June  2020, 14(3): 489-510. doi: 10.3934/ipi.2020023

Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency

1. 

Beijing Computational Science Research Center, Beijing 100193, China

2. 

LSEC, NCMIS and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

3. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

4. 

NCMIS and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

5. 

Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany

* Corresponding author: Haiwen Zhang

Received  July 2019 Revised  November 2019 Published  March 2020

Fund Project: This work is partly supported by the NNSF of China grants 91630309, 11871466 and 11571355, the NSAF grant U1930402, and the China Postdoctoral Science Foundation grant 2019TQ0023

This paper is concerned with uniqueness results in inverse acoustic and electromagnetic scattering problems with phaseless total-field data at a fixed frequency. We use superpositions of two point sources as the incident fields at a fixed frequency and measure the modulus of the acoustic total-field (called phaseless acoustic near-field data) on two spheres containing the scatterers generated by such incident fields on the two spheres. Based on this idea, we prove that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined from the phaseless acoustic near-field data at a fixed frequency. Moreover, the idea is also applied to the electromagnetic case, and it is proved that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined by the phaseless electric near-field data at a fixed frequency, that is, the modulus of the tangential component with the orientations $ \boldsymbol{e}_\phi $ and $ \boldsymbol{e}_\theta $, respectively, of the electric total-field measured on a sphere enclosing the scatters and generated by superpositions of two electric dipoles at a fixed frequency located on the measurement sphere and another bigger sphere with the polarization vectors $ \boldsymbol{e}_\phi $ and $ \boldsymbol{e}_\theta $, respectively. As far as we know, this is the first uniqueness result for three-dimensional inverse electromagnetic scattering with phaseless near-field data.

Citation: Xiaoxu Xu, Bo Zhang, Haiwen Zhang. Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency. Inverse Problems & Imaging, 2020, 14 (3) : 489-510. doi: 10.3934/ipi.2020023
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T. BrownI. Jeffrey and P. Mojabi, Multiplicatively regularized source reconstruction method for phaseless planar near-field antenna measurements, IEEE Trans. Antennas and Propagation, 65 (2017), 2020-2031.  doi: 10.1109/TAP.2017.2670518.  Google Scholar

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Z. Chen and G. Huang, Phaseless imaging by reverse time migration: Acoustic waves, Numer. Math. Theory Methods Appl., 10 (2017), 1-21.  doi: 10.4208/nmtma.2017.m1617.  Google Scholar

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T. IserniaG. Leone and R. Pierri, Radiation pattern evaluation from near-field intensities on planes, IEEE Trans. on Antennas and Propagation, 44 (1996), 701-710.  doi: 10.1109/8.496257.  Google Scholar

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

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X. JiX. Liu and B. Zhang, Target reconstruction with a reference point scatterer using phaseless far field patterns, SIAM J. Imaging Sci., 12 (2019), 372-391.  doi: 10.1137/18M1205789.  Google Scholar

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M. V. Klibanov, A phaseless inverse scattering problem for the 3-D Helmholtz equation, Inverse Probl. Imaging, 11 (2017), 263-276.  doi: 10.3934/ipi.2017013.  Google Scholar

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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

[27]

M. V. Klibanov and V. G. Romanov, Uniqueness of a 3-D coefficient inverse scattering problem without the phase information, Inverse Problems, 33 (2017), 095007, 10pp. doi: 10.1088/1361-6420/aa7a18.  Google Scholar

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J. Li, H. Liu and Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Problems, 33 (2017), 035011, 20pp. doi: 10.1088/1361-6420/aa5bf3.  Google Scholar

[30]

X. Liu, B. Zhang and G. Hu, Uniqueness in the inverse scattering problem in a piecewise homogeneous medium, Inverse Problems, 26 (2010), 015002, 14pp. doi: 10.1088/0266-5611/26/1/015002.  Google Scholar

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S. Maretzke and T. Hohage, Stability estimates for linearized near-field phase retrieval in X-ray phase contrast imaging, SIAM J. Appl. Math., 77 (2017), 384-408.  doi: 10.1137/16M1086170.  Google Scholar

[32]

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

[33]

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

[34]

L. PanY. ZhongX. Chen and S. P. Yeo, Subspace-based optimization method for inverse scattering problems utilizing phaseless data, IEEE Trans. Geoscience and Remote Sensing, 49 (2011), 981-987.  doi: 10.1109/TGRS.2010.2070512.  Google Scholar

[35]

R. PierriG. D'Elia and F. Soldovieri, A two probes scanning phaseless near-field far-field transformation technique, IEEE Trans. Antennas and Propagation, 47 (1999), 792-802.  doi: 10.1109/8.774132.  Google Scholar

