Abstract

Effective transmission of information through scattering media has been of great importance in imaging systems and beneficial to high capacity wireless communication. Despite numerous attempts to achieve high-resolution sub-diffraction-limited imaging through employing the engineered structures such as the so-called metamaterials or utilizing techniques like time reversal methods, the proposed ideas suffer from the fundamental limitations for design and practical realization. In this paper, we investigate disorder-based engineered scattering structures and introduce a novel technique for achieving super-resolution based on designing and employing engineered all-dielectric medium. We show that disorder in the proposed design can be exploited to significantly modify the information content of scattered fields in the far-field region. Under the presence of the designed structures, using computational methods, signals associated with ultra sub-wavelength features of the illuminating sources can be enhanced and extracted from the far-field image. Not only can the presented approach lead to remarkable enhancement of resolution in such systems, but also orthogonal transmission channels are attainable when the closely-packed sources are excited properly. The latter provides a new scheme for encoding and multiplexing signals leading to the enhancement of information capacity in emerging information processing systems. The design procedure and physical constraints are studied and discussed.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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2017 (1)

2016 (5)

A. Ozcan and E. McLeod, “Lensless imaging and sensing,” Annu. Rev. Biomed. Eng. 18(1), 77–102 (2016).
[Crossref] [PubMed]

M. Haghtalab, R. Faraji-Dana, and S. Safavi-Naeini, “Design and Analysis of Disordered Optical Nanoantenna Structures,” J. Lightwave Technol. 34(11), 2838–2847 (2016).
[Crossref]

M. Lis, M. Plaut, A. Zai, D. Cipolle, J. Russo, J. Lewis, and T. Fedynyshyn, “Polymer Dielectrics for 3D‐Printed RF Devices in the Ka Band,” Adv. Mater. Technol. 1(2), 1600027 (2016).

S. Peng, R. Zhang, V. H. Chen, E. T. Khabiboulline, P. Braun, and H. A. Atwater, “Three-dimensional single gyroid photonic crystals with a mid-infrared bandgap,” ACS Photonics 3(6), 1131–1137 (2016).
[Crossref]

A. Porat, E. R. Andresen, H. Rigneault, D. Oron, S. Gigan, and O. Katz, “Widefield lensless imaging through a fiber bundle via speckle correlations,” Opt. Express 24(15), 16835–16855 (2016).
[Crossref] [PubMed]

2015 (2)

2014 (6)

L. Lianlin, “Subwavelength imaging of sparse broadband sources in an open disordered medium from a single antenna,” IEEE Antennas Wirel. Propag. Lett. 13, 1461–1464 (2014).
[Crossref]

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89(22), 224202 (2014).
[Crossref]

L. Tian, X. Li, K. Ramchandran, and L. Waller, “Multiplexed coded illumination for Fourier Ptychography with an LED array microscope,” Biomed. Opt. Express 5(7), 2376–2389 (2014).
[Crossref] [PubMed]

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

L. Li, F. Li, and T. J. Cui, “Feasibility of resonant metalens for the subwavelength imaging using a single sensor in the far field,” Opt. Express 22(15), 18688–18697 (2014).
[Crossref] [PubMed]

2013 (3)

L. Cherkezyan, I. Capoglu, H. Subramanian, J. D. Rogers, D. Damania, A. Taflove, and V. Backman, “Interferometric spectroscopy of scattered light can quantify the statistics of subdiffractional refractive-index fluctuations,” Phys. Rev. Lett. 111(3), 033903 (2013).
[Crossref] [PubMed]

W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, and W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white light-emitting diode lighting,” Appl. Opt. 52(12), 2602–2609 (2013).
[Crossref] [PubMed]

S. Shahir, M. Mohajer, A. Rohani, and S. Safavi-Naeini, “Permittivity profile estimation based on non-radiating equivalent source (2d case),” Prog. Electromagnetics Research B 50, 157–175 (2013).
[Crossref]

2012 (3)

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109(12), 127701 (2012).
[Crossref] [PubMed]

