February  2015, 9(1): 189-210. doi: 10.3934/ipi.2015.9.189

Near-field imaging of obstacles

1. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States, United States

Received  January 2014 Revised  June 2014 Published  January 2015

A novel method is developed for solving the inverse obstacle scattering problem in near-field imaging. The obstacle surface is assumed to be a small and smooth deformation of a circle. Using the transformed field expansion, the direct obstacle scattering problem is reduced to a successive sequence of two-point boundary value problems. Analytical solutions of these problems are derived by a Green's function method. The nonlinear inverse problem is linearized by dropping the higher order terms in the power series expansion. Based on the linear model and analytical solutions, an explicit reconstruction formula is obtained. In addition, a nonlinear correction scheme is devised to improve the results dramatically when the deformation is large. The method requires only a single incident wave at a fixed frequency. Numerical tests show that the method is stable and effective for near-field imaging of obstacles with subwavelength resolution.
Citation: Peijun Li, Yuliang Wang. Near-field imaging of obstacles. Inverse Problems & Imaging, 2015, 9 (1) : 189-210. doi: 10.3934/ipi.2015.9.189
References:
[1]

G. Bao, T. Cui and P. Li, Inverse diffraction grating of Maxwell's equations in biperiodic structures,, Optics Express, 22 (2014), 4799. doi: 10.1364/OE.22.004799. Google Scholar

[2]

G. Bao and P. Li, Inverse medium scattering problems in near-field optics,, J. Comput. Math., 25 (2007), 252. Google Scholar

[3]

G. Bao and P. Li, Numerical solution of inverse scattering for near-field optics,, Optics Lett., 32 (2007), 1465. doi: 10.1364/OL.32.001465. Google Scholar

[4]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces,, SIAM J. Appl. Math., 73 (2013), 2162. doi: 10.1137/130916266. Google Scholar

[5]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces in dielectric media,, SIAM J. Imaging Sci., 7 (2014), 867. doi: 10.1137/130944485. Google Scholar

[6]

G. Bao and P. Li, Convergence analysis in near-field imaging,, Inverse Problems, 30 (2014). doi: 10.1088/0266-5611/30/8/085008. Google Scholar

[7]

G. Bao and J. Lin, Imaging of reflective surfaces by near-field optics,, Optics Lett., 37 (2012), 5027. doi: 10.1364/OL.37.005027. Google Scholar

[8]

G. Bao and J. Lin, Near-field imaging of the surface displacement on an infinite ground plane,, Inverse Probl. Imag., 7 (2013), 377. doi: 10.3934/ipi.2013.7.377. Google Scholar

[9]

O. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries,, J. Opt. Soc. Am. A, 10 (1993), 1168. doi: 10.1364/JOSAA.10.001168. Google Scholar

[10]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction,, Springer, (2006). Google Scholar

[11]

P. Carney and J. Schotland, Near-field tomography,, in Inside Out: Inverse Problems and Applications (ed. G. Uhlmann), 47 (2003), 133. Google Scholar

[12]

T. Cheng, P. Li and Y. Wang, Near-field imaging of perfectly conducting grating surfaces,, J. Opt. Soc. Am. A, 30 (2013), 2473. doi: 10.1364/JOSAA.30.002473. Google Scholar

[13]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inverse Problems, 12 (1996), 383. doi: 10.1088/0266-5611/12/4/003. Google Scholar

[14]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory,, Wiley, (1983). Google Scholar

[15]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-03537-5. Google Scholar

[16]

D. Courjon, Near-field Microscopy and Near-field Optics,, Imperial College Press, (2003). doi: 10.1088/0034-4885/57/10/002. Google Scholar

[17]

D. Courjon and C. Bainier, Near field microscopy and near field optics,, Rep. Prog. Phys., 57 (1994), 989. doi: 10.1088/0034-4885/57/10/002. Google Scholar

[18]

F. Hettlich, Frechét derivatives in inverse obstacle scattering,, Inverse Problems, 11 (1995), 371. doi: 10.1088/0266-5611/11/2/007. Google Scholar

[19]

M. Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency,, Inverse Problems, 14 (1998), 949. doi: 10.1088/0266-5611/14/4/012. Google Scholar