[36]

S. F. Razavi and Y. Rahmat-Samii, Phaseless measurements over nonrectangular planar near-field systems without probe corotation, IEEE Trans. Antennas and Propagation, 61 (2013), 143-152.  doi: 10.1109/TAP.2012.2218211.  Google Scholar

[37]

V. G. Romanov and M. Yamamoto, Phaseless inverse problems with interference waves, J. Inverse Ill-Posed Probl., 26 (2018), 681-688.  doi: 10.1515/jiip-2018-0037.  Google Scholar

[38]

C. H. SchmidtS. F. RazaviT. F. Eibert and Y. Rahmat-Samii, Phaseless spherical near-field antenna measurements for low and medium gain antennas, Adv. Radio Sci., 8 (2010), 43-48.  doi: 10.5194/ars-8-43-2010.  Google Scholar

[39]

J. Shin, Inverse obstacle backscattering problems with phaseless data, European J. Appl. Math., 27 (2016), 111-130.  doi: 10.1017/S0956792515000406.  Google Scholar

[40]

F. Sun, D. Zhang and Y. Guo, Uniqueness in phaseless inverse scattering problems with known superposition of incident point sources, Inverse Problems, 35 (2019), 105007. doi: 10.1088/1361-6420/ab3373.  Google Scholar

[41]

T. TakenakaD. J. N. WallH. Harada and M. Tanaka, Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field, Microwave and Optical Technology Letters, 14 (1998), 182-188.  doi: 10.1002/(SICI)1098-2760(19970220)14:3<182::AID-MOP15>3.0.CO;2-A.  Google Scholar

[42]

X. XuB. Zhang and H. Zhang, Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data, SIAM J. Imaging Sci., 12 (2019), 119-152.  doi: 10.1137/18M1210204.  Google Scholar

[43]

X. Xu, Uniqueness in Phaseless Inverse Acoustic and Electromagnetic Scattering Problems, Ph.D thesis, University of Chinese Academy of Sciences, Beijing, 2019. Google Scholar

[44]

X. XuB. Zhang and H. Zhang, Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency, SIAM J. Appl. Math., 78 (2018), 1737-1753.  doi: 10.1137/17M1149699.  Google Scholar

[45]

X. XuB. Zhang and H. Zhang, Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency. II, SIAM J. Appl. Math., 78 (2018), 3024-3039.  doi: 10.1137/18M1196820.  Google Scholar

[46]

B. Zhang and H. Zhang, Imaging of locally rough surfaces from intensity-only far-field or near-field data, Inverse Problems, 33 (2017), 055001, 28pp. doi: 10.1088/1361-6420/aa5fc8.  Google Scholar

[47]

B. Zhang and H. Zhang, Recovering scattering obstacles by multi-frequency phaseless far-field data, J. Comput. Phys., 345 (2017), 58-73.  doi: 10.1016/j.jcp.2017.05.022.  Google Scholar

[48]

B. Zhang and H. Zhang, Fast imaging of scattering obstacles from phaseless far-field measurements at a fixed frequency, Inverse Problems, 34 (2018), 104005, 24pp. doi: 10.1088/1361-6420/aad81f.  Google Scholar

[49]

D. Zhang and Y. Guo, Uniqueness results on phaseless inverse acoustic scattering with a reference ball, Inverse Problems, 34 (2018), 085002, 12pp. doi: 10.1088/1361-6420/aac53c.  Google Scholar

[50]

D. Zhang, F. Sun, Y. Guo and H. Liu, Uniqueness in inverse acoustic scattering with phaseless near-field measurements, preprint, 2019, arXiv: 1905.08242v1. Google Scholar

[51]

D. Zhang, Y. Wang, Y. Guo and J. Li, Uniqueness in inverse cavity scattering problems with phaseless near-field data, Inverse Problems, 36 (2020), 025004, 10pp. doi: 10.1088/1361-6420/ab53ee.  Google Scholar

show all references

References:
[1]

H. AmmariY. T. Chow and J. Zou, Phased and phaseless domain reconstructions in the 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. Opt. Soc. Am. 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]

E. Blåsten, The Inverse Problem of the Schrödinger Equation in the Plane, Licentiate thesis, University of Helsinki, 2011, arXiv: 1103.6200v1. Google Scholar

[5]

T. BrownI. Jeffrey and P. Mojabi, Multiplicatively regularized source reconstruction method for phaseless planar near-field antenna measurements, IEEE Trans. Antennas and Propagation, 65 (2017), 2020-2031.  doi: 10.1109/TAP.2017.2670518.  Google Scholar

