone, and R. Pierri, “Resolution in two-dimensional tomographic reconstructions in the Fresnel zone from Born scattered fields,” J. Opt. A: Pure Appl. Opt. 6, 529–536 (2004).
[CrossRef]

R. Pierri, A. Liseno, F. Soldovieri, and R. Solimene, “In-depth resolution for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
[CrossRef]

Sotthivirat, S.

Spajer, M.

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

Sterling, W. C.

T. K. Moon and W. C. Sterling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, 2000).

Sun, J.

J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography based on volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
[CrossRef]

J. Sun, J. C. Schotland, and P. S. Carney, “Strong probe effects in near-field optics,” J. Appl. Phys. 102, 103103 (2007).
[CrossRef]

Synge, E.

E. Synge, “A suggested method for extending microscopic resolution into the ultra-microscopic region,” Phil. Mag. 6, 356–362 (1928).

Trautman, J. K.

E. Betzig and J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

Volkov, V. S.

P. S. Carney, R. A. Frazin, S. Bozhevolnyi, V. S. Volkov, A. Boltasseva, and J. C. Schotland, “A computational lens for the near field,” Phys. Rev. Lett. 92, 163903 (2004).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

Appl. Phys. Lett.

P. S. Carney and J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

G. Y. Panasyuk, V. A. Markel, P. S. Carney, and J. C. Schotland, “Nonlinear inverse scattering and three-dimensional near-field optical imaging,” Appl. Phys. Lett. 89, 221116 (2006).
[CrossRef]

J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography based on volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

Inverse Probl.

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
[CrossRef]

A. K. Louis, “A unified approach to regularization methods for linear ill-posed problems,” Inverse Probl. 15, 489–498 (1999).
[CrossRef]

J. Appl. Phys.

J. Sun, J. C. Schotland, and P. S. Carney, “Strong probe effects in near-field optics,” J. Appl. Phys. 102, 103103 (2007).
[CrossRef]

J. Mod. Opt.

D. G. Fischer, “The information content of weakly scattered fields: implications for near-field imaging of three-dimensional structures,” J. Mod. Opt. 47, 1359–1374 (2000).

J. Opt. A: Pure Appl. Opt.

P. S. Carney and J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A: Pure Appl. Opt. 4, S140–S144 (2002).
[CrossRef]

R. Solimene, G. Leone, and R. Pierri, “Resolution in two-dimensional tomographic reconstructions in the Fresnel zone from Born scattered fields,” J. Opt. A: Pure Appl. Opt. 6, 529–536 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Nature

E. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

Opt. Commun.

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

Opt. Express

Opt. Lett.

Phil. Mag.

E. Synge, “A suggested method for extending microscopic resolution into the ultra-microscopic region,” Phil. Mag. 6, 356–362 (1928).

Phys. Rev. Lett.

P. S. Carney, R. A. Frazin, S. Bozhevolnyi, V. S. Volkov, A. Boltasseva, and J. C. Schotland, “A computational lens for the near field,” Phys. Rev. Lett. 92, 163903 (2004).
[CrossRef] [PubMed]

A. Sentenac, P. C. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97, 243901 (2006).
[CrossRef]

Prog. Surf. Sci.

J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Science

E. Betzig and J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Ultramicroscopy

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Ä spatial resolution light microscope. I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

T. K. Moon and W. C. Sterling, Mathematical Methods and Algorithms for Signal Processing (Prentice-Hall, 2000).

J. W. Goodman, Statistical Optics (Wiley, 1985).

C. W. Helstrom, Elements of Signal Detection & Estimation (Prentice Hall, 1995).

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd revised ed. (McGraw-Hill, 1986).

F. Natterer, The Mathematics of Computerized Tomography (SIAM, 2001).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Log of the singular values of the weighted measurement operator K as a function of index j for cases (a) A1 and (c) A2 is shown. In both cases, the singular value spectra are plotted for three spatial frequencies, | q | = 0.5 k 0 , 3 k 0 , and 9 k 0 . The log of the first-, third-, and ninth-largest singular values of the weighted measurement operator K as a function of spatial frequency | q | for cases (b) A1 and (d) A2 is also shown.

