Abstract

This paper investigates the feasibility of the resonant metalens for the imaging beyond the diffraction limit using a single sensor in the far-field. It is shown that the resonant metalens can be related to the super-resonance phenomenon. We demonstrate that the super-resonance supports the enhancement of the information capacity of an imaging system, which is responsible for the subwavelength imaging of the probed objects by using a single sensor in combination with a broadband illumination. Such imaging concept has its unique advantage of producing real-time data when an object is illuminated by broadband waves, without the harsh requirements such as near-field scanning, mechanical scanning, or antenna arrays. The proposed method is expected to find its applications in nanolithography, detection, sensing, and subwavelength imaging in the near future.

© 2014 Optical Society of America

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References

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  1. L. Rayleigh,“On pin-hole photography,” The London, Edinburg and Dublin philosophical magazine and journal of science, 5, 31 (1891).
  2. http://en.wikipedia.org/wiki/Near-field_scanning_optical_microscope
  3. E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15(9), 094008 (2013).
    [CrossRef]
  4. F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010).
    [CrossRef] [PubMed]
  5. F. Lemoult, M. Fink, and G. Lerosey, “Far-field sub-wavelength imaging and focusing using a wire medium based resonant metalens,” Waves in Random and Complex Media 21(4), 614–627 (2011).
    [CrossRef]
  6. F. Lemoult, M. Fink, and G. Lerosey, “Revisiting the wire medium: an ideal resonant metalens,” Waves in Random and Complex Media 21(4), 591–613 (2011).
    [CrossRef]
  7. F. Lemoult, M. Fink, and G. Lerosey, “Dispersion in media containing resonant inclusions: where does it come from,” 2012 Conference on, Lasers and Electro-Optics (2012).
    [CrossRef]
  8. F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun. 3, 1885 (2012).
  9. D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).
  10. R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).
  11. P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036606 (2004).
    [CrossRef] [PubMed]
  12. P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express 16(25), 20157–20165 (2008).
    [CrossRef] [PubMed]
  13. P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447–466 (1998).
    [CrossRef]
  14. F. Simonetti, M. Fleming, and E. A. Marengo, “Illustration of the role of multiple scattering in subwavelength imaging from far-field measurements,” J. Opt. Soc. Am. A 25(2), 292–303 (2008).
    [CrossRef] [PubMed]
  15. O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antenn. Propag. 37(7), 918–926 (1989).
    [CrossRef]
  16. I. J. Cox and C. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 3(8), 1152–1158 (1986).
    [CrossRef]
  17. I. Tolstoy, “Superresonant systems of scatters I,” J. Acoust. Soc. Am. 80(1), 282–294 (1986).
    [CrossRef]
  18. G. S. Sammelmann and R. H. Hackman, “Acoustic scattering in a homogeneous waveguide,” J. Acoust. Soc. Am. 82(1), 324–336 (1987).
    [CrossRef]
  19. The super-resonance is mathematically that the matrix B=I−k2α(ω)RgG0lens→lens(ω)in Eq. (3) is strongly ill-posed, which means that the ratio σ1σN(i.e., the condition number) is very large, whereσ1is the first singular value (the maximum) of the matrix, and σNis the final (the minimum) singular value.
  20. L. Li and B. Jafarpour, “Effective solution of nonlinear subsurface flow inverse problems in sparse bases,” Inverse Probl. 26(10), 105016 (2010).
    [CrossRef]
  21. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer Press 2010).

2013

E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15(9), 094008 (2013).
[CrossRef]

R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).

2012

F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun. 3, 1885 (2012).

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).