[6]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33.  doi: 10.1515/jiip.2008.002.  Google Scholar

[7]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences, Vol. 188, Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar

[8]

F. CakoniD. Colton and P. Monk, The electromagnetic inverse-scattering problem for partly coated Lipschitz domains, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 661-682.  doi: 10.1017/S0308210500003413.  Google Scholar

[9]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 80, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9780898719406.  Google Scholar

[10]

X. Chen, Computational Methods for Electromagnetic Inverse Scattering, John Wiley & Sons, Singapore, 2018. doi: 10.1002/9781119311997.  Google Scholar

[11]

Z. ChenS. Fang and G. Huang, A direct imaging method for the half-space inverse scattering problem with phaseless data, Inverse Probl. Imaging, 11 (2017), 901-916.  doi: 10.3934/ipi.2017042.  Google Scholar

[12]

Z. Chen and G. Huang, A direct imaging method for electromagnetic scattering data without phase information, SIAM J. Imaging Sci., 9 (2016), 1273-1297.  doi: 10.1137/15M1053475.  Google Scholar

[13]

Z. Chen and G. Huang, Phaseless imaging by reverse time migration: Acoustic waves, Numer. Math. Theory Methods Appl., 10 (2017), 1-21.  doi: 10.4208/nmtma.2017.m1617.  Google Scholar

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^rd$ edition, Applied Mathematical Sciences, Vol. 93, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[15]

P. Hähner, On Acoustic, Electromagnetic, and Elastic Scattering Problems in Inhomogeneous Media, Habilitation thesis, Göttingen, 1998. Google Scholar

[16] J. E. Hansen, Spherical Near-Field Antenna Measurements, IET Electromagnetic Waves Series, Vol. 26, IET Press, London, 1988.   Google Scholar
[17]

T. IserniaG. Leone and R. Pierri, Radiation pattern evaluation from near-field intensities on planes, IEEE Trans. on Antennas and Propagation, 44 (1996), 701-710.  doi: 10.1109/8.496257.  Google Scholar

[18]

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

[19]

X. JiX. Liu and B. Zhang, Target reconstruction with a reference point scatterer using phaseless far field patterns, SIAM J. Imaging Sci., 12 (2019), 372-391.  doi: 10.1137/18M1205789.  Google Scholar

[20]

X. Ji, X. Liu and B. Zhang, Phaseless inverse source scattering problem: Phase retrieval, uniqueness and direct sampling methods, Journal of Computational Physics: X, 1 (2019), 100003. doi: 10.1016/j.jcpx.2019.100003.  Google Scholar

[21]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, 2$^nd$ edition, Applied Mathematical Sciences, Vol. 120, Springer, New York, 2011. doi: 10.1007/978-1-4419-8474-6.  Google Scholar

[22] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, Vol. 36, Oxford University Press, Oxford, 2008.   Google Scholar
[23]

A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations, Expansion-, Integral-, and Variational Methods, Applied Mathematical Sciences, Vol. 190, Springer, Cham, 2015. doi: 10.1007/978-3-319-11086-8.  Google Scholar

[24]

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

[25]

M. V. Klibanov, A phaseless inverse scattering problem for the 3-D Helmholtz equation, Inverse Probl. Imaging, 11 (2017), 263-276.  doi: 10.3934/ipi.2017013.  Google Scholar

[26]

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

[27]

M. V. Klibanov and V. G. Romanov, Uniqueness of a 3-D coefficient inverse scattering problem without the phase information, Inverse Problems, 33 (2017), 095007, 10pp. doi: 10.1088/1361-6420/aa7a18.  Google Scholar

[28]

R. Leis, Initial-Boundary Value Problems in Mathematical Physics, B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986. doi: 10.1007/978-3-663-10649-4.  Google Scholar

[29]

J. Li, H. Liu and Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Problems, 33 (2017), 035011, 20pp. doi: 10.1088/1361-6420/aa5bf3.  Google Scholar

[30]

X. Liu, B. Zhang and G. Hu, Uniqueness in the inverse scattering problem in a piecewise homogeneous medium, Inverse Problems, 26 (2010), 015002, 14pp. doi: 10.1088/0266-5611/26/1/015002.  Google Scholar

[31]

S. Maretzke and T. Hohage, Stability estimates for linearized near-field phase retrieval in X-ray phase contrast imaging, SIAM J. Appl. Math., 77 (2017), 384-408.  doi: 10.1137/16M1086170.  Google Scholar

[32]

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

[33]

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

[34]