Fig. 2
Fig. 2

Absolute values of the singular functions | f + ( z ) | for the first-, third-, and ninth-largest singular values of the weighted measurement operator K for case A1 and spatial frequencies (a) | q | = 0.5 k 0 and (b) | q | = 9 k 0 , and case A2 and spatial frequencies (c) | q | = 0.5 k 0 and (d) | q | = 9 k 0 .

Fig. 3
Fig. 3

Absolute value of the δ function response for case A1 and two values of the truncation threshold, SNR j 1 and SNR j 100 . The left-hand column illustrates the point response for SNR j 1 at three different locations in the sample: (a) lower, z / λ = 0 ; (c) middle, z / λ = 0.25 ; and (e) upper, z / λ = 0.5 , as well as (g) the number of singular values used in the reconstruction as a function of spatial frequency. The right-hand column illustrates the corresponding quantities for the case SNR j 100 .

Fig. 4
Fig. 4

Absolute value of the δ function response for case A2 and two values of the truncation threshold, SNR j 1 and SNR j 100 . The left-hand column illustrates the point response for SNR j 1 at three different locations in the sample: (a) lower, z / λ = 0 ; (c) middle, z / λ = 0.25 ; and (e) upper, z / λ = 0.5 , as well as (g) the number of singular values used in the reconstruction as a function of spatial frequency.

Fig. 5
Fig. 5

Log of the singular values of the weighted measurement operator K as a function of index j for cases (a) B1 and (c) B2 is shown. In both cases, the singular value spectra are plotted for three spatial frequencies, | q | = 0.5 k 0 , 3 k 0 , and 9 k 0 . The log of the first-, third-, and ninth-largest singular values of the weighted measurement operator K as a function of spatial frequency | q | for cases (b) B1 and (d) B2 is also shown.

Fig. 6
Fig. 6

Absolute values of the singular functions | f + ( z ) | for the first-, third-, and ninth-largest singular values of the weighted measurement operator K for case B1 and spatial frequencies (a) | q | = 0.5 k 0 and (b) | q | = 9 k 0 , and case B2 and spatial frequencies (c) | q | = 0.5 k 0 and (d) | q | = 9 k 0 .

Fig. 7
Fig. 7

Absolute value of the point response for case B1 and two values of the truncation threshold, SNR j 1 and SNR j 100 . The left-hand column illustrates the point response for SNR j 1 at three different locations in the sample: (a) lower, z / λ = 0 ; (c) middle, z / λ = 0.25 ; and (e) upper, z / λ = 0.5 , as well as (g) the number of singular values used in the reconstruction as a function of spatial frequency. The right-hand column illustrates the corresponding quantities for the case SNR j 100 .

Fig. 8
Fig. 8

Absolute value of the point response for case B2 and two values of the truncation threshold, SNR j 1 and SNR j 100 . The left-hand column illustrates the point response for SNR j 1 at three different locations in the sample: (a) lower, z / λ = 0 ; (c) middle, z / λ = 0.25 ; and (e) upper, z / λ = 0.5 , as well as (g) the number of singular values used in the reconstruction as a function of spatial frequency.

Tables (1)

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Table 1 Parameters for the Numerical Examples Discussed in the Text a

Equations (52)

