April  2017, 11(2): 263-276. doi: 10.3934/ipi.2017013

A phaseless inverse scattering problem for the 3-D Helmholtz equation

Department of Mathematics and Statistics, University of North Carolina at Charlotte Charlotte, NC 28213, USA

Received  August 2016 Revised  November 2016 Published  March 2017

Fund Project: This work was supported by US Army Research Laboratory and US Army Research Office grant W911NF-15-1-0233 as well as by the Office of Naval Research grant N00014-15-1-2330.

An inverse scattering problem for the 3-D Helmholtz equation is considered. Only the modulus of the complex valued scattered wave field is assumed to be measured and the phase is not measured. This problem naturally arises in the lensless quality control of fabricated nanostructures. Uniqueness theorem is proved.

Citation: 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
References:
[1]

H. AmmariY. T. Chow and J. Zou, Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients, SIAM J. Appl. Math., 76 (2016), 1000-1030.  doi: 10.1137/15M1043959.  Google Scholar

[2]

G. BaoP. Li and J. Lv, Numerical solution of an inverse diffraction grating problem from phaseless data, J.Optical Society of America A, 30 (2013), 293-299.  doi: 10.1364/JOSAA.30.000293.  Google Scholar

[3]

G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002, 16pp.  doi: 10.1088/0266-5611/32/8/085002.  Google Scholar

[4]

G. BaoP. LiJ. Lin and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Problems, 31 (2015), 093001, 21pp.  doi: 10.1088/0266-5611/31/9/093001.  Google Scholar

[5] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Pergamon Press, New York-London-Paris-Los Angeles, 1959.   Google Scholar
[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, New York, 1992. doi: 10.1007/978-3-662-02835-3.  Google Scholar

[7]

A. V. DarahanauA. Y. NikulinA. SouvorovY. NishinoB. C. Muddle and T. Ishikawa, Nano-resolution profiling of micro-structures using quantitative X-ray phase retrieval from Fraunhofer diffraction data, Physics Letters A, 335 (2005), 494-498.  doi: 10.1016/j.physleta.2004.10.084.  Google Scholar

[8]

M. DierolfO. BankS. KyndeP. ThibaultI. JohnsonA. MenzelK. JefimovsC. DavidO. Marti and F. Pfeiffer, Ptychography & lenseless X-ray imaging, Europhysics News, 39 (2008), 22-24.   Google Scholar

[9]

G. HuJ. LiH. Liu and H. Sun, Inverse elastic scattering for multiscale rigid bodies with a single far-field pattern, SIAM J. Imaging Sciences, 7 (2014), 1799-1825.  doi: 10.1137/130944187.  Google Scholar

[10]

V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006.  Google Scholar

[11]

O. IvanyshynR. Kress and P. Serranho, Huygens' principle and iterative methods in inverse obstacle scattering, Advances in Computational Mathematics, 33 (2010), 413-429.  doi: 10.1007/s10444-009-9135-6.  Google Scholar

[12]

O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phaseless far field data, J. Computational Physics, 230 (2011), 3443-3452.  doi: 10.1016/j.jcp.2011.01.038.  Google Scholar

[13]

M. V. Klibanov and P. E. Sacks, Phaseless inverse scattering and the phase problem in optics, J. Math. Physics, 33 (1992), 3813-3821.  doi: 10.1063/1.529990.  Google Scholar

[14]

M. V. KlibanovP. E. Sacks and A. V. Tikhonravov, The phase retrieval problem. Topical Review, Inverse Problems, 11 (1995), 1-28.  doi: 10.1088/0266-5611/11/1/001.  Google Scholar

[15]

M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74 (2014), 392-410.  doi: 10.1137/130926250.  Google Scholar

[16]

M. V. Klibanov, On the first solution of a long standing problem: Uniqueness of the phaseless quantum inverse scattering problem in 3-d, Applied Mathematics Letters, 37 (2014), 82-85.  doi: 10.1016/j.aml.2014.06.005.  Google Scholar

[17]

M. V. Klibanov, Uniqueness of two phaseless non-overdetermined inverse acoustics problems in 3-d, Applicable Analysis, 93 (2014), 1135-1149.  doi: 10.1080/00036811.2013.818136.  Google Scholar

[18]

M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, SIAM J. Appl. Math., 76 (2016), 178-196.  doi: 10.1137/15M1022367.  Google Scholar

[19]

M. V. Klibanov and V. G. Romanov, Two reconstruction procedures for a 3-D phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32 (2016), 015005, 16pp.  doi: 10.1088/0266-5611/32/1/015005.  Google Scholar

[20]

M. V. Klibanov and V. G. Romanov, The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation, J. Inverse and Ill-Posed Problems, 23 (2015), 415-428.  doi: 10.1515/jiip-2015-0025.  Google Scholar

[21]

M. V. Klibanov and V. G. Romanov, Explicit formula for the solution of the phaseless inverse scattering problem of imaging of nano structures, J. Inverse and Ill-Posed Problems, 23 (2015), 187-193.  doi: 10.1515/jiip-2015-0004.  Google Scholar