S. A. Alexandrov, S. Uttam, R. K. Bista, K. Staton, and Y. Liu, “Spectral encoding of spatial frequency approach for characterization of nanoscale structures,” Appl. Phys. Lett. 101(3), 33702 (2012).
[Crossref] [PubMed]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

2010 (2)

B. Khademhosseinieh, I. Sencan, G. Biener, T. W. Su, A. F. Coskun, D. Tseng, and A. Ozcan, “Lensfree on-chip imaging using nanostructured surfaces,” Appl. Phys. Lett. 96(17), 171106 (2010).
[Crossref] [PubMed]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4(5), 320–322 (2010).
[Crossref]

2009 (1)

2008 (3)

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[Crossref] [PubMed]

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[Crossref] [PubMed]

J. A. Lewis, “Novel inks for direct-write assembly of 3-D periodic structures,” Mater. Matters 3(1), 4–7 (2008).

2007 (3)

S. W. Hell, “Far-field optical nanoscopy,” Science 316(5828), 1153–1158 (2007).
[Crossref] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[Crossref] [PubMed]

2005 (1)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

1987 (1)

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the “simulated annealing”,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[Crossref]

1982 (1)

Alexandrov, S. A.

S. A. Alexandrov, S. Uttam, R. K. Bista, K. Staton, and Y. Liu, “Spectral encoding of spatial frequency approach for characterization of nanoscale structures,” Appl. Phys. Lett. 101(3), 33702 (2012).
[Crossref] [PubMed]

Andresen, E. R.

Atwater, H. A.

S. Peng, R. Zhang, V. H. Chen, E. T. Khabiboulline, P. Braun, and H. A. Atwater, “Three-dimensional single gyroid photonic crystals with a mid-infrared bandgap,” ACS Photonics 3(6), 1131–1137 (2016).
[Crossref]

Backman, V.

L. Cherkezyan, I. Capoglu, H. Subramanian, J. D. Rogers, D. Damania, A. Taflove, and V. Backman, “Interferometric spectroscopy of scattered light can quantify the statistics of subdiffractional refractive-index fluctuations,” Phys. Rev. Lett. 111(3), 033903 (2013).
[Crossref] [PubMed]

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, J. D. Rogers, H. K. Roy, R. E. Brand, and V. Backman, “Partial-wave microscopic spectroscopy detects subwavelength refractive index fluctuations: an application to cancer diagnosis,” Opt. Lett. 34(4), 518–520 (2009).
[Crossref] [PubMed]

Bates, M.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[Crossref] [PubMed]

Bertolotti, J.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Biener, G.

B. Khademhosseinieh, I. Sencan, G. Biener, T. W. Su, A. F. Coskun, D. Tseng, and A. Ozcan, “Lensfree on-chip imaging using nanostructured surfaces,” Appl. Phys. Lett. 96(17), 171106 (2010).
[Crossref] [PubMed]

Bista, R. K.

S. A. Alexandrov, S. Uttam, R. K. Bista, K. Staton, and Y. Liu, “Spectral encoding of spatial frequency approach for characterization of nanoscale structures,” Appl. Phys. Lett. 101(3), 33702 (2012).
[Crossref] [PubMed]

Blochet, B.

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Boniface, A.

Brand, R. E.

Braun, P.

S. Peng, R. Zhang, V. H. Chen, E. T. Khabiboulline, P. Braun, and H. A. Atwater, “Three-dimensional single gyroid photonic crystals with a mid-infrared bandgap,” ACS Photonics 3(6), 1131–1137 (2016).
[Crossref]

Cao, H.

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89(22), 224202 (2014).
[Crossref]

Capoglu, I.

L. Cherkezyan, I. Capoglu, H. Subramanian, J. D. Rogers, D. Damania, A. Taflove, and V. Backman, “Interferometric spectroscopy of scattered light can quantify the statistics of subdiffractional refractive-index fluctuations,” Phys. Rev. Lett. 111(3), 033903 (2013).
[Crossref] [PubMed]

Capoglu, I. R.

Carron, I.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

Chardon, G.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

Chen, V. H.

