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Nearfield 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 
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), 10571072. doi: 10.1002/10991476(200008)23:12<1057::AIDMMA151>3.0.CO;26. 
[2] 
H. Ammari, G. Bao and A. W. Wood, Analysis of the electromagnetic scattering from a cavity, Japan J. Indust. Appl. Math., 19 (2002), 301310. doi: 10.1007/BF03167458. 
[3] 
H. Ammari, J. Garnier and K. Sølna, Resolution and stability analysis in fullaperture, 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 nearfield microscopy, Appl. Phys. Lett., 77 (2000), 2798800. 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, SpringerVerlag, Berlin, 1998. 
[8] 
D. Courjon and C. Bainier, Near field microscopy and near field optics, Rep. Prog. Phys., 57 (1994), 9891028. 
[9] 
G. Derveaux, G. Papanicolaou and C. Tsogka, Resolution and denoising in nearfield imaging, Inverse Problems, 22 (2006), 14371456. doi: 10.1088/02665611/22/4/017. 
[10] 
H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Mathematics and Its Application, Kluwer Academic Pubishers, New York, 1996. doi: 10.1007/9789400917408. 
[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, SpringerVerlag, New York, 1996. doi: 10.1007/9781461253389. 
[13] 
R. Kress and T. Tran, Inverse scattering for a locally perturbed halfplane, Inverse Problems, 16 (2000), 15411559. doi: 10.1088/02665611/16/5/323. 
[14] 
L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Am. J. Math., 73 (1951), 615624. doi: 10.2307/2372313. 
[15] 
L Novotny and B. Hecht, "Principles of NanoOptics," Cambridge University Press, 2006. 
[16] 
L. Rayleigh, On the theory of optical images with special reference to the optical microscope, Phil. Mag., 5 (1896), 167195. 
[17] 
F. Reitich and C. Turc, Highorder solutions of threedimensional roughsurface scattering problems at highfrequencies. I: The scalar case, Waves Random and Complex Media, 15 (2005), 116. doi: 10.1080/17455030500053393. 
[18] 
J. Sun, P. Carney and J. Schotland, Nearfield scanning optical tomography: A nondestructive method for threedimensional nanoscale imaging, IEEE J. Sel. Top. Quant., 12 (2006), 10721082. 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), 10351038. 
[20] 
A. Willers, The Helmholtz equation in disturbed halfspaces, Math. Meth. Appl. Sci., 9 (1987), 312323. doi: 10.1002/mma.1670090124. 
[21] 
B. Zhang and S. N. ChandlerWilde, Integral equation methods for scattering by infinite rough surfaces, Math. Meth. Appl. Sci., 26 (2003), 463488. 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), 10571072. doi: 10.1002/10991476(200008)23:12<1057::AIDMMA151>3.0.CO;26. 
[2] 
H. Ammari, G. Bao and A. W. Wood, Analysis of the electromagnetic scattering from a cavity, Japan J. Indust. Appl. Math., 19 (2002), 301310. doi: 10.1007/BF03167458. 
[3] 
H. Ammari, J. Garnier and K. Sølna, Resolution and stability analysis in fullaperture, 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 nearfield microscopy, Appl. Phys. Lett., 77 (2000), 2798800. 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, SpringerVerlag, Berlin, 1998. 
[8] 
D. Courjon and C. Bainier, Near field microscopy and near field optics, Rep. Prog. Phys., 57 (1994), 9891028. 
[9] 
G. Derveaux, G. Papanicolaou and C. Tsogka, Resolution and denoising in nearfield imaging, Inverse Problems, 22 (2006), 14371456. doi: 10.1088/02665611/22/4/017. 
[10] 
H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Mathematics and Its Application, Kluwer Academic Pubishers, New York, 1996. doi: 10.1007/9789400917408. 
[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, SpringerVerlag, New York, 1996. doi: 10.1007/9781461253389. 
[13] 
R. Kress and T. Tran, Inverse scattering for a locally perturbed halfplane, Inverse Problems, 16 (2000), 15411559. doi: 10.1088/02665611/16/5/323. 
[14] 
L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Am. J. Math., 73 (1951), 615624. doi: 10.2307/2372313. 
[15] 
L Novotny and B. Hecht, "Principles of NanoOptics," Cambridge University Press, 2006. 
[16] 
L. Rayleigh, On the theory of optical images with special reference to the optical microscope, Phil. Mag., 5 (1896), 167195. 
[17] 
F. Reitich and C. Turc, Highorder solutions of threedimensional roughsurface scattering problems at highfrequencies. I: The scalar case, Waves Random and Complex Media, 15 (2005), 116. doi: 10.1080/17455030500053393. 
[18] 
J. Sun, P. Carney and J. Schotland, Nearfield scanning optical tomography: A nondestructive method for threedimensional nanoscale imaging, IEEE J. Sel. Top. Quant., 12 (2006), 10721082. 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), 10351038. 
[20] 
A. Willers, The Helmholtz equation in disturbed halfspaces, Math. Meth. Appl. Sci., 9 (1987), 312323. doi: 10.1002/mma.1670090124. 
[21] 
B. Zhang and S. N. ChandlerWilde, Integral equation methods for scattering by infinite rough surfaces, Math. Meth. Appl. Sci., 26 (2003), 463488. doi: 10.1002/mma.361. 
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