## 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 *l _{p}*-norm-based sparsity regularization ($\text{0}\le p\le \text{1}$). 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

Full Article | PDF Article**OSA Recommended Articles**

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### References

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- Publication

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

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

[Crossref] [PubMed] -
S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).

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

- L. Li and 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 and G. V. Eleftheriades, “2D and 3D sub-diffraction source imaging with a superoscillatory filter,” Opt. Express 21(7), 8142–8156 (2013).

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

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

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

[PubMed] - A. M. H. Wong and 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 and 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]

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

[Crossref]
[PubMed]

#### 2010 (2)

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

[Crossref]

A. M. H. Wong and 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)

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).

[Crossref]
[PubMed]

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

[Crossref]
[PubMed]

#### 2008 (1)

X. Zhang and 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.

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

[Crossref]
[PubMed]

#### Candes, E. J.

E. J. Candes and 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.

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).

[Crossref]
[PubMed]

#### Eleftheriades, G. V.

R. K. Amineh and 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 and 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 and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” (submitted) (2014).

#### Gazit, S.

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).

[Crossref]
[PubMed]

#### Huang, F. M.

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

[Crossref]
[PubMed]

#### 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]

#### Li, F.

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

[Crossref]

#### Liu, Z.

X. Zhang and 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 and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15(9), 094008 (2013).

[Crossref]

#### Segev, M.

[Crossref]
[PubMed]

#### Szameit, A.

[Crossref]
[PubMed]

#### Wong, A. M. H.

A. M. H. Wong and 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 and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).

[Crossref]
[PubMed]

#### 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]

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

[Crossref]
[PubMed]

#### IEEE Antennas Wirel. Propag. Lett. (1)

[Crossref]

#### Inverse Probl. (1)

[Crossref]

#### J. Opt. (1)

[Crossref]

#### Nano Lett. (1)

[Crossref]
[PubMed]

#### Nat. Mater. (1)

[Crossref]
[PubMed]

#### Opt. Express (2)

[Crossref]
[PubMed]

[Crossref]
[PubMed]

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

[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 and 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]

### Cited By

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### Figures (9)

**Fig. 1**

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

**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), _{
$\gamma ={10}^{-5}$
}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**

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**

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**

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

**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

**Fig. 7**

The reconstructed images by the sparsity-promoted superoscillation approach (a) and conventional sparsity-promoted reconstruction approach, where

**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**

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)

**Table 1** The procedure of adaptive super oscillation imaging algorithm (

### Equations (9)

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