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× × E ( r ) n 2 ( z ) k 0 2 E ( r ) = 4 π k 0 2 η ( r ) E ( r ) ,
k ( q ) = [ q , k z ( q ) ] ,
k z ( q ) = k 0 2 | q | 2 .
E i ( r ) = A i e ^ i exp [ i k ( q i ) · r ] ,
E α s ( r ) = k 0 2 A i d 3 r η ( r ) G α β ( r , r ) e ^ β i exp [ i q i · ρ + i k z ( q i ) z ] ,
E r ( r ) = A r e ^ r exp [ i k ( q r ) · r ] .
I total ( ρ , z d ) = E α r E α r * + E α s E α s * + E α s E α r * + E α r E α s * ,
I ( ρ , z d ) E α s E α r * + E α r E α s * .
I ˜ ( q , z d ) 1 2 π d 2 ρ I ( ρ , z d ) exp [ i q · ρ ] .
I ˜ ( q ) = 0 Δ z [ H ( q , q r , q i , z ) η ˜ ( q + q i q r , z ) + H * ( q , q r , q i , z ) η ˜ * ( q + q i q r , z ) ] d z ,
H ( q , q r , q i , z ) = i 2 π k 0 2 A r A i e ^ β r * e ^ α i h α β ( q r q , z ) k z ( q r q ) r × exp { i [ k z ( q r q ) k z * ( q r ) ] z d + i [ k z ( q i ) k z ( q r q ) ] z } ,
H ( q , z ) [ H ( q , q 0 i , q 0 i , z ) H * ( q , q 0 i , q 0 i , z ) H ( q , q P 1 i , q P 1 i , z ) H * ( q , q P 1 i , q P 1 i , z ) ] ,
I ˜ ( q ) I ˜ 0 ( q ) I ˜ P 1 ( q ) ,
η ˜ ( q , z ) η ˜ ( q , z ) η ˜ * ( q , z ) .
η ˜ H ( q , z ) η ˜ ( q , z ) = 0 Δ z d z { | η ˜ ( q , z ) | 2 + | η ˜ ( q , z ) | 2 } ,
I ˜ ( q ) = [ H η ˜ ] ( q ) = 0 Δ z d z H ( q , z ) η ˜ ( q , z ) ,
I ˜ ( q ) = H ( q ) η ˜ + ( q ) .
β n = 2 π L n , 0 n M 1 2 , = 2 π L [ n M ] , M 1 2 + 1 n M 1 ,
c ˜ j , m n = 1 M 2 k = 0 M 1 l = 0 M 1 c j , k l exp { 2 π i [ k m M + l n M ] } .
ln P c ˜ ( c ˜ j ) = M 2 2 σ j 2 [ ( c ˜ j , 00 c ˜ j , 00 ¯ ) ( c ˜ j , 00 * c ˜ j , 00 * ¯ ) + 2 m = 1 M 1 2 [ ( c ˜ j , 0 m c ˜ j , 0 m ¯ ) ( c ˜ j , 0 m * c ˜ j , 0 m * ¯ ) + ( c ˜ j , m 0 c ˜ j , m 0 ¯ ) ( c ˜ j , m 0 * c ˜ j , m 0 * ¯ ) ] + 2 n = 1 M 1 m = 1 M 1 2 ( c ˜ j , m n c ˜ j , m n ¯ ) ( c ˜ j , m n * c ˜ j , m n * ¯ ) ] + const . ,
P X ( x ) = 1 ( 2 π ) m | C | exp [ 1 2 ( x x ¯ ) H C 1 ( x x ¯ ) ] ,
( x x ¯ ) ( x x ¯ ) H ¯ = 2 C .
P total ( c ˜ 0 , , c ˜ P 1 ) = j = 0 P 1 P c ˜ ( c ˜ j ) .
ln P total ( c ˜ 0 , , c ˜ P 1 ) = m = 0 M 1 n = 0 M 1 ln P m n ( c ˜ m n ) ,
ln P m n ( c ˜ m n ) = ( c ˜ m n c ˜ m n ¯ ) H Σ H Σ ( c ˜ m n c ˜ m n ¯ ) + const . ,
Σ M [ 1 / σ 0 1 / σ P 1 ] .
( c ˜ c ˜ ¯ ) ( c ˜ c ˜ ¯ ) H ¯ = [ Σ 2 Σ 2 ] .
