Abstract

Far-field imaging beyond the Rayleigh limit is one of the most important challenges in optics, microwave, and ultrasonics. We propose a novel sparsity-promoted super-oscillation imaging scheme for reconstructing more universal objects in subwavelength scales, which solves a weighted optimization problem constrained by lp-norm-based sparsity regularization (0p1). We demonstrate numerically that the proposed imaging technique improves the resolution related to existing approaches remarkably for the case of very high signal-to-noise ratio (SNR), including the traditional super-oscillation imaging and sparsity-based super-resolution imaging. The standard superoscillation based super-resolution imaging approach can be regarded as the first-iteration solution of the proposed scheme. Numerical results for one- and two-dimensional super-resolution imaging are presented for validation.

© 2014 Optical Society of America

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References

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  2. X. Zhang, Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
    [CrossRef] [PubMed]
  3. S. Gazit, A. Szameit, Y. C. Eldar, M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
    [CrossRef] [PubMed]
  4. E. J. Candes, C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” (submitted) (2014).
  5. L. Li, F. Li, “Beating the Rayleigh limit: orbital-angular-momentum-based super-resolution diffraction tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(3), 033205 (2013).
    [CrossRef] [PubMed]
  6. R. K. Amineh, G. V. Eleftheriades, “2D and 3D sub-diffraction source imaging with a superoscillatory filter,” Opt. Express 21(7), 8142–8156 (2013).
    [CrossRef] [PubMed]
  7. L. Li, X. Xu, and F. Li, “Towards super-resolution microwave imaging: general framework,” 10th International Symposium On Antenna, Propagation & EM Theory (2012).
    [CrossRef]
  8. E. T. F. Rogers, N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15(9), 094008 (2013).
    [CrossRef]
  9. F. M. Huang, N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009).
    [CrossRef] [PubMed]
  10. L. Li, B. Jafarpour, “Effective solution of nonlinear subsurface flow inverse problems in sparse bases,” Inverse Probl. 26(10), 105016 (2010).
    [CrossRef]
  11. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
    [PubMed]
  12. A. M. H. Wong, G. V. Eleftheriades, “Adaptation of Schelkunoff’s superdirective antenna theory for the realization of superoscillatory antenna arrays,” IEEE Antennas Wirel. Propag. Lett. 9, 315–318 (2010).
    [CrossRef]

2013 (3)

L. Li, F. Li, “Beating the Rayleigh limit: orbital-angular-momentum-based super-resolution diffraction tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(3), 033205 (2013).
[CrossRef] [PubMed]

R. K. Amineh, G. V. Eleftheriades, “2D and 3D sub-diffraction source imaging with a superoscillatory filter,” Opt. Express 21(7), 8142–8156 (2013).
[CrossRef] [PubMed]

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

2010 (2)

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

A. M. H. Wong, G. V. Eleftheriades, “Adaptation of Schelkunoff’s superdirective antenna theory for the realization of superoscillatory antenna arrays,” IEEE Antennas Wirel. Propag. Lett. 9, 315–318 (2010).
[CrossRef]

2009 (2)

2008 (1)

X. Zhang, Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

1982 (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
[PubMed]

Amineh, R. K.

Candes, E. J.

E. J. Candes, C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” (submitted) (2014).

Devaney, A. J.

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
[PubMed]

Eldar, Y. C.

Eleftheriades, G. V.

R. K. Amineh, G. V. Eleftheriades, “2D and 3D sub-diffraction source imaging with a superoscillatory filter,” Opt. Express 21(7), 8142–8156 (2013).
[CrossRef] [PubMed]

A. M. H. Wong, G. V. Eleftheriades, “Adaptation of Schelkunoff’s superdirective antenna theory for the realization of superoscillatory antenna arrays,” IEEE Antennas Wirel. Propag. Lett. 9, 315–318 (2010).
[CrossRef]

Fernandez-Granda, C.

E. J. Candes, C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” (submitted) (2014).

Gazit, S.

Huang, F. M.

F. M. Huang, N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009).
[CrossRef] [PubMed]

Jafarpour, B.

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

Li, F.

L. Li, F. Li, “Beating the Rayleigh limit: orbital-angular-momentum-based super-resolution diffraction tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(3), 033205 (2013).
[CrossRef] [PubMed]

Li, L.

L. Li, F. Li, “Beating the Rayleigh limit: orbital-angular-momentum-based super-resolution diffraction tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(3), 033205 (2013).
[CrossRef] [PubMed]

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

Liu, Z.

X. Zhang, Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Rogers, E. T. F.

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

Segev, M.

Szameit, A.

Wong, A. M. H.

A. M. H. Wong, G. V. Eleftheriades, “Adaptation of Schelkunoff’s superdirective antenna theory for the realization of superoscillatory antenna arrays,” IEEE Antennas Wirel. Propag. Lett. 9, 315–318 (2010).
[CrossRef]

Zhang, X.

X. Zhang, Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[CrossRef] [PubMed]

Zheludev, N. I.

