American Institute of Mathematical Sciences

Advanced
May  2013, 7(2): 377-396. doi: 10.3934/ipi.2013.7.377

Near-field imaging of the surface displacement on an infinite ground plane

Gang Bao 1, and  Junshan Lin 2,

1. 

Department of Mathematics, Zhejiang University, Hangzhou, China
2. 

Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, United States

Received  November 2011 Revised  June 2012 Published  May 2013

This paper is concerned with the inverse diffraction problem for an unbounded obstacle which is a ground plane with some local disturbance. The data is collected in the near-field regime with a distance above the surface displacement that is smaller than the wavelength. In this regime, the evanescent modes carried by the scattered wave are significant, which makes it different from the far-field measurement. We formulate explicitly the connection between the evanescent wave modes and the high frequency components of the surface displacement, and present a new numerical scheme to reconstruct the surface displacement from the boundary measurements. By extracting the information carried by the evanescent modes effectively, it is shown that the resolution of the reconstructed image is significantly improved in the near field. Numerical examples show that images with a resolution of $\lambda/10$ are obtained.
Keywords: Inverse scattering, near-field imaging, Helmholtz equation..
Mathematics Subject Classification: 35J05, 35R30, 65N2.
Citation: Gang Bao, Junshan Lin. Near-field imaging of the surface displacement on an infinite ground plane. Inverse Problems & Imaging, 2013, 7 (2) : 377-396. doi: 10.3934/ipi.2013.7.377
References:
[1]

H. Ammari, G. Bao and A. W. Wood, An integral equation method for the electromagnetic scattering from cavities, Math. Meth. Appl. Sci., 23 (2000), 1057-1072. doi: 10.1002/1099-1476(200008)23:12<1057::AID-MMA151>3.0.CO;2-6.  Google Scholar

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H. Ammari, G. Bao and A. W. Wood, Analysis of the electromagnetic scattering from a cavity, Japan J. Indust. Appl. Math., 19 (2002), 301-310. doi: 10.1007/BF03167458.  Google Scholar

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H. Ammari, J. Garnier and K. Sølna, Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,, Proc. Amer. Math. Soc., ().   Google Scholar

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[5]

P. Carney and J. Schotland, Inverse scattering for near-field microscopy, Appl. Phys. Lett., 77 (2000), 2798-800. doi: 10.1063/1.1320844.  Google Scholar

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D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics, Wiley, New York, 1983.  Google Scholar

[7]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998.  Google Scholar

[8]

D. Courjon and C. Bainier, Near field microscopy and near field optics, Rep. Prog. Phys., 57 (1994), 989-1028. Google Scholar

[9]

G. Derveaux, G. Papanicolaou and C. Tsogka, Resolution and denoising in near-field imaging, Inverse Problems, 22 (2006), 1437-1456. doi: 10.1088/0266-5611/22/4/017.  Google Scholar

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H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Mathematics and Its Application, Kluwer Academic Pubishers, New York, 1996. doi: 10.1007/978-94-009-1740-8.  Google Scholar

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L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, 1997.  Google Scholar

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A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-5338-9.  Google Scholar

[13]

R. Kress and T. Tran, Inverse scattering for a locally perturbed half-plane, Inverse Problems, 16 (2000), 1541-1559. doi: 10.1088/0266-5611/16/5/323.  Google Scholar

[14]

L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Am. J. Math., 73 (1951), 615-624. doi: 10.2307/2372313.  Google Scholar

[15]

L Novotny and B. Hecht, "Principles of Nano-Optics," Cambridge University Press, 2006. Google Scholar

[16]

L. Rayleigh, On the theory of optical images with special reference to the optical microscope, Phil. Mag., 5 (1896), 167-195. Google Scholar

[17]

F. Reitich and C. Turc, High-order solutions of three-dimensional roughsurface scattering problems at high-frequencies. I: The scalar case, Waves Random and Complex Media, 15 (2005), 1-16. doi: 10.1080/17455030500053393.  Google Scholar

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J. Sun, P. Carney and J. Schotland, Near-field scanning optical tomography: A nondestructive method for three-dimensional nanoscale imaging, IEEE J. Sel. Top. Quant., 12 (2006), 1072-1082. doi: 10.1109/JSTQE.2006.879567.  Google Scholar

[19]

