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

[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.

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

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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.

[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.

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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.

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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.

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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.

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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.

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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.

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

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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.

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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.

[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.

[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.

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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.

[24]

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

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[39]

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

[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.

[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.

[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.

[43]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

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

[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.

[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.

[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.

[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.

[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. 
[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[39]

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

[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.

[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.

[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.

[43]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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