American Institute of Mathematical Sciences

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

Near-field imaging of obstacles

Peijun Li 1, and  Yuliang Wang 1,

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.
Keywords: inverse obstacle scattering, Near-field imaging, transformed field expansion..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Peijun Li, Yuliang Wang. Near-field imaging of obstacles. Inverse Problems and 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), 4799-4816. doi: 10.1364/OE.22.004799.

[2]

G. Bao and P. Li, Inverse medium scattering problems in near-field optics, J. Comput. Math., 25 (2007), 252-265.

[3]

G. Bao and P. Li, Numerical solution of inverse scattering for near-field optics, Optics Lett., 32 (2007), 1465-1467. doi: 10.1364/OL.32.001465.

[4]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 2162-2187. doi: 10.1137/130916266.

[5]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imaging Sci., 7 (2014), 867-899. doi: 10.1137/130944485.

[6]

G. Bao and P. Li, Convergence analysis in near-field imaging, Inverse Problems, 30 (2014), 085008, 26PP. doi: 10.1088/0266-5611/30/8/085008.

[7]

G. Bao and J. Lin, Imaging of reflective surfaces by near-field optics, Optics Lett., 37 (2012), 5027-5029. doi: 10.1364/OL.37.005027.

[8]

G. Bao and J. Lin, Near-field imaging of the surface displacement on an infinite ground plane, Inverse Probl. Imag., 7 (2013), 377-396. doi: 10.3934/ipi.2013.7.377.

[9]

O. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries, J. Opt. Soc. Am. A, 10 (1993), 1168-1175. doi: 10.1364/JOSAA.10.001168.

[10]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Springer, Berlin, 2006.

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P. Carney and J. Schotland, Near-field tomography, in Inside Out: Inverse Problems and Applications (ed. G. Uhlmann), Cambridge University Press, 47 (2003), 133-168.

[12]

T. Cheng, P. Li and Y. Wang, Near-field imaging of perfectly conducting grating surfaces, J. Opt. Soc. Am. A, 30 (2013), 2473-2481. doi: 10.1364/JOSAA.30.002473.

[13]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.

[14]

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

[15]

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

[16]

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

[17]

D. Courjon and C. Bainier, Near field microscopy and near field optics, Rep. Prog. Phys., 57 (1994), 989-1028. doi: 10.1088/0034-4885/57/10/002.

[18]

F. Hettlich, Frechét derivatives in inverse obstacle scattering, Inverse Problems, 11 (1995), 371-382. doi: 10.1088/0266-5611/11/2/007.

[19]

M. Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency, Inverse Problems, 14 (1998), 949-954. doi: 10.1088/0266-5611/14/4/012.

[20]

A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 81-96. doi: 10.1088/0266-5611/9/1/005.

[21]

A. Kirsch, The music algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306.

[22]

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

[23]

R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Problems, 19 (2003), S91-S104. doi: 10.1088/0266-5611/19/6/056.

[24]

R. Kress and W. Rundell, A quasi-Newton method in inverse obstacle scattering, Inverse Problems, 10 (1994), 1145-1157. doi: 10.1088/0266-5611/10/5/011.

[25]

P. Li and J. Shen, Analysis of the scattering by an unbounded rough surface, Math. Meth. Appl. Sci., 35 (2012), 2166-2184. doi: 10.1002/mma.2560.

[26]

A. Malcolm and D. P. Nicholls, A field expansions method for scattering by periodic multilayered media, J. Acout. Soc. Am., 129 (2011), 1783-1793. doi: 10.1121/1.3531931.

[27]

A. Malcolm and D. P. Nicholls, A boundary perturbation method for recovering interface shapes in layered media, Inverse Problems, 27 (2011), 095009, 18pp. doi: 10.1088/0266-5611/27/9/095009.

[28]

D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: Cancellations, conditioning, and convergence, J. Opt. Soc. Am. A, 21 (2004), 590-605. doi: 10.1364/JOSAA.21.000590.

[29]

D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: improved algorithms, J. Opt. Soc. Am. A, 21 (2004), 606-621. doi: 10.1364/JOSAA.21.000606.

[30]

D. P. Nicholls and J. Shen, A Stable High-Order Method for Two-Dimensional Bounded-Obstacle Scattering, SIAM J. Sci. Comput., 28 (2006), 1398-1419. doi: 10.1137/050632920.

[31]

R. Potthast, Stability estimates and reconstructions in inverse acoustic scattering using singular sources, J. Comp. Appl. Math., 114 (2000), 247-274. doi: 10.1016/S0377-0427(99)00201-0.