A. V. Tikhonov, On the solution of incorrectly formulated problems and the regularization method, Soviet Math. Doklady, 4 (1963), 1035-1038. Google Scholar

[20]

A. Willers, The Helmholtz equation in disturbed half-spaces, Math. Meth. Appl. Sci., 9 (1987), 312-323. doi: 10.1002/mma.1670090124.  Google Scholar

[21]

B. Zhang and S. N. Chandler-Wilde, Integral equation methods for scattering by infinite rough surfaces, Math. Meth. Appl. Sci., 26 (2003), 463-488. doi: 10.1002/mma.361.  Google Scholar

show all references

References:
[1]

H. Ammari, G. Bao and A. W. Wood, An integral equation method for the electromagnetic scattering from cavities, Math. Meth. Appl. Sci., 23 (2000), 1057-1072. doi: 10.1002/1099-1476(200008)23:12<1057::AID-MMA151>3.0.CO;2-6.  Google Scholar

[2]

H. Ammari, G. Bao and A. W. Wood, Analysis of the electromagnetic scattering from a cavity, Japan J. Indust. Appl. Math., 19 (2002), 301-310. doi: 10.1007/BF03167458.  Google Scholar

[3]

H. Ammari, J. Garnier and K. Sølna, Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,, Proc. Amer. Math. Soc., ().   Google Scholar

[4]

M. Born and E. Wolf, "Principles of Optics," (6th ed.), Cambridge University Press, 1980. doi: 10.1017/CBO9781139644181.  Google Scholar

[5]

P. Carney and J. Schotland, Inverse scattering for near-field microscopy, Appl. Phys. Lett., 77 (2000), 2798-800. doi: 10.1063/1.1320844.  Google Scholar

[6]

D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics, Wiley, New York, 1983.  Google Scholar

[7]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998.  Google Scholar

[8]

D. Courjon and C. Bainier, Near field microscopy and near field optics, Rep. Prog. Phys., 57 (1994), 989-1028. Google Scholar

[9]

G. Derveaux, G. Papanicolaou and C. Tsogka, Resolution and denoising in near-field imaging, Inverse Problems, 22 (2006), 1437-1456. doi: 10.1088/0266-5611/22/4/017.  Google Scholar

[10]

H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Mathematics and Its Application, Kluwer Academic Pubishers, New York, 1996. doi: 10.1007/978-94-009-1740-8.  Google Scholar

[11]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, 1997.  Google Scholar

[12]

A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-5338-9.  Google Scholar

[13]

R. Kress and T. Tran, Inverse scattering for a locally perturbed half-plane, Inverse Problems, 16 (2000), 1541-1559. doi: 10.1088/0266-5611/16/5/323.  Google Scholar

[14]

L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Am. J. Math., 73 (1951), 615-624. doi: 10.2307/2372313.  Google Scholar

[15]

L Novotny and B. Hecht, "Principles of Nano-Optics," Cambridge University Press, 2006. Google Scholar

[16]

L. Rayleigh, On the theory of optical images with special reference to the optical microscope, Phil. Mag., 5 (1896), 167-195. Google Scholar

[17]

F. Reitich and C. Turc, High-order solutions of three-dimensional roughsurface scattering problems at high-frequencies. I: The scalar case, Waves Random and Complex Media, 15 (2005), 1-16. doi: 10.1080/17455030500053393.  Google Scholar

[18]

J. Sun, P. Carney and J. Schotland, Near-field scanning optical tomography: A nondestructive method for three-dimensional nanoscale imaging, IEEE J. Sel. Top. Quant., 12 (2006), 1072-1082. doi: 10.1109/JSTQE.2006.879567.  Google Scholar

[19]

A. V. Tikhonov, On the solution of incorrectly formulated problems and the regularization method, Soviet Math. Doklady, 4 (1963), 1035-1038. Google Scholar

[20]

A. Willers, The Helmholtz equation in disturbed half-spaces, Math. Meth. Appl. Sci., 9 (1987), 312-323. doi: 10.1002/mma.1670090124.  Google Scholar

[21]

B. Zhang and S. N. Chandler-Wilde, Integral equation methods for scattering by infinite rough surfaces, Math. Meth. Appl. Sci., 26 (2003), 463-488. doi: 10.1002/mma.361.  Google Scholar

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