S. Peng, R. Zhang, V. H. Chen, E. T. Khabiboulline, P. Braun, and H. A. Atwater, “Three-dimensional single gyroid photonic crystals with a mid-infrared bandgap,” ACS Photonics 3(6), 1131–1137 (2016).
[Crossref]

Cherkezyan, L.

L. Cherkezyan, I. Capoglu, H. Subramanian, J. D. Rogers, D. Damania, A. Taflove, and V. Backman, “Interferometric spectroscopy of scattered light can quantify the statistics of subdiffractional refractive-index fluctuations,” Phys. Rev. Lett. 111(3), 033903 (2013).
[Crossref] [PubMed]

Cipolle, D.

M. Lis, M. Plaut, A. Zai, D. Cipolle, J. Russo, J. Lewis, and T. Fedynyshyn, “Polymer Dielectrics for 3D‐Printed RF Devices in the Ka Band,” Adv. Mater. Technol. 1(2), 1600027 (2016).

Corana, A.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the “simulated annealing”,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[Crossref]

Coskun, A. F.

B. Khademhosseinieh, I. Sencan, G. Biener, T. W. Su, A. F. Coskun, D. Tseng, and A. Ozcan, “Lensfree on-chip imaging using nanostructured surfaces,” Appl. Phys. Lett. 96(17), 171106 (2010).
[Crossref] [PubMed]

Cui, T. J.

Damania, D.

L. Cherkezyan, I. Capoglu, H. Subramanian, J. D. Rogers, D. Damania, A. Taflove, and V. Backman, “Interferometric spectroscopy of scattered light can quantify the statistics of subdiffractional refractive-index fluctuations,” Phys. Rev. Lett. 111(3), 033903 (2013).
[Crossref] [PubMed]

Daudet, L.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

de Rosny, J.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[Crossref] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Faraji-Dana, R.

Fedynyshyn, T.

M. Lis, M. Plaut, A. Zai, D. Cipolle, J. Russo, J. Lewis, and T. Fedynyshyn, “Polymer Dielectrics for 3D‐Printed RF Devices in the Ka Band,” Adv. Mater. Technol. 1(2), 1600027 (2016).

Feichtner, T.

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109(12), 127701 (2012).
[Crossref] [PubMed]

Feld, M. S.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[Crossref] [PubMed]

Fienup, J. R.

Fink, M.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[Crossref] [PubMed]

Gigan, S.

A. Boniface, M. Mounaix, B. Blochet, R. Piestun, and S. Gigan, “Transmission-matrix-based point-spread-function engineering through a complex medium,” Optica 4(1), 54–59 (2017).
[Crossref]

A. Porat, E. R. Andresen, H. Rigneault, D. Oron, S. Gigan, and O. Katz, “Widefield lensless imaging through a fiber bundle via speckle correlations,” Opt. Express 24(15), 16835–16855 (2016).
[Crossref] [PubMed]

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

Haghtalab, M.

Hecht, B.

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109(12), 127701 (2012).
[Crossref] [PubMed]

Heidmann, P.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

Hell, S. W.

S. W. Hell, “Far-field optical nanoscopy,” Science 316(5828), 1153–1158 (2007).
[Crossref] [PubMed]

Huang, B.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[Crossref] [PubMed]

IJzerman, W. L.

Katz, O.

A. Porat, E. R. Andresen, H. Rigneault, D. Oron, S. Gigan, and O. Katz, “Widefield lensless imaging through a fiber bundle via speckle correlations,” Opt. Express 24(15), 16835–16855 (2016).
[Crossref] [PubMed]

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

Khabiboulline, E. T.

S. Peng, R. Zhang, V. H. Chen, E. T. Khabiboulline, P. Braun, and H. A. Atwater, “Three-dimensional single gyroid photonic crystals with a mid-infrared bandgap,” ACS Photonics 3(6), 1131–1137 (2016).
[Crossref]

Khademhosseinieh, B.

B. Khademhosseinieh, I. Sencan, G. Biener, T. W. Su, A. F. Coskun, D. Tseng, and A. Ozcan, “Lensfree on-chip imaging using nanostructured surfaces,” Appl. Phys. Lett. 96(17), 171106 (2010).
[Crossref] [PubMed]

Kiunke, M.