[32]

R. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Trans. Antennas Propag., 34 (1986), 276-280. doi: 10.1109/TAP.1986.1143830.

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), 4799-4816. doi: 10.1364/OE.22.004799.

[2]

G. Bao and P. Li, Inverse medium scattering problems in near-field optics, J. Comput. Math., 25 (2007), 252-265.

[3]

G. Bao and P. Li, Numerical solution of inverse scattering for near-field optics, Optics Lett., 32 (2007), 1465-1467. doi: 10.1364/OL.32.001465.

[4]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 2162-2187. doi: 10.1137/130916266.

[5]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imaging Sci., 7 (2014), 867-899. doi: 10.1137/130944485.

[6]

G. Bao and P. Li, Convergence analysis in near-field imaging, Inverse Problems, 30 (2014), 085008, 26PP. doi: 10.1088/0266-5611/30/8/085008.

[7]

G. Bao and J. Lin, Imaging of reflective surfaces by near-field optics, Optics Lett., 37 (2012), 5027-5029. doi: 10.1364/OL.37.005027.

[8]

G. Bao and J. Lin, Near-field imaging of the surface displacement on an infinite ground plane, Inverse Probl. Imag., 7 (2013), 377-396. doi: 10.3934/ipi.2013.7.377.

[9]

O. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of variation of boundaries, J. Opt. Soc. Am. A, 10 (1993), 1168-1175. doi: 10.1364/JOSAA.10.001168.

[10]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Springer, Berlin, 2006.

[11]

P. Carney and J. Schotland, Near-field tomography, in Inside Out: Inverse Problems and Applications (ed. G. Uhlmann), Cambridge University Press, 47 (2003), 133-168.

[12]

T. Cheng, P. Li and Y. Wang, Near-field imaging of perfectly conducting grating surfaces, J. Opt. Soc. Am. A, 30 (2013), 2473-2481. doi: 10.1364/JOSAA.30.002473.

[13]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.

[14]

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

[15]

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

[16]

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

[17]

D. Courjon and C. Bainier, Near field microscopy and near field optics, Rep. Prog. Phys., 57 (1994), 989-1028. doi: 10.1088/0034-4885/57/10/002.

[18]

F. Hettlich, Frechét derivatives in inverse obstacle scattering, Inverse Problems, 11 (1995), 371-382. doi: 10.1088/0266-5611/11/2/007.

[19]

M. Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency, Inverse Problems, 14 (1998), 949-954. doi: 10.1088/0266-5611/14/4/012.

[20]

A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 81-96. doi: 10.1088/0266-5611/9/1/005.

[21]

A. Kirsch, The music algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306.

[22]

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

[23]

R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Problems, 19 (2003), S91-S104. doi: 10.1088/0266-5611/19/6/056.

[24]

R. Kress and W. Rundell, A quasi-Newton method in inverse obstacle scattering, Inverse Problems, 10 (1994), 1145-1157. doi: 10.1088/0266-5611/10/5/011.

[25]

P. Li and J. Shen, Analysis of the scattering by an unbounded rough surface, Math. Meth. Appl. Sci., 35 (2012), 2166-2184. doi: 10.1002/mma.2560.

[26]

A. Malcolm and D. P. Nicholls, A field expansions method for scattering by periodic multilayered media, J. Acout. Soc. Am., 129 (2011), 1783-1793. doi: 10.1121/1.3531931.

[27]

A. Malcolm and D. P. Nicholls, A boundary perturbation method for recovering interface shapes in layered media, Inverse Problems, 27 (2011), 095009, 18pp. doi: 10.1088/0266-5611/27/9/095009.

[28]

D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: Cancellations, conditioning, and convergence, J. Opt. Soc. Am. A, 21 (2004), 590-605. doi: 10.1364/JOSAA.21.000590.

[29]

D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering: improved algorithms, J. Opt. Soc. Am. A, 21 (2004), 606-621. doi: 10.1364/JOSAA.21.000606.

[30]

D. P. Nicholls and J. Shen, A Stable High-Order Method for Two-Dimensional Bounded-Obstacle Scattering, SIAM J. Sci. Comput., 28 (2006), 1398-1419. doi: 10.1137/050632920.

[31]

R. Potthast, Stability estimates and reconstructions in inverse acoustic scattering using singular sources, J. Comp. Appl. Math., 114 (2000), 247-274. doi: 10.1016/S0377-0427(99)00201-0.

[32]

R. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Trans. Antennas Propag., 34 (1986), 276-280. doi: 10.1109/TAP.1986.1143830.

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