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 and 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.

[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.

[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.

[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.

[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. 
[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.

[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.

[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. 

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[34]

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

[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.

[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. 

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

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.

[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.

[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.

[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.

[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. 
[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.

[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.

[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. 

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[34]

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

[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.

[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. 

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