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109(12), 127701 (2012).
[Crossref] [PubMed]

Lagendijk, A.

W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, and W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white light-emitting diode lighting,” Appl. Opt. 52(12), 2602–2609 (2013).
[Crossref] [PubMed]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4(5), 320–322 (2010).
[Crossref]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Lerosey, G.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[Crossref] [PubMed]

Lewis, J.

M. Lis, M. Plaut, A. Zai, D. Cipolle, J. Russo, J. Lewis, and T. Fedynyshyn, “Polymer Dielectrics for 3D‐Printed RF Devices in the Ka Band,” Adv. Mater. Technol. 1(2), 1600027 (2016).

Lewis, J. A.

J. A. Lewis, “Novel inks for direct-write assembly of 3-D periodic structures,” Mater. Matters 3(1), 4–7 (2008).

Li, F.

Li, L.

Li, X.

Lianlin, L.

L. Lianlin, “Subwavelength imaging of sparse broadband sources in an open disordered medium from a single antenna,” IEEE Antennas Wirel. Propag. Lett. 13, 1461–1464 (2014).
[Crossref]

Liew, S. F.

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89(22), 224202 (2014).
[Crossref]

Lis, M.

M. Lis, M. Plaut, A. Zai, D. Cipolle, J. Russo, J. Lewis, and T. Fedynyshyn, “Polymer Dielectrics for 3D‐Printed RF Devices in the Ka Band,” Adv. Mater. Technol. 1(2), 1600027 (2016).

Liu, Y.

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

Liutkus, A.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

Marchesi, M.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the “simulated annealing”,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[Crossref]

Martina, D.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

Martini, C.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the “simulated annealing”,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[Crossref]

McLeod, E.

A. Ozcan and E. McLeod, “Lensless imaging and sensing,” Annu. Rev. Biomed. Eng. 18(1), 77–102 (2016).
[Crossref] [PubMed]

Mohajer, M.

S. Shahir, M. Mohajer, A. Rohani, and S. Safavi-Naeini, “Permittivity profile estimation based on non-radiating equivalent source (2d case),” Prog. Electromagnetics Research B 50, 157–175 (2013).
[Crossref]

Mosk, A. P.

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89(22), 224202 (2014).
[Crossref]

W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, and W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white light-emitting diode lighting,” Appl. Opt. 52(12), 2602–2609 (2013).
[Crossref] [PubMed]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4(5), 320–322 (2010).
[Crossref]

Mounaix, M.

Oron, D.

Ozcan, A.

A. Ozcan and E. McLeod, “Lensless imaging and sensing,” Annu. Rev. Biomed. Eng. 18(1), 77–102 (2016).
[Crossref] [PubMed]

B. Khademhosseinieh, I. Sencan, G. Biener, T. W. Su, A. F. Coskun, D. Tseng, and A. Ozcan, “Lensfree on-chip imaging using nanostructured surfaces,” Appl. Phys. Lett. 96(17), 171106 (2010).
[Crossref] [PubMed]

Peng, S.

S. Peng, R. Zhang, V. H. Chen, E. T. Khabiboulline, P. Braun, and H. A. Atwater, “Three-dimensional single gyroid photonic crystals with a mid-infrared bandgap,” ACS Photonics 3(6), 1131–1137 (2016).
[Crossref]

Piestun, R.

Plaut, M.

M. Lis, M. Plaut, A. Zai, D. Cipolle, J. Russo, J. Lewis, and T. Fedynyshyn, “Polymer Dielectrics for 3D‐Printed RF Devices in the Ka Band,” Adv. Mater. Technol. 1(2), 1600027 (2016).

Popoff, S.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

Popoff, S. M.

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89(22), 224202 (2014).
[Crossref]

Porat, A.

Pradhan, P.

Psaltis, D.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[Crossref] [PubMed]

Ramchandran, K.

Ridella, S.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the “simulated annealing”,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[Crossref]

Rigneault, H.

Rogers, J. D.

