Abstract

In this paper, we propose an approach to overcome the well-known “diffraction limit” when imaging sources several wavelengths away. We employ superdirectivity antenna concepts to design a well-controlled superoscillatory filter (SOF) based on the properties of Tschebyscheff polynomials. The SOF is applied to the reconstructed images from holographic algorithms which are based on the back-propagation principle. We demonstrate the capability of this approach when imaging point-sources several wavelengths away in one-, two-, and three-dimensional imaging with super-resolution. We also investigate the robustness of the proposed algorithm with the sharpness of the SOF, the presence of noise, the imaging distance, and the size of the scanning aperture.

© 2013 OSA

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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. A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett.92(11), 117403 (2004).
    [CrossRef] [PubMed]
  3. A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: Subdiffraction focusing with patterned surfaces,” Science320(5875), 511–513 (2008).
    [CrossRef] [PubMed]
  4. L. Markley and G. V. Eleftheriades, “Meta-screens and near-field antenna-arrays: a new perspective on subwavelength focusing and imaging,” Metamaterials; Elsevier 5(2–3), 97–106 (2011).
  5. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14(18), 8247–8256 (2006).
    [CrossRef] [PubMed]
  6. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74(7), 075103 (2006).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  8. J. L. Harris, “Diffraction and resolving power,” J. Opt. Soc. Am.54(7), 931–933 (1964).
    [CrossRef]
  9. C. W. Barnes, “Object restoration in a diffraction-limited imaging system,” J. Opt. Soc. Am.56(5), 575–578 (1966).
    [CrossRef]
  10. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta (Lond.)21(9), 709720 (1974).
    [CrossRef]
  11. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett.9(3), 1249–1254 (2009).
    [CrossRef] [PubMed]
  12. D. Slepian and H. O. Pollak, “Prolate spheroidal wavefunctions, quadrature and uncertainty-I,” Bell Syst. Tech. J.40, 43–63 (1961).
  13. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater.11(5), 432–435 (2012).
    [CrossRef] [PubMed]
  14. 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]
  15. A. M. H. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antenn. Propag.59(12), 4766–4776 (2011).
    [CrossRef]
  16. N. Yaru, “A note on super-gain antenna arrays,” Proc. of I.R.E. 39(9), 1081–1085 (1951).
    [CrossRef]
  17. D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech.49(9), 1581–1592 (2001).
    [CrossRef]
  18. M. Ravan, R. K. Amineh, and N. K. Nikolova, “Two-dimensional near-field microwave holography,” Inverse Probl.26(5), 055011 (2010).
    [CrossRef]
  19. R. K. Amineh, M. Ravan, A. Khalatpour, and N. K. Nikolova, “Three-dimensional near-field microwave holography using reflected and transmitted signals,” IEEE Trans. Antenn. Propag.59(12), 4777–4789 (2011).
    [CrossRef]
  20. B. E. Saleh and C. T. Malvin, Fundamentals of Photonics (Wiley, 1991).
  21. S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944), 24, 46–156 (1945).
  22. C. M. Grinstead and J. L. Snell, Introduction to Probability (American Mathematical Society, 1997).

2012

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater.11(5), 432–435 (2012).
[CrossRef] [PubMed]

2011

A. M. H. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antenn. Propag.59(12), 4766–4776 (2011).
[CrossRef]

R. K. Amineh, M. Ravan, A. Khalatpour, and N. K. Nikolova, “Three-dimensional near-field microwave holography using reflected and transmitted signals,” IEEE Trans. Antenn. Propag.59(12), 4777–4789 (2011).
[CrossRef]

2010

M. Ravan, R. K. Amineh, and N. K. Nikolova, “Two-dimensional near-field microwave holography,” Inverse Probl.26(5), 055011 (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

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

2008

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: Subdiffraction focusing with patterned surfaces,” Science320(5875), 511–513 (2008).
[CrossRef] [PubMed]

2006

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14(18), 8247–8256 (2006).
[CrossRef] [PubMed]

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74(7), 075103 (2006).
[CrossRef]

2004

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett.92(11), 117403 (2004).
[CrossRef] [PubMed]

2001

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech.49(9), 1581–1592 (2001).
[CrossRef]

2000

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

1974

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta (Lond.)21(9), 709720 (1974).
[CrossRef]

1966

1964

1961

D. Slepian and H. O. Pollak, “Prolate spheroidal wavefunctions, quadrature and uncertainty-I,” Bell Syst. Tech. J.40, 43–63 (1961).

Alekseyev, L. V.

Amineh, R. K.

R. K. Amineh, M. Ravan, A. Khalatpour, and N. K. Nikolova, “Three-dimensional near-field microwave holography using reflected and transmitted signals,” IEEE Trans. Antenn. Propag.59(12), 4777–4789 (2011).
[CrossRef]

M. Ravan, R. K. Amineh, and N. K. Nikolova, “Two-dimensional near-field microwave holography,” Inverse Probl.26(5), 055011 (2010).
[CrossRef]

Barnes, C. W.