[20]

A. Kirsch, The domain derivative and two applications in inverse scattering theory,, Inverse Problems, 9 (1993), 81. doi: 10.1088/0266-5611/9/1/005. Google Scholar

[21]

A. Kirsch, The music algorithm and the factorization method in inverse scattering theory for inhomogeneous media,, Inverse Problems, 18 (2002), 1025. doi: 10.1088/0266-5611/18/4/306. Google Scholar

[22]

NIST Digital Library of Mathematical Functions., http://dlmf.nist.gov/,, Release 1.0.6 of 2013-05-06., (): 2013. Google Scholar

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R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares,, Inverse Problems, 19 (2003). doi: 10.1088/0266-5611/19/6/056. Google Scholar

[24]

R. Kress and W. Rundell, A quasi-Newton method in inverse obstacle scattering,, Inverse Problems, 10 (1994), 1145. doi: 10.1088/0266-5611/10/5/011. Google Scholar

[25]

P. Li and J. Shen, Analysis of the scattering by an unbounded rough surface,, Math. Meth. Appl. Sci., 35 (2012), 2166. doi: 10.1002/mma.2560. Google Scholar

[26]

A. Malcolm and D. P. Nicholls, A field expansions method for scattering by periodic multilayered media,, J. Acout. Soc. Am., 129 (2011), 1783. doi: 10.1121/1.3531931. Google Scholar

[27]

A. Malcolm and D. P. Nicholls, A boundary perturbation method for recovering interface shapes in layered media,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/9/095009. Google Scholar

[28]

D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: Cancellations, conditioning, and convergence,, J. Opt. Soc. Am. A, 21 (2004), 590. doi: 10.1364/JOSAA.21.000590. Google Scholar

[29]

D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: improved algorithms,, J. Opt. Soc. Am. A, 21 (2004), 606. doi: 10.1364/JOSAA.21.000606. Google Scholar

[30]

D. P. Nicholls and J. Shen, A Stable High-Order Method for Two-Dimensional Bounded-Obstacle Scattering,, SIAM J. Sci. Comput., 28 (2006), 1398. doi: 10.1137/050632920. Google Scholar

[31]

R. Potthast, Stability estimates and reconstructions in inverse acoustic scattering using singular sources,, J. Comp. Appl. Math., 114 (2000), 247. doi: 10.1016/S0377-0427(99)00201-0. Google Scholar

[32]

R. Schmidt, Multiple emitter location and signal parameter estimation,, IEEE Trans. Antennas Propag., 34 (1986), 276. doi: 10.1109/TAP.1986.1143830. Google Scholar

show all references

References:
[1]

G. Bao, T. Cui and P. Li, Inverse diffraction grating of Maxwell's equations in biperiodic structures,, Optics Express, 22 (2014), 4799. doi: 10.1364/OE.22.004799. Google Scholar

[2]

G. Bao and P. Li, Inverse medium scattering problems in near-field optics,, J. Comput. Math., 25 (2007), 252. Google Scholar

[3]

G. Bao and P. Li, Numerical solution of inverse scattering for near-field optics,, Optics Lett., 32 (2007), 1465. doi: 10.1364/OL.32.001465. Google Scholar

[4]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces,, SIAM J. Appl. Math., 73 (2013), 2162. doi: 10.1137/130916266. Google Scholar

[5]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces in dielectric media,, SIAM J. Imaging Sci., 7 (2014), 867. doi: 10.1137/130944485. Google Scholar

[6]

G. Bao and P. Li, Convergence analysis in near-field imaging,, Inverse Problems, 30 (2014). doi: 10.1088/0266-5611/30/8/085008. Google Scholar

[7]

G. Bao and J. Lin, Imaging of reflective surfaces by near-field optics,, Optics Lett., 37 (2012), 5027. doi: 10.1364/OL.37.005027. Google Scholar

[8]

G. Bao and J. Lin, Near-field imaging of the surface displacement on an infinite ground plane,, Inverse Probl. Imag., 7 (2013), 377. doi: 10.3934/ipi.2013.7.377. Google Scholar

[9]

O. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries,, J. Opt. Soc. Am. A, 10 (1993), 1168. doi: 10.1364/JOSAA.10.001168. Google Scholar

[10]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction,, Springer, (2006). Google Scholar

[11]

P. Carney and J. Schotland, Near-field tomography,, in Inside Out: Inverse Problems and Applications (ed. G. Uhlmann), 47 (2003), 133. Google Scholar

[12]

T. Cheng, P. Li and Y. Wang, Near-field imaging of perfectly conducting grating surfaces,, J. Opt. Soc. Am. A, 30 (2013), 2473. doi: 10.1364/JOSAA.30.002473. Google Scholar

[13]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inverse Problems, 12 (1996), 383. doi: 10.1088/0266-5611/12/4/003. Google Scholar

[14]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory,, Wiley, (1983). Google Scholar

[15]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-03537-5. Google Scholar

[16]

D. Courjon, Near-field Microscopy and Near-field Optics,, Imperial College Press, (2003). doi: 10.1088/0034-4885/57/10/002. Google Scholar

[17]

D. Courjon and C. Bainier, Near field microscopy and near field optics,, Rep. Prog. Phys., 57 (1994), 989. doi: 10.1088/0034-4885/57/10/002. Google Scholar

[18]

F. Hettlich, Frechét derivatives in inverse obstacle scattering,, Inverse Problems, 11 (1995), 371. doi: 10.1088/0266-5611/11/2/007. Google Scholar

[19]

M. Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency,, Inverse Problems, 14 (1998), 949. doi: 10.1088/0266-5611/14/4/012. Google Scholar

[20]

A. Kirsch, The domain derivative and two applications in inverse scattering theory,, Inverse Problems, 9 (1993), 81. doi: 10.1088/0266-5611/9/1/005. Google Scholar

[21]

A. Kirsch, The music algorithm and the factorization method in inverse scattering theory for inhomogeneous media,, Inverse Problems, 18 (2002), 1025. doi: 10.1088/0266-5611/18/4/306. Google Scholar

[22]

NIST Digital Library of Mathematical Functions., http://dlmf.nist.gov/,, Release 1.0.6 of 2013-05-06., (): 2013. Google Scholar

[23]

R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares,, Inverse Problems, 19 (2003). doi: 10.1088/0266-5611/19/6/056. Google Scholar

[24]

R. Kress and W. Rundell, A quasi-Newton method in inverse obstacle scattering,, Inverse Problems, 10 (1994), 1145. doi: 10.1088/0266-5611/10/5/011. Google Scholar

[25]

P. Li and J. Shen, Analysis of the scattering by an unbounded rough surface,, Math. Meth. Appl. Sci., 35 (2012), 2166. doi: 10.1002/mma.2560. Google Scholar

[26]

A. Malcolm and D. P. Nicholls, A field expansions method for scattering by periodic multilayered media,, J. Acout. Soc. Am., 129 (2011), 1783. doi: 10.1121/1.3531931. Google Scholar

[27]

A. Malcolm and D. P. Nicholls, A boundary perturbation method for recovering interface shapes in layered media,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/9/095009. Google Scholar

[28]

D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: Cancellations, conditioning, and convergence,, J. Opt. Soc. Am. A, 21 (2004), 590. doi: 10.1364/JOSAA.21.000590. Google Scholar

[29]

D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: improved algorithms,, J. Opt. Soc. Am. A, 21 (2004), 606. doi: 10.1364/JOSAA.21.000606. Google Scholar

[30]

D. P. Nicholls and J. Shen, A Stable High-Order Method for Two-Dimensional Bounded-Obstacle Scattering,, SIAM J. Sci. Comput., 28 (2006), 1398. doi: 10.1137/050632920. Google Scholar

[31]

R. Potthast, Stability estimates and reconstructions in inverse acoustic scattering using singular sources,, J. Comp. Appl. Math., 114 (2000), 247. doi: 10.1016/S0377-0427(99)00201-0. Google Scholar

[32]

R. Schmidt, Multiple emitter location and signal parameter estimation,, IEEE Trans. Antennas Propag., 34 (1986), 276. doi: 10.1109/TAP.1986.1143830. Google Scholar

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