L. Cherkezyan, I. Capoglu, H. Subramanian, J. D. Rogers, D. Damania, A. Taflove, and V. Backman, “Interferometric spectroscopy of scattered light can quantify the statistics of subdiffractional refractive-index fluctuations,” Phys. Rev. Lett. 111(3), 033903 (2013).
[Crossref] [PubMed]

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, J. D. Rogers, H. K. Roy, R. E. Brand, and V. Backman, “Partial-wave microscopic spectroscopy detects subwavelength refractive index fluctuations: an application to cancer diagnosis,” Opt. Lett. 34(4), 518–520 (2009).
[Crossref] [PubMed]

Rohani, A.

S. Shahir, M. Mohajer, A. Rohani, and S. Safavi-Naeini, “Permittivity profile estimation based on non-radiating equivalent source (2d case),” Prog. Electromagnetics Research B 50, 157–175 (2013).
[Crossref]

Roy, H. K.

Russo, J.

M. Lis, M. Plaut, A. Zai, D. Cipolle, J. Russo, J. Lewis, and T. Fedynyshyn, “Polymer Dielectrics for 3D‐Printed RF Devices in the Ka Band,” Adv. Mater. Technol. 1(2), 1600027 (2016).

Safavi-Naeini, S.

M. Haghtalab, R. Faraji-Dana, and S. Safavi-Naeini, “Design and Analysis of Disordered Optical Nanoantenna Structures,” J. Lightwave Technol. 34(11), 2838–2847 (2016).
[Crossref]

S. Shahir, M. Mohajer, A. Rohani, and S. Safavi-Naeini, “Permittivity profile estimation based on non-radiating equivalent source (2d case),” Prog. Electromagnetics Research B 50, 157–175 (2013).
[Crossref]

Selig, O.

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109(12), 127701 (2012).
[Crossref] [PubMed]

Sencan, I.

B. Khademhosseinieh, I. Sencan, G. Biener, T. W. Su, A. F. Coskun, D. Tseng, and A. Ozcan, “Lensfree on-chip imaging using nanostructured surfaces,” Appl. Phys. Lett. 96(17), 171106 (2010).
[Crossref] [PubMed]

Shahir, S.

S. Shahir, M. Mohajer, A. Rohani, and S. Safavi-Naeini, “Permittivity profile estimation based on non-radiating equivalent source (2d case),” Prog. Electromagnetics Research B 50, 157–175 (2013).
[Crossref]

Staton, K.

S. A. Alexandrov, S. Uttam, R. K. Bista, K. Staton, and Y. Liu, “Spectral encoding of spatial frequency approach for characterization of nanoscale structures,” Appl. Phys. Lett. 101(3), 33702 (2012).
[Crossref] [PubMed]

Su, T. W.

B. Khademhosseinieh, I. Sencan, G. Biener, T. W. Su, A. F. Coskun, D. Tseng, and A. Ozcan, “Lensfree on-chip imaging using nanostructured surfaces,” Appl. Phys. Lett. 96(17), 171106 (2010).
[Crossref] [PubMed]

Subramanian, H.

L. Cherkezyan, I. Capoglu, H. Subramanian, J. D. Rogers, D. Damania, A. Taflove, and V. Backman, “Interferometric spectroscopy of scattered light can quantify the statistics of subdiffractional refractive-index fluctuations,” Phys. Rev. Lett. 111(3), 033903 (2013).
[Crossref] [PubMed]

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, J. D. Rogers, H. K. Roy, R. E. Brand, and V. Backman, “Partial-wave microscopic spectroscopy detects subwavelength refractive index fluctuations: an application to cancer diagnosis,” Opt. Lett. 34(4), 518–520 (2009).
[Crossref] [PubMed]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Taflove, A.

L. Cherkezyan, I. Capoglu, H. Subramanian, J. D. Rogers, D. Damania, A. Taflove, and V. Backman, “Interferometric spectroscopy of scattered light can quantify the statistics of subdiffractional refractive-index fluctuations,” Phys. Rev. Lett. 111(3), 033903 (2013).
[Crossref] [PubMed]

Tian, L.

Tourin, A.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[Crossref] [PubMed]

Tseng, D.