2011

F. Lemoult, M. Fink, and G. Lerosey, “Far-field sub-wavelength imaging and focusing using a wire medium based resonant metalens,” Waves in Random and Complex Media 21(4), 614–627 (2011).
[CrossRef]

F. Lemoult, M. Fink, and G. Lerosey, “Revisiting the wire medium: an ideal resonant metalens,” Waves in Random and Complex Media 21(4), 591–613 (2011).
[CrossRef]

2010

F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010).
[CrossRef] [PubMed]

L. Li and B. Jafarpour, “Effective solution of nonlinear subsurface flow inverse problems in sparse bases,” Inverse Probl. 26(10), 105016 (2010).
[CrossRef]

2008

2004

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036606 (2004).
[CrossRef] [PubMed]

1998

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447–466 (1998).
[CrossRef]

1989

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antenn. Propag. 37(7), 918–926 (1989).
[CrossRef]

1987

G. S. Sammelmann and R. H. Hackman, “Acoustic scattering in a homogeneous waveguide,” J. Acoust. Soc. Am. 82(1), 324–336 (1987).
[CrossRef]

1986

Belkebir, K.

Bucci, O. M.

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antenn. Propag. 37(7), 918–926 (1989).
[CrossRef]

Carminat, R.

R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).

Chaumet, P. C.

P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express 16(25), 20157–20165 (2008).
[CrossRef] [PubMed]

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036606 (2004).
[CrossRef] [PubMed]

Cox, I. J.

de Rosny, J.

F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010).
[CrossRef] [PubMed]

de Vries, P.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447–466 (1998).
[CrossRef]

Fink, M.

R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).

F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun. 3, 1885 (2012).

F. Lemoult, M. Fink, and G. Lerosey, “Revisiting the wire medium: an ideal resonant metalens,” Waves in Random and Complex Media 21(4), 591–613 (2011).
[CrossRef]

F. Lemoult, M. Fink, and G. Lerosey, “Far-field sub-wavelength imaging and focusing using a wire medium based resonant metalens,” Waves in Random and Complex Media 21(4), 614–627 (2011).
[CrossRef]

F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010).
[CrossRef] [PubMed]

Fleming, M.

Franceschetti, G.

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antenn. Propag. 37(7), 918–926 (1989).
[CrossRef]

Hackman, R. H.

G. S. Sammelmann and R. H. Hackman, “Acoustic scattering in a homogeneous waveguide,” J. Acoust. Soc. Am. 82(1), 324–336 (1987).
[CrossRef]

Jafarpour, B.

L. Li and B. Jafarpour, “Effective solution of nonlinear subsurface flow inverse problems in sparse bases,” Inverse Probl. 26(10), 105016 (2010).
[CrossRef]

Lagendijk, A.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447–466 (1998).
[CrossRef]

Lemoult, F.

F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun. 3, 1885 (2012).

F. Lemoult, M. Fink, and G. Lerosey, “Revisiting the wire medium: an ideal resonant metalens,” Waves in Random and Complex Media 21(4), 591–613 (2011).
[CrossRef]

F. Lemoult, M. Fink, and G. Lerosey, “Far-field sub-wavelength imaging and focusing using a wire medium based resonant metalens,” Waves in Random and Complex Media 21(4), 614–627 (2011).
[CrossRef]

F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010).
[CrossRef] [PubMed]

Lerosey, G.

F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun. 3, 1885 (2012).

F. Lemoult, M. Fink, and G. Lerosey, “Revisiting the wire medium: an ideal resonant metalens,” Waves in Random and Complex Media 21(4), 591–613 (2011).
[CrossRef]

F. Lemoult, M. Fink, and G. Lerosey, “Far-field sub-wavelength imaging and focusing using a wire medium based resonant metalens,” Waves in Random and Complex Media 21(4), 614–627 (2011).
[CrossRef]

F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010).
[CrossRef] [PubMed]

Li, L.

L. Li and B. Jafarpour, “Effective solution of nonlinear subsurface flow inverse problems in sparse bases,” Inverse Probl. 26(10), 105016 (2010).
[CrossRef]

Liu, Z.

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).

Lu, D.

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).

Marengo, E. A.

Pierrat, R.

R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).

Rahmani, A.