[22]

M. V. KlibanovL. H. Nguyen and K. Pan, Nanostructures imaging via numerical solution of a 3-D inverse scattering problem without the phase information, Applied Numerical Mathematics, 110 (2016), 190-203.  doi: 10.1016/j.apnum.2016.08.014.  Google Scholar

[23]

O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Springer, New York, 1985. doi: 10.1007/978-1-4757-4317-3.  Google Scholar

[24]

M. M. Lavrentiev, V. G. Romanov and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis, AMS, Providence, RI, 1986. Google Scholar

[25]

J. LiH. LiuZ. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scattereres, SIAM J. Appl. Math., 73 (2013), 1721-1746.  doi: 10.1137/130907690.  Google Scholar

[26]

J. LiH. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, SIAM J. Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X.  Google Scholar

[27]

R. G. Novikov, A multidimensional inverse spectral problem for the equation $ -Δ ψ +(v(x)-Eu(x))ψ =0$, Funct. Anal. Appl., 22 (1988), 263-272.  doi: 10.1007/BF01077418.  Google Scholar

[28]

R. G. Novikov, The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator, J.Functional Analysis, 103 (1992), 409-463.  doi: 10.1016/0022-1236(92)90127-5.  Google Scholar

[29]

R. G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geometrical Analysis, 26 (2016), 346-359.  doi: 10.1007/s12220-014-9553-7.  Google Scholar

[30]

R. G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bulletin des Sciences Mathé matiques, 139 (2015), 923-936.  doi: 10.1016/j.bulsci.2015.04.005.  Google Scholar

[31]

T. C. PetersenaV. J. Keastb and D. M. Paganinc, Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation, Ultramisroscopy, 108 (2008), 805-815.  doi: 10.1016/j.ultramic.2008.01.001.  Google Scholar

[32]

V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1987.  Google Scholar

[33]

V. G. Romanov, Inverse problems for differential equations with memory, Eurasian J. of Mathematical and Computer Applications, 2 (2014), 51-80.   Google Scholar

[34]

A. RuhlandtM. KrenkelM. Bartels and T. Salditt, Three-dimensional phase retrieval in propagation-based phase-contrast imaging, Physical Review A, 89 (2014), 033847.   Google Scholar

[35]

U. Schröder and T. Schuster, An iterative method to reconstruct the refractive index of a medium from time-of-flight-measurements, Inverse Problems, 32 (2016), 085009, 35pp.  doi: 10.1088/0266-5611/32/8/085009.  Google Scholar

[36]

B. R. Vainberg, Principles of radiation, limiting absorption and limiting amplitude in the general theory of partial differential equations, Russian Math. Surveys, 21 (1966), 115-194.   Google Scholar

[37] B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach Science Publishers, New York, 1989.   Google Scholar

show all references

References:
[1]

H. AmmariY. T. Chow and J. Zou, Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients, SIAM J. Appl. Math., 76 (2016), 1000-1030.  doi: 10.1137/15M1043959.  Google Scholar

[2]

G. BaoP. Li and J. Lv, Numerical solution of an inverse diffraction grating problem from phaseless data, J.Optical Society of America A, 30 (2013), 293-299.  doi: 10.1364/JOSAA.30.000293.  Google Scholar

[3]

G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002, 16pp.  doi: 10.1088/0266-5611/32/8/085002.  Google Scholar

[4]

G. BaoP. LiJ. Lin and F. Triki, Inverse scattering problems with multi-frequencies, Inverse Problems, 31 (2015), 093001, 21pp.  doi: 10.1088/0266-5611/31/9/093001.  Google Scholar

[5] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Pergamon Press, New York-London-Paris-Los Angeles, 1959.   Google Scholar
[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, New York, 1992. doi: 10.1007/978-3-662-02835-3.  Google Scholar

[7]

A. V. DarahanauA. Y. NikulinA. SouvorovY. NishinoB. C. Muddle and T. Ishikawa, Nano-resolution profiling of micro-structures using quantitative X-ray phase retrieval from Fraunhofer diffraction data, Physics Letters A, 335 (2005), 494-498.  doi: 10.1016/j.physleta.2004.10.084.  Google Scholar

[8]

M. DierolfO. BankS. KyndeP. ThibaultI. JohnsonA. MenzelK. JefimovsC. DavidO. Marti and F. Pfeiffer, Ptychography & lenseless X-ray imaging, Europhysics News, 39 (2008), 22-24.   Google Scholar

[9]

G. HuJ. LiH. Liu and H. Sun, Inverse elastic scattering for multiscale rigid bodies with a single far-field pattern, SIAM J. Imaging Sciences, 7 (2014), 1799-1825.  doi: 10.1137/130944187.  Google Scholar

[10]

V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006.  Google Scholar

[11]

O. IvanyshynR. Kress and P. Serranho, Huygens' principle and iterative methods in inverse obstacle scattering, Advances in Computational Mathematics, 33 (2010), 413-429.  doi: 10.1007/s10444-009-9135-6.  Google Scholar