L. PanY. ZhongX. Chen and S. P. Yeo, Subspace-based optimization method for inverse scattering problems utilizing phaseless data, IEEE Trans. Geoscience and Remote Sensing, 49 (2011), 981-987.  doi: 10.1109/TGRS.2010.2070512.  Google Scholar

[35]

R. PierriG. D'Elia and F. Soldovieri, A two probes scanning phaseless near-field far-field transformation technique, IEEE Trans. Antennas and Propagation, 47 (1999), 792-802.  doi: 10.1109/8.774132.  Google Scholar

[36]

S. F. Razavi and Y. Rahmat-Samii, Phaseless measurements over nonrectangular planar near-field systems without probe corotation, IEEE Trans. Antennas and Propagation, 61 (2013), 143-152.  doi: 10.1109/TAP.2012.2218211.  Google Scholar

[37]

V. G. Romanov and M. Yamamoto, Phaseless inverse problems with interference waves, J. Inverse Ill-Posed Probl., 26 (2018), 681-688.  doi: 10.1515/jiip-2018-0037.  Google Scholar

[38]

C. H. SchmidtS. F. RazaviT. F. Eibert and Y. Rahmat-Samii, Phaseless spherical near-field antenna measurements for low and medium gain antennas, Adv. Radio Sci., 8 (2010), 43-48.  doi: 10.5194/ars-8-43-2010.  Google Scholar

[39]

J. Shin, Inverse obstacle backscattering problems with phaseless data, European J. Appl. Math., 27 (2016), 111-130.  doi: 10.1017/S0956792515000406.  Google Scholar

[40]

F. Sun, D. Zhang and Y. Guo, Uniqueness in phaseless inverse scattering problems with known superposition of incident point sources, Inverse Problems, 35 (2019), 105007. doi: 10.1088/1361-6420/ab3373.  Google Scholar

[41]

T. TakenakaD. J. N. WallH. Harada and M. Tanaka, Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field, Microwave and Optical Technology Letters, 14 (1998), 182-188.  doi: 10.1002/(SICI)1098-2760(19970220)14:3<182::AID-MOP15>3.0.CO;2-A.  Google Scholar

[42]

X. XuB. Zhang and H. Zhang, Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data, SIAM J. Imaging Sci., 12 (2019), 119-152.  doi: 10.1137/18M1210204.  Google Scholar

[43]

X. Xu, Uniqueness in Phaseless Inverse Acoustic and Electromagnetic Scattering Problems, Ph.D thesis, University of Chinese Academy of Sciences, Beijing, 2019. Google Scholar

[44]

X. XuB. Zhang and H. Zhang, Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency, SIAM J. Appl. Math., 78 (2018), 1737-1753.  doi: 10.1137/17M1149699.  Google Scholar

[45]

X. XuB. Zhang and H. Zhang, Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency. II, SIAM J. Appl. Math., 78 (2018), 3024-3039.  doi: 10.1137/18M1196820.  Google Scholar

[46]

B. Zhang and H. Zhang, Imaging of locally rough surfaces from intensity-only far-field or near-field data, Inverse Problems, 33 (2017), 055001, 28pp. doi: 10.1088/1361-6420/aa5fc8.  Google Scholar

[47]

B. Zhang and H. Zhang, Recovering scattering obstacles by multi-frequency phaseless far-field data, J. Comput. Phys., 345 (2017), 58-73.  doi: 10.1016/j.jcp.2017.05.022.  Google Scholar

[48]

B. Zhang and H. Zhang, Fast imaging of scattering obstacles from phaseless far-field measurements at a fixed frequency, Inverse Problems, 34 (2018), 104005, 24pp. doi: 10.1088/1361-6420/aad81f.  Google Scholar

[49]

D. Zhang and Y. Guo, Uniqueness results on phaseless inverse acoustic scattering with a reference ball, Inverse Problems, 34 (2018), 085002, 12pp. doi: 10.1088/1361-6420/aac53c.  Google Scholar

[50]

D. Zhang, F. Sun, Y. Guo and H. Liu, Uniqueness in inverse acoustic scattering with phaseless near-field measurements, preprint, 2019, arXiv: 1905.08242v1. Google Scholar

[51]

D. Zhang, Y. Wang, Y. Guo and J. Li, Uniqueness in inverse cavity scattering problems with phaseless near-field data, Inverse Problems, 36 (2020), 025004, 10pp. doi: 10.1088/1361-6420/ab53ee.  Google Scholar

Figure 1.  Acoustic scattering by an obstacle (left) or a medium (right)
Figure 2.  Electromagnetic scattering by an obstacle (left) or a medium (right)
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