Chad, J. E.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater.11(5), 432–435 (2012).
[CrossRef] [PubMed]

Dennis, M. R.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater.11(5), 432–435 (2012).
[CrossRef] [PubMed]

Eleftheriades, G. V.

A. M. H. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antenn. Propag.59(12), 4766–4776 (2011).
[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]

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett.92(11), 117403 (2004).
[CrossRef] [PubMed]

Engheta, N.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74(7), 075103 (2006).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta (Lond.)21(9), 709720 (1974).
[CrossRef]

Grbic, A.

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: Subdiffraction focusing with patterned surfaces,” Science320(5875), 511–513 (2008).
[CrossRef] [PubMed]

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett.92(11), 117403 (2004).
[CrossRef] [PubMed]

Hall, T. E.

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech.49(9), 1581–1592 (2001).
[CrossRef]

Harris, J. L.

Huang, F. M.

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

Jacob, Z.

Jiang, L.

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: Subdiffraction focusing with patterned surfaces,” Science320(5875), 511–513 (2008).
[CrossRef] [PubMed]

Khalatpour, A.

R. K. Amineh, M. Ravan, A. Khalatpour, and N. K. Nikolova, “Three-dimensional near-field microwave holography using reflected and transmitted signals,” IEEE Trans. Antenn. Propag.59(12), 4777–4789 (2011).
[CrossRef]

Lindberg, J.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater.11(5), 432–435 (2012).
[CrossRef] [PubMed]

McMakin, D. L.

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech.49(9), 1581–1592 (2001).
[CrossRef]

Merlin, R.

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: Subdiffraction focusing with patterned surfaces,” Science320(5875), 511–513 (2008).
[CrossRef] [PubMed]

Narimanov, E.

Nikolova, N. K.

R. K. Amineh, M. Ravan, A. Khalatpour, and N. K. Nikolova, “Three-dimensional near-field microwave holography using reflected and transmitted signals,” IEEE Trans. Antenn. Propag.59(12), 4777–4789 (2011).
[CrossRef]

M. Ravan, R. K. Amineh, and N. K. Nikolova, “Two-dimensional near-field microwave holography,” Inverse Probl.26(5), 055011 (2010).
[CrossRef]

Pendry, J. B.

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

Pollak, H. O.

D. Slepian and H. O. Pollak, “Prolate spheroidal wavefunctions, quadrature and uncertainty-I,” Bell Syst. Tech. J.40, 43–63 (1961).

Ravan, M.

R. K. Amineh, M. Ravan, A. Khalatpour, and N. K. Nikolova, “Three-dimensional near-field microwave holography using reflected and transmitted signals,” IEEE Trans. Antenn. Propag.59(12), 4777–4789 (2011).
[CrossRef]

M. Ravan, R. K. Amineh, and N. K. Nikolova, “Two-dimensional near-field microwave holography,” Inverse Probl.26(5), 055011 (2010).
[CrossRef]

Rogers, E. T. F.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater.11(5), 432–435 (2012).
[CrossRef] [PubMed]

Roy, T.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater.11(5), 432–435 (2012).
[CrossRef] [PubMed]

Salandrino, A.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74(7), 075103 (2006).
[CrossRef]

Savo, S.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater.11(5), 432–435 (2012).
[CrossRef] [PubMed]

Sheen, D. M.

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech.49(9), 1581–1592 (2001).
[CrossRef]

Slepian, D.

D. Slepian and H. O. Pollak, “Prolate spheroidal wavefunctions, quadrature and uncertainty-I,” Bell Syst. Tech. J.40, 43–63 (1961).

Wong, A. M. H.

A. M. H. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antenn. Propag.59(12), 4766–4776 (2011).
[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]

Zheludev, N. I.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater.11(5), 432–435 (2012).
[CrossRef] [PubMed]

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

Bell Syst. Tech. J.

D. Slepian and H. O. Pollak, “Prolate spheroidal wavefunctions, quadrature and uncertainty-I,” Bell Syst. Tech. J.40, 43–63 (1961).

IEEE Antennas Wirel. Propag. Lett.

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]

IEEE Trans. Antenn. Propag.

A. M. H. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antenn. Propag.59(12), 4766–4776 (2011).
[CrossRef]

R. K. Amineh, M. Ravan, A. Khalatpour, and N. K. Nikolova, “Three-dimensional near-field microwave holography using reflected and transmitted signals,” IEEE Trans. Antenn. Propag.59(12), 4777–4789 (2011).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech.49(9), 1581–1592 (2001).
[CrossRef]

Inverse Probl.