[12]

O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phaseless far field data, J. Computational Physics, 230 (2011), 3443-3452.  doi: 10.1016/j.jcp.2011.01.038.  Google Scholar

[13]

M. V. Klibanov and P. E. Sacks, Phaseless inverse scattering and the phase problem in optics, J. Math. Physics, 33 (1992), 3813-3821.  doi: 10.1063/1.529990.  Google Scholar

[14]

M. V. KlibanovP. E. Sacks and A. V. Tikhonravov, The phase retrieval problem. Topical Review, Inverse Problems, 11 (1995), 1-28.  doi: 10.1088/0266-5611/11/1/001.  Google Scholar

[15]

M. V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74 (2014), 392-410.  doi: 10.1137/130926250.  Google Scholar

[16]

M. V. Klibanov, On the first solution of a long standing problem: Uniqueness of the phaseless quantum inverse scattering problem in 3-d, Applied Mathematics Letters, 37 (2014), 82-85.  doi: 10.1016/j.aml.2014.06.005.  Google Scholar

[17]

M. V. Klibanov, Uniqueness of two phaseless non-overdetermined inverse acoustics problems in 3-d, Applicable Analysis, 93 (2014), 1135-1149.  doi: 10.1080/00036811.2013.818136.  Google Scholar

[18]

M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, SIAM J. Appl. Math., 76 (2016), 178-196.  doi: 10.1137/15M1022367.  Google Scholar

[19]

M. V. Klibanov and V. G. Romanov, Two reconstruction procedures for a 3-D phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32 (2016), 015005, 16pp.  doi: 10.1088/0266-5611/32/1/015005.  Google Scholar

[20]

M. V. Klibanov and V. G. Romanov, The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation, J. Inverse and Ill-Posed Problems, 23 (2015), 415-428.  doi: 10.1515/jiip-2015-0025.  Google Scholar

[21]

M. V. Klibanov and V. G. Romanov, Explicit formula for the solution of the phaseless inverse scattering problem of imaging of nano structures, J. Inverse and Ill-Posed Problems, 23 (2015), 187-193.  doi: 10.1515/jiip-2015-0004.  Google Scholar

[22]

M. V. KlibanovL. H. Nguyen and K. Pan, Nanostructures imaging via numerical solution of a 3-D inverse scattering problem without the phase information, Applied Numerical Mathematics, 110 (2016), 190-203.  doi: 10.1016/j.apnum.2016.08.014.  Google Scholar

[23]

O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Springer, New York, 1985. doi: 10.1007/978-1-4757-4317-3.  Google Scholar

[24]

M. M. Lavrentiev, V. G. Romanov and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis, AMS, Providence, RI, 1986. Google Scholar

[25]

J. LiH. LiuZ. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scattereres, SIAM J. Appl. Math., 73 (2013), 1721-1746.  doi: 10.1137/130907690.  Google Scholar

[26]

J. LiH. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, SIAM J. Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X.  Google Scholar

[27]

R. G. Novikov, A multidimensional inverse spectral problem for the equation $ -Δ ψ +(v(x)-Eu(x))ψ =0$, Funct. Anal. Appl., 22 (1988), 263-272.  doi: 10.1007/BF01077418.  Google Scholar

[28]

R. G. Novikov, The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator, J.Functional Analysis, 103 (1992), 409-463.  doi: 10.1016/0022-1236(92)90127-5.  Google Scholar

[29]

R. G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geometrical Analysis, 26 (2016), 346-359.  doi: 10.1007/s12220-014-9553-7.  Google Scholar

[30]

R. G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bulletin des Sciences Mathé matiques, 139 (2015), 923-936.  doi: 10.1016/j.bulsci.2015.04.005.  Google Scholar

[31]

T. C. PetersenaV. J. Keastb and D. M. Paganinc, Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation, Ultramisroscopy, 108 (2008), 805-815.  doi: 10.1016/j.ultramic.2008.01.001.  Google Scholar

[32]

V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1987.  Google Scholar

[33]

V. G. Romanov, Inverse problems for differential equations with memory, Eurasian J. of Mathematical and Computer Applications, 2 (2014), 51-80.   Google Scholar

[34]

A. RuhlandtM. KrenkelM. Bartels and T. Salditt, Three-dimensional phase retrieval in propagation-based phase-contrast imaging, Physical Review A, 89 (2014), 033847.   Google Scholar

[35]

U. Schröder and T. Schuster, An iterative method to reconstruct the refractive index of a medium from time-of-flight-measurements, Inverse Problems, 32 (2016), 085009, 35pp.  doi: 10.1088/0266-5611/32/8/085009.  Google Scholar

[36]

B. R. Vainberg, Principles of radiation, limiting absorption and limiting amplitude in the general theory of partial differential equations, Russian Math. Surveys, 21 (1966), 115-194.   Google Scholar

[37] B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach Science Publishers, New York, 1989.   Google Scholar
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