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

F. M. Huang, N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009).
[CrossRef] [PubMed]

IEEE Antennas Wirel. Propag. Lett. (1)

A. M. H. Wong, G. V. Eleftheriades, “Adaptation of Schelkunoff’s superdirective antenna theory for the realization of superoscillatory antenna arrays,” IEEE Antennas Wirel. Propag. Lett. 9, 315–318 (2010).
[CrossRef]

Inverse Probl. (1)

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

J. Opt. (1)

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

Nano Lett. (1)

F. M. Huang, N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009).
[CrossRef] [PubMed]

Nat. Mater. (1)

X. Zhang, Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[CrossRef] [PubMed]

Opt. Express (2)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

L. Li, F. Li, “Beating the Rayleigh limit: orbital-angular-momentum-based super-resolution diffraction tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(3), 033205 (2013).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Ultrason. Imaging (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
[PubMed]

Other (2)

E. J. Candes, C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” (submitted) (2014).

L. Li, X. Xu, and F. Li, “Towards super-resolution microwave imaging: general framework,” 10th International Symposium On Antenna, Propagation & EM Theory (2012).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

The sketch maps of imaging setups.(a) The 1D case. (b) The 2Dcase.

Fig. 2
Fig. 2

The reconstructed images of double point sources centered at (−0.45, 0) and (0.45, 0) by five different methods. For this set of simulation, the Rayleigh limit is around 5.0. (a) Traditional super-oscillation imaging technique in [2], (b)the improved super-oscillation imaging approach by Eq. (8), (c)the standard back propagation imaging algorithm, (d) the proposed sparsity-promoted super-oscillation approach as demonstrated in Table 1, and (e) the conventional sparsity-promoted reconstruction approach. For Figs. 2(d) and 2(e), γ = 10 5 is used. In these figures, the x-axis denotes the location of the probed source along x-direction, while the y-axis corresponds to the amplitude of reconstructed image.

Fig. 3
Fig. 3

The reconstructed images of two point sources are centered at (−0.15,0) and (0.15,0) by five different imaging methods. The setup of other parameters is the same as that used in Fig. 2.

Fig. 4
Fig. 4

The reconstructed images of two groups of two-point sources, which are located at (−1.3, 0), (−0.8, 0), (0.7,0) and (1.2,0) by five different methods. The setup of other simulation parameters is the same as that used in Fig. 2.

Fig. 5
Fig. 5

The reconstructed images of two point sources with SNR being 40dB. The setup of other simulation parameters is the same as that in Fig. 2 but with γ = 0.001 .

Fig. 6
Fig. 6

The reconstructed images of double point sources with SNR being 30dB. The setup of other simulation parameters is the same as that in Fig. 2 but with γ = 0.01 .

Fig. 7
Fig. 7

The reconstructed images by the sparsity-promoted superoscillation approach (a) and conventional sparsity-promoted reconstruction approach, where p = 1, SNR = 30dB, and other parameters are the same as those used in Figs. 5(c) and (d).

Fig. 8
Fig. 8

The reconstructed images of four point sources are located at (−0.3,-0.3, 0), (−0.3,0.2,0), (0.2,-0.3,0) and (0.2,0.2,0) by five different methods. For this set of simulation, SNR = 45dB, p = 0.5, and the Rayleigh limit is around 5.0. (a) Traditional super-oscillation imaging technique in [7], (b)the improved super-oscillation imaging approach by Eq. (8), (c)the standard back propagation algorithm, (d) the proposed sparsity-promoted super-oscillation approach as demonstrated in Table 1, and (e) the conventional sparsity-promoted reconstruction approach. In these figures, the horizontal and vertical axes are along the x and y directions, respectively.

Fig. 9
Fig. 9

The reconstructed images of four point sources are located at (−0.6,-0.6, 0), (−0.6,0.4,0), (0.4,-0.6,0) and (0.4,0.4,0) by five different methods, where SNR = 30dB and p = 1.0. The setup of other parameters is the same as that in Fig. 8.

Tables (1)

Tables Icon

Table 1 The procedure of adaptive super oscillation imaging algorithm ( Λ is a diagonal matrix involved in the iteratively reweighted algorithm)

Equations (9)

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S ( x ) = D x D x J ( x ) G ( x x ) d x
G ( x x ) = j 4 π 1 k z e j k x ( x x ) + j k z z 0 d k x
S ^ ( k x ) = e j k z z 0 k z J ^ ( k x ) , k 0 sin θ k x k 0 sin θ ,
J ^ ( x ) = k 0 sin θ k 0 sin θ S ^ ( k x ) T ( k x ) e j k x x d k x
h ( x , x ) = - k 0 sin θ k 0 sin θ e j k z | z - z | k z T ( k x ) e j k x | x - x | d k x
min J , W | | S F J | | W 2 s . t . , F W F = Δ .
J ^ = F W S
J ^ = ( F W F ) 1 F W S
min J , W [ | | S F J | | W 2 + γ | | J | | P P ] , s . t . , F W F = Δ .

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