P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express 16(25), 20157–20165 (2008).
[CrossRef] [PubMed]

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036606 (2004).
[CrossRef] [PubMed]

Rogers, E. T. F.

E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15(9), 094008 (2013).
[CrossRef]

Sammelmann, G. S.

G. S. Sammelmann and R. H. Hackman, “Acoustic scattering in a homogeneous waveguide,” J. Acoust. Soc. Am. 82(1), 324–336 (1987).
[CrossRef]

Sentenac, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036606 (2004).
[CrossRef] [PubMed]

Sheppard, C. R.

Simonetti, F.

Tolstoy, I.

I. Tolstoy, “Superresonant systems of scatters I,” J. Acoust. Soc. Am. 80(1), 282–294 (1986).
[CrossRef]

van Coevorden, D. V.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447–466 (1998).
[CrossRef]

Vandenbem, C.

R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).

Zheludev, N. I.

E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15(9), 094008 (2013).
[CrossRef]

IEEE Trans. Antenn. Propag.

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antenn. Propag. 37(7), 918–926 (1989).
[CrossRef]

Inverse Probl.

L. Li and B. Jafarpour, “Effective solution of nonlinear subsurface flow inverse problems in sparse bases,” Inverse Probl. 26(10), 105016 (2010).
[CrossRef]

J. Acoust. Soc. Am.

I. Tolstoy, “Superresonant systems of scatters I,” J. Acoust. Soc. Am. 80(1), 282–294 (1986).
[CrossRef]

G. S. Sammelmann and R. H. Hackman, “Acoustic scattering in a homogeneous waveguide,” J. Acoust. Soc. Am. 82(1), 324–336 (1987).
[CrossRef]

J. Opt.

E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15(9), 094008 (2013).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Commun.

F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun. 3, 1885 (2012).

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).

Opt. Express

Phys. Rev. A.

R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036606 (2004).
[CrossRef] [PubMed]

Phys. Rev. Lett.

F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010).
[CrossRef] [PubMed]

Rev. Mod. Phys.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447–466 (1998).
[CrossRef]

Waves in Random and Complex Media

F. Lemoult, M. Fink, and G. Lerosey, “Far-field sub-wavelength imaging and focusing using a wire medium based resonant metalens,” Waves in Random and Complex Media 21(4), 614–627 (2011).
[CrossRef]

F. Lemoult, M. Fink, and G. Lerosey, “Revisiting the wire medium: an ideal resonant metalens,” Waves in Random and Complex Media 21(4), 591–613 (2011).
[CrossRef]

Other

F. Lemoult, M. Fink, and G. Lerosey, “Dispersion in media containing resonant inclusions: where does it come from,” 2012 Conference on, Lasers and Electro-Optics (2012).
[CrossRef]

L. Rayleigh,“On pin-hole photography,” The London, Edinburg and Dublin philosophical magazine and journal of science, 5, 31 (1891).

http://en.wikipedia.org/wiki/Near-field_scanning_optical_microscope

The super-resonance is mathematically that the matrix B=I−k2α(ω)RgG0lens→lens(ω)in Eq. (3) is strongly ill-posed, which means that the ratio σ1σN(i.e., the condition number) is very large, whereσ1is the first singular value (the maximum) of the matrix, and σNis the final (the minimum) singular value.

M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer Press 2010).

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Figures (6)

Fig. 1
Fig. 1

The sketch map for illustrating the principle of the resonant metalens for the subwavelength imaging from far-field measurements. In this figure, the metalens is made of a lattice of 3 × 10 metallic cylinders characterized by Eq. (1). The distance between two neighboring scatterers is d.

Fig. 2
Fig. 2

The normalized amplitude of the frequency-dependent (a) and time-dependent (b) responses acquired at r d . Here, a point source (as shown in the inset in Fig. 2(b)) is centered at (−11.7, 0) µm. This set of figures shows many abrupt changes within a very small frequency separation, implying that the response is highly sensitivity to frequency. The results are generated by applying the full-wave solver to the Maxwell’s equations, i.e., the coupled dipole method.