c ˜ m n ¯ = L 2 W H m n η ˜ m n ,
ln P m n ( c ˜ m n ) = ( c ˜ m n L 2 W H m n η ˜ m n ) H Σ H Σ × ( c ˜ m n L 2 W H m n η ˜ m n ) + const .
ln P m n ( c ˜ ) η ˜ = 0 .
L 2 W K H K η ^ = K H Σ c ˜ .
K + = k = 0 k r 1 1 s k f k ( z ) g k H ,
η ˜ + = L 2 W K + Σ c ˜ ¯ .
ln P m n ( c ˜ ) = ( Σ c ˜ L 2 W K η ˜ ) H ( Σ c ˜ L 2 W K η ˜ ) 1 2 ( η ˜ η ˜ 0 ) H P ( η ˜ η ˜ 0 ) + const . ,
( L 2 W K H K 1 2 P ) η ^ = K H Σ c ˜ 1 2 P η ˜ 0 .
η ^ reg = L 2 W K reg + Σ c ˜ = k = 0 k t 1 d k f k ( z ) ,
( η ^ + η ˜ + ) ( η ^ + η ˜ + ) H ¯ = L 4 W 2 K + K + H = L 4 W 2 k = 0 k r 1 1 s k 2 f k ( z 1 ) f k H ( z 2 ) ,
PSE ( η ^ + η ˜ + ) H ( η ^ + η ˜ + ) ¯ = L 4 W 2 k = 0 k r 1 1 s k 2 .
TSE = η ˜ H η ˜ 2 η ˜ H η ^ + + η ^ + H η ^ + ¯ = [ ( η ˜ η ˜ + ) H ( η ˜ η ˜ + ) ] + PSE ,
( η ^ + η ˜ + ) = l = 0 k r 1 ( d l d l ¯ ) f k ( z ) ,
( d r d r ¯ ) ( d q * d q * ¯ ) ¯ = L 4 W 2 1 s q 2 δ q r ,
η ˜ ( q , z ) = η s 2 z a z 2 J 1 ( | q | 2 z a z 2 ) | q | ,
d j ( q m n ) ¯ = f j H ( z ) η ˜ ( z ) .
SNR j ( q m n ) = W M s j η s L 2 | q m n | | 0 Δ z d z [ f j + * ( q m n , z ) + f j * ( q m n , z ) ] 2 z a z 2 J 1 ( | q m n | 2 z a z 2 ) | .
G α β ( r , r ) = d 2 q k z ( q ) h α β ( q , z ) exp [ i k ( q ) · ( r r ) ] .
h α β ( q , z ) = 1 | q | 2 q x 2 p x x + q y 2 p y y q x q y ( p x x p y y ) | q | q x p x z q x q y ( p x x p y y ) q y 2 p x x + q x 2 p y y | q | q y p x z | q | q x p z x | q | q y p z x | q | 2 p z z ,
p x x = k z 2 ( q ) k 0 2 [ 1 + R 2 exp ( 2 i k z ( q ) z ) ] , p x z = | q | k z ( q ) k 0 2 [ 1 R 2 exp ( 2 i k z ( q ) z ) ] , p y y = 1 + R 1 exp ( 2 i k z ( q ) z ) ] , p z x = | q | k z ( q ) k 0 2 [ 1 + R 2 exp ( 2 i k z ( q ) z ) ] , p z z = | q | 2 k 0 2 [ 1 R 2 exp ( 2 i k z ( q ) z ) ] .
R 1 ( q ) = k z ( q ) k z ( q ) k z ( q ) + k z ( q ) ,
R 2 ( q ) = k z ( q ) n 2 k z ( q ) k z ( q ) + n 2 k z ( q ) ,
K = Σ H = k = 0 P 1 s k g k f k ( z ) H ,
f k ( z ) = f k + ( z ) f k ( z )
( K K H ) j k = M 2 σ j σ k 0 Δ z [ H ( q q j i + q j r , q j r , q j i , z ) H * ( q q k i + q k r , q k r , q k i , z ) + H * ( q q j i + q j r , q j r , q j i , z ) H ( q q k i + q k r , q k r , q k i , z ) ] d z .

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