B. Khademhosseinieh, I. Sencan, G. Biener, T. W. Su, A. F. Coskun, D. Tseng, and A. Ozcan, “Lensfree on-chip imaging using nanostructured surfaces,” Appl. Phys. Lett. 96(17), 171106 (2010).
[Crossref] [PubMed]

Tukker, T. W.

Uttam, S.

S. A. Alexandrov, S. Uttam, R. K. Bista, K. Staton, and Y. Liu, “Spectral encoding of spatial frequency approach for characterization of nanoscale structures,” Appl. Phys. Lett. 101(3), 33702 (2012).
[Crossref] [PubMed]

van Putten, E. G.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4(5), 320–322 (2010).
[Crossref]

Vos, W. L.

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89(22), 224202 (2014).
[Crossref]

W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, and W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white light-emitting diode lighting,” Appl. Opt. 52(12), 2602–2609 (2013).
[Crossref] [PubMed]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Waller, L.

Wang, W.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[Crossref] [PubMed]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

Yang, C.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[Crossref] [PubMed]

Yao, K.

Yaqoob, Z.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[Crossref] [PubMed]

Zai, A.

M. Lis, M. Plaut, A. Zai, D. Cipolle, J. Russo, J. Lewis, and T. Fedynyshyn, “Polymer Dielectrics for 3D‐Printed RF Devices in the Ka Band,” Adv. Mater. Technol. 1(2), 1600027 (2016).

Zhang, R.

S. Peng, R. Zhang, V. H. Chen, E. T. Khabiboulline, P. Braun, and H. A. Atwater, “Three-dimensional single gyroid photonic crystals with a mid-infrared bandgap,” ACS Photonics 3(6), 1131–1137 (2016).
[Crossref]

Zhang, X.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Zhuang, X.

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[Crossref] [PubMed]

ACM Trans. Math. Softw. (1)

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the “simulated annealing”,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[Crossref]

ACS Photonics (1)

S. Peng, R. Zhang, V. H. Chen, E. T. Khabiboulline, P. Braun, and H. A. Atwater, “Three-dimensional single gyroid photonic crystals with a mid-infrared bandgap,” ACS Photonics 3(6), 1131–1137 (2016).
[Crossref]

Adv. Mater. Technol. (1)

M. Lis, M. Plaut, A. Zai, D. Cipolle, J. Russo, J. Lewis, and T. Fedynyshyn, “Polymer Dielectrics for 3D‐Printed RF Devices in the Ka Band,” Adv. Mater. Technol. 1(2), 1600027 (2016).

Annu. Rev. Biomed. Eng. (1)

A. Ozcan and E. McLeod, “Lensless imaging and sensing,” Annu. Rev. Biomed. Eng. 18(1), 77–102 (2016).
[Crossref] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

B. Khademhosseinieh, I. Sencan, G. Biener, T. W. Su, A. F. Coskun, D. Tseng, and A. Ozcan, “Lensfree on-chip imaging using nanostructured surfaces,” Appl. Phys. Lett. 96(17), 171106 (2010).
[Crossref] [PubMed]

S. A. Alexandrov, S. Uttam, R. K. Bista, K. Staton, and Y. Liu, “Spectral encoding of spatial frequency approach for characterization of nanoscale structures,” Appl. Phys. Lett. 101(3), 33702 (2012).
[Crossref] [PubMed]

Biomed. Opt. Express (1)

IEEE Antennas Wirel. Propag. Lett. (1)

L. Lianlin, “Subwavelength imaging of sparse broadband sources in an open disordered medium from a single antenna,” IEEE Antennas Wirel. Propag. Lett. 13, 1461–1464 (2014).
[Crossref]

J. Lightwave Technol. (1)

Mater. Matters (1)

J. A. Lewis, “Novel inks for direct-write assembly of 3-D periodic structures,” Mater. Matters 3(1), 4–7 (2008).

Nat. Photonics (3)

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[Crossref] [PubMed]

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4(5), 320–322 (2010).
[Crossref]

Nature (2)

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491(7423), 232–234 (2012).
[Crossref] [PubMed]

L. Waller and L. Tian, “Computational imaging: Machine learning for 3D microscopy,” Nature 523(7561), 416–417 (2015).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (1)