Fig. 3
Fig. 3

The dependence of the amplitude of electrical field scattered from the super-resonance lens on k in,y / k p and ω/ ω p . The electrical field is acquired at r d . In this figure, the x-axis is k in,y / k p and the y-axis denotes ω/ ω p . This set of results is generated by applying the full-wave solver to the Maxwell’s equations, i.e., the coupled dipole method.

Fig. 4
Fig. 4

(a) The black line corresponds to the normalized amplitude of electric fields measured by the sensor located at r d as a function of operational wavelength. The red line corresponds to the logarithm of the condition number of B=I k 2 α( ω )Rg G 0 lenslens ( ω ) . (b) The reconstruction results using the super-resonance (SR) lens in combination with a broadband illumination with the operational wavelength ranging from 578 µm to 588 µm by a step of 0.002 µm, denoted by the red line. For comparison, the ground truth (black line) and the results without the super-resonance lens (blue line) are also provided. (The reader of interest can get the Matlab Code for reproducing above results by sending a request email to lianlin.li@pku.edu.cn)

Fig. 5
Fig. 5

The reconstruction results using the super-resonance (SR) lens for four different working distances of sensor, where the simulation parameters are the same as those used in Fig. 4. The ground truth is also provided.

Fig. 6
Fig. 6

The reconstruction results using the super-resonance (SR) lens for different noise levels of 40dB, 35dB, 30dB, 25dB and 20dB, where the simulation parameters are the same as those used in Fig. 4. The ground truth is also provided.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

G( r d , r s ;ω )= G 0 ( r d , r s ;ω )+ k 2 α( ω ) n=1 N G 0 ( r d , r n ;ω ) g n
and g m = G 0 ( r m , r s ;ω )+ k 2 α( ω ) n=1,nm N G 0 ( r m , r n ;ω ) g n (m=1, 2, ,N)
G( r d , r s ;ω )= G 0 ( r d , r s ;ω )+ k 2 α( ω ) G 0 lensfar ( r d ;ω )×   ( I k 2 α( ω )Rg G 0 lenslens ( ω ) ) 1 G 0 sourcelens ( r s ;ω )
d dω g m = d dω G 0 ( r m , r s ;ω )+ d k 2 α( ω ) dω n=1,nm N G 0 ( r m , r n ;ω ) g n + k 2 α( ω ) n=1,nm N d G 0 ( r m , r n ;ω ) dω g n + k 2 α( ω ) n=1,nm N G 0 ( r m , r n ;ω ) d g n dω
d dω g m = d k 2 α( ω ) dω n=1,nm N G 0 ( r m , r n ;ω ) g n + k 2 α( ω ) n=1,nm N G 0 ( r m , r n ;ω ) d g n dω
 dG( r s ;ω ) dω = d k 2 α( ω ) dω   ( I k 2 α( ω )Rg G 0 lenslens ( ω ) ) 1 Rg G 0 lenslens ( ω )G( r s ;ω ) = d k 2 α( ω ) dω   ( I k 2 α( ω )Rg G 0 lenslens ( ω ) ) 1 ( G( r s ;ω ) G 0 sourcelens ( r s ;ω ) )
ΔG( ω )=G( r s1 ;ω )G( r s2 ;ω ),
Δ G 0 sourcelens ( ω )= G 0 sourcelens ( r s1 ;ω ) G 0 sourcelens ( r s2 ;ω )
 d  dω ΔG( ω )= d k 2 α( ω ) dω   ( I k 2 α( ω )Rg G 0 lenslens ( ω ) ) 1 ( ΔG( ω )Δ G 0 sourcelens ( ω ) )
O ^ ( r )= argmin O( r' ) [ | E( ω ) D G( r d ,r';ω ) E in ( r';ω )O( r' )d r | 2 dω+ γ D | O( r' ) |dr' ]

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