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

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.
Citation: Gang Bao, Junshan Lin. Near-field imaging of the surface displacement on an infinite ground plane. Inverse Problems and 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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[19]

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

[20]

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

[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.

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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[19]

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

[20]

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

[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.

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