# American Institute of Mathematical Sciences

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 & 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. 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. 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.), (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. doi: 10.1063/1.1320844. Google Scholar [6] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", Pure and Applied Mathematics, (1983). Google Scholar [7] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Applied Mathematical Sciences, 93 (1998). Google Scholar [8] D. Courjon and C. Bainier, Near field microscopy and near field optics,, Rep. Prog. Phys., 57 (1994), 989. Google Scholar [9] G. Derveaux, G. Papanicolaou and C. Tsogka, Resolution and denoising in near-field imaging,, Inverse Problems, 22 (2006), 1437. 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, (1996). doi: 10.1007/978-94-009-1740-8. Google Scholar [11] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1997). Google Scholar [12] A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems,", Applied Mathematical Sciences, 120 (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. 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. 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. 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. 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. 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. Google Scholar [20] A. Willers, The Helmholtz equation in disturbed half-spaces,, Math. Meth. Appl. Sci., 9 (1987), 312. 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. 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. 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. 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.), (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. doi: 10.1063/1.1320844. Google Scholar [6] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", Pure and Applied Mathematics, (1983). Google Scholar [7] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Applied Mathematical Sciences, 93 (1998). Google Scholar [8] D. Courjon and C. Bainier, Near field microscopy and near field optics,, Rep. Prog. Phys., 57 (1994), 989. Google Scholar [9] G. Derveaux, G. Papanicolaou and C. Tsogka, Resolution and denoising in near-field imaging,, Inverse Problems, 22 (2006), 1437. 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, (1996). doi: 10.1007/978-94-009-1740-8. Google Scholar [11] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1997). Google Scholar [12] A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems,", Applied Mathematical Sciences, 120 (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. 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. 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. 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. 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. 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. Google Scholar [20] A. Willers, The Helmholtz equation in disturbed half-spaces,, Math. Meth. Appl. Sci., 9 (1987), 312. 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. doi: 10.1002/mma.361. Google Scholar
 [1] Peijun Li, Yuliang Wang. Near-field imaging of obstacles. Inverse Problems & Imaging, 2015, 9 (1) : 189-210. doi: 10.3934/ipi.2015.9.189 [2] Ming Li, Ruming Zhang. Near-field imaging of sound-soft obstacles in periodic waveguides. Inverse Problems & Imaging, 2017, 11 (6) : 1091-1105. doi: 10.3934/ipi.2017050 [3] Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 [4] Giovanni Bozza, Massimo Brignone, Matteo Pastorino, Andrea Randazzo, Michele Piana. Imaging of unknown targets inside inhomogeneous backgrounds by means of qualitative inverse scattering. Inverse Problems & Imaging, 2009, 3 (2) : 231-241. doi: 10.3934/ipi.2009.3.231 [5] Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757 [6] Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems & Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033 [7] Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027 [8] Roland Griesmaier. Reciprocity gap music imaging for an inverse scattering problem in two-layered media. Inverse Problems & Imaging, 2009, 3 (3) : 389-403. doi: 10.3934/ipi.2009.3.389 [9] Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042 [10] Kaitlyn (Voccola) Muller. SAR correlation imaging and anisotropic scattering. Inverse Problems & Imaging, 2018, 12 (3) : 697-731. doi: 10.3934/ipi.2018030 [11] Peter Monk, Virginia Selgas. Near field sampling type methods for the inverse fluid--solid interaction problem. Inverse Problems & Imaging, 2011, 5 (2) : 465-483. doi: 10.3934/ipi.2011.5.465 [12] John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181 [13] S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604 [14] Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems & Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073 [15] John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems & Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333 [16] Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159 [17] Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343 [18] Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048 [19] Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 [20] Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183

2018 Impact Factor: 1.469

## Metrics

• PDF downloads (7)
• HTML views (0)
• Cited by (18)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]