Optica (1)

Phys. Rev. B (1)

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89(22), 224202 (2014).
[Crossref]

Phys. Rev. Lett. (2)

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109(12), 127701 (2012).
[Crossref] [PubMed]

L. Cherkezyan, I. Capoglu, H. Subramanian, J. D. Rogers, D. Damania, A. Taflove, and V. Backman, “Interferometric spectroscopy of scattered light can quantify the statistics of subdiffractional refractive-index fluctuations,” Phys. Rev. Lett. 111(3), 033903 (2013).
[Crossref] [PubMed]

Prog. Electromagnetics Research B (1)

S. Shahir, M. Mohajer, A. Rohani, and S. Safavi-Naeini, “Permittivity profile estimation based on non-radiating equivalent source (2d case),” Prog. Electromagnetics Research B 50, 157–175 (2013).
[Crossref]

Sci. Rep. (1)

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep. 4(1), 5552 (2014).
[Crossref] [PubMed]

Science (5)

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[Crossref] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

S. W. Hell, “Far-field optical nanoscopy,” Science 316(5828), 1153–1158 (2007).
[Crossref] [PubMed]

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[Crossref] [PubMed]

Other (4)

H. Stark, ed., Image recovery: theory and application (Elsevier, 1987).

L. Tsang and J. A. Kong, Scattering of electromagnetic waves, vol. 26 (John Wiley & Sons, 2004).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (New York: McGraw-Hill, 1961).

M. P. Bendsoe and O. Sigmund, Topology optimization: theory, methods and applications (Springer Science & Business Media, 2013).

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Figures (14)

Fig. 1
Fig. 1 The geometry of problem. The radiated fields emitted from N point sources inside the source region propagate through the engineered scattering structure which is composed of a plurality of dielectric wires with radius “a” and dielectric constant of εd distributed inside a rectangular region with side length of td and ld. The field measurement is performed at M points over the image region which is located at far-field region. In this work, we only consider the case where the point sources and measurement samples are located on two lines with length of lo and li, located at do and di from the scattering medium, respectively.
Fig. 2
Fig. 2 The location map of a randomly configured structure composed of 340 dielectric wires.
Fig. 3
Fig. 3 The highest order transmittance (dB) as a function of N and M (MN) and di = 20 λ, li = 400 λ. For (a) lo = 0.25 λ, (b) lo = 0.5 λ, (c) lo = λ, and (d) lo = 2 λ.
Fig. 4
Fig. 4 The highest order transmittance (dB) as a function of N and M (MN) for di = 20 λ, lo = 2 λ. For (a) li = 20 λ, (b) li = 40 λ, (c) li = 400 λ, and (d) li = 800 λ.
Fig. 5
Fig. 5 The highest order transmittance (dB) as a function of N and M (MN) and lo = 2 λ. For (a) d = di + do = 20 λ, li = 40 λ, and (b) d = di + do = 200 λ, li = 400 λ. The two cases, (a) and (b), represent the results for a same numerical aperture.
Fig. 6
Fig. 6 (a) and (b) show the highest order transmittance (dB) of 10 and 15 point sources, respectively, distributed uniformly on lo = 0.58λ when illuminating random scattering structures. The horizontal axis shows the number of dielectric wires in the structure which are randomly configured in a rectangular region of side lengths td = 0.34λ and ld = 5.85λ. The calculations are performed for a large number of random configurations, each denoted by a number on the vertical axis (number of simulations). (c) and (d) demonstrate the average value of the transmittance calculated in (a) and (b), respectively.
Fig. 7
Fig. 7 The highest order transmittance (dB) for (a) 10 point sources and (b) 15 point sources distributed uniformly on lo = 0.58λ when illuminating scattering structures populated uniformly by periodic configuration of dielectric wires. The horizontal and vertical axis shows the number of dielectric wires at each column and row, respectively.
Fig. 8
Fig. 8 The highest order transmittance (dB) for disordered metalenses designed for 10 [(a), (c), (e)] and 15 [(b), (d), (f)] point sources distributed uniformly over lo = 0.58λ. (I) The highest order transmittance for N = 1,2,...,30 and 0.003λ<lo<4.57λ when there is no scattering medium introduced. (II) The highest order transmittance when the engineered scattering medium composed of 692, 888, 530, 520, 507, and 512 dielectric wires (εr = 9, a = 0.02λ), for (a) – (f), respectively, is introduced in front of the radiating point sources. (III) The enhancement achieved for the transmittance of the highest order radiated fields. The black square shows the target point in our design where an enhancement of about 78 dB, 100 dB, 80 dB, 90 dB, 78 dB, and 85 dB, for (a) – (f), respectively, is achieved. tan (φ) indicates the corresponding resolution, which is near 0.064λ [(a), (c), (e)] and 0.041λ [(b), (d), (f)] for 10 and 15 point sources, respectively.
Fig. 9
Fig. 9 The transmittance (dB) of radiating modes versus the extinction coefficient (ĸ) of dielectric constant (the horizontal axis) for n = 3. (a), (c), and (e) correspond to the structures of Figs. 8(a), 8(c), and 8(e), respectively, designed for the ten point sources (ten modes), while (b), (d), and (f) correspond to those of Figs. 8(b), 8(d), and 8(f), respectively, designed for the fifteen point sources (fifteen modes). The black solid lines are the relative average transmittances (dB) for each configuration, the value of which can be read from the right vertical axes in the plots (a)-(f).
Fig. 10
Fig. 10 (a) Orthogonal source basis vectors for 10 numbers of point sources distributed uniformly over lo = 0.58λ and do = 0.16λ, when there is no scattering medium introduced. The field measurements are performed at M=200 points over the image plane of li = 400λ and di = 20 λ. (b) The corresponding orthogonal basis vectors over the image plane.
Fig. 11
Fig. 11 (a) The modified orthogonal source basis vectors for 10 numbers of point sources distributed uniformly over lo = 0.58λ and do = 0.16λ, when the scattering medium of Fig. 8(c) is introduced. The field measurements are performed at M = 200 points over the image plane of li = 400λ and di = 20 λ. (b) The corresponding modified orthogonal basis vectors over the image plane.
Fig. 12
Fig. 12 The electric field distribution (absolute value) (dB) for exciting (a) the highest order mode in free space and (b) the modified highest order mode at the presence of designed scattering structure [Fig. 8(c)].
Fig. 13
Fig. 13 The root mean square error versus the SNR of receiver for propagation through both free space (red and black) and engineered scattering structure (blue and green) of Fig. 8(c). The results show the average values taken over hundreds of random source distributions for both 10 point sources and 7 point sources distributed uniformly over lo = 0.58λ and do = 0.16λ. The field measurements are performed at M = 200 points over the image plane of li = 400λ and di = 20 λ.
Fig. 14
Fig. 14 The flowchart of the proposed algorithm used for engineering the scattering features of the dielectric medium.

Equations (9)

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E y = m= + j m C m H m ( 2 ) ( k 0 ρ ) e jm( φ φ i ) , 
C m = η 0 η d J m ( k 0 a ) J m ( k d a ) J m ( k 0 a ) J m ( k d a ) J m ( k d a )H ' m ( 2 ) ( k 0 a ) η 0 η d J m ( k d a ) H m ( 2 ) ( k 0 a )  .
E k,inc y = E external sources y ( r k )+ l=1 lk L E l y ( r k )= n=1 N s n H 0 ( 2 ) ( k 0 | r k r n | )+ l=1 lk L E l y ( r k ) ,
[ E image ]=[ t ][ E source ] ,
[ t ]=[ U ]Σ [ V ] T .
[ E image ]= n=1 N σ n   U n ( V n T [ E source ] ) = n=1 N σ n   U n E source n , 
[ E source ]= n=1 N ( V n T [ E source ] ) V n = n=1 N E source n V n . 
Threshol d new =γ×sign( | γ |Crossover value ) , γ=( n1+ α β )×Threshol d old
α=( 1 n +1n ), β= 1±sign( Threshol d old ) 2  , ( +( ):No( at least one ) dielectric inclusion added ).

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