M. Ravan, R. K. Amineh, and N. K. Nikolova, “Two-dimensional near-field microwave holography,” Inverse Probl.26(5), 055011 (2010).
[CrossRef]

J. Opt. Soc. Am.

Nano Lett.

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

Nat. Mater.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater.11(5), 432–435 (2012).
[CrossRef] [PubMed]

Opt. Acta (Lond.)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta (Lond.)21(9), 709720 (1974).
[CrossRef]

Opt. Express

Phys. Rev. B

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74(7), 075103 (2006).
[CrossRef]

Phys. Rev. Lett.

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

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett.92(11), 117403 (2004).
[CrossRef] [PubMed]

Science

A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: Subdiffraction focusing with patterned surfaces,” Science320(5875), 511–513 (2008).
[CrossRef] [PubMed]

Other

L. Markley and G. V. Eleftheriades, “Meta-screens and near-field antenna-arrays: a new perspective on subwavelength focusing and imaging,” Metamaterials; Elsevier 5(2–3), 97–106 (2011).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

N. Yaru, “A note on super-gain antenna arrays,” Proc. of I.R.E. 39(9), 1081–1085 (1951).
[CrossRef]

B. E. Saleh and C. T. Malvin, Fundamentals of Photonics (Wiley, 1991).

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944), 24, 46–156 (1945).

C. M. Grinstead and J. L. Snell, Introduction to Probability (American Mathematical Society, 1997).

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Figures (12)

Fig. 1
Fig. 1

(a) 3D Illustration of the imaging setup for 2D and 3D imaging and (b) simplified illustration of the setup for 1D imaging.

Fig. 2
Fig. 2

Block diagram of applying a SOF in the spectral domain to improve the resolution in the holographic imaging.

Fig. 3
Fig. 3

Spectrum of the designed SOF (all normalized to 1) with (a) five spectral lines (N = 2), (b) seven spectral lines (N = 3), (c) nine spectral lines (N = 4) and (d) spatial variation of the designed SOFs compared with the diffracted limited “sinc” function.

Fig. 4
Fig. 4

Reconstructed images of double point-sources at x=±0.35λ when L x =10λ and z 0 =5λ without applying SOF and when applying SOFs designed with N = 2, N = 3, and N = 4.

Fig. 5
Fig. 5

Reconstruction of double point-sources at x=±0.35λ for L x =10λ and z 0 =5λ when adding WGN with (a) SNR = 80 dB (almost noiseless), (b) SNR = 30 dB, (c) SNR = 15 dB, and (d) SNR = 5 dB.

Fig. 6
Fig. 6

Reconstruction of double point-sources at x=±0.35λ with SNR = 30 dB when (a) L x =10λ and z 0 =3λ , (b) and z 0 =5λ , (c) and z 0 =7λ , and (d) L x =20λ and .

Fig. 7
Fig. 7

2D Variation of the magnitude of (a) the diffraction limited “sinc” function and (b) the designed SOF.

Fig. 8
Fig. 8

Reconstruction of four point-sources at (−0.43,0,5)λ, (0.43,0,5)λ, (0,−0.43,5)λ, and (0,0.43,5)λ when Lx = Ly = 10λ and z0 = 5λ (a) without applying SOF and with SNR = 30 dB, (b) with applying SOF and SNR = 30 dB, (c) without applying SOF and with SNR = 20 dB, and (d) with applying SOF and SNR = 20 dB.

Fig. 9
Fig. 9

Reconstruction of four point-sources at (−0.43,0,5)λc, (0.43,0,5)λc, (0,−0.43,4)λc, and (0,0.43,4)λc when Lx = Ly = 10λc (a) without applying SOF and with SNR = 30 dB, (b) with applying SOF and SNR = 30 dB, (c) without applying SOF and with SNR = 10 dB, (d) with applying SOF and SNR = 10 dB,(e) without applying SOF and with SNR = –12 dB, and (f) with applying SOF and SNR = –12 dB. The horizontal and vertical axes are along the x and y directions, respectively, and they have been normalized with respect to λc. The colorbar is the same as those in Fig. 8.

Fig. 10
Fig. 10

Reconstruction of double point-sources at x=±0.35λ for L x =10λ and z 0 =5λ when adding WGN with SNR = 30 dB and when the sampling step for the probe along the x axis is (a) Δx=0.4λ and (b) Δx=0.52λ .

Fig. 11
Fig. 11

Reconstruction of double point-sources at x=±0.35λ for L x =10λ and z 0 =5λ when adding PN with (a) SNR = 80 dB (almost noiseless), (b) SNR = 30 dB, (c) SNR = 15 dB, and (d) SNR = 5 dB.

Fig. 12
Fig. 12

Reconstruction of double point-sources for L x =10λ and z 0 =5λ for (a) sources at x=±0.39λ and adding WGN with SNR of 30 dB, (b) sources at x=±0.32λ and adding WGN with SNR of 30 dB, (c) sources at x=±0.39λ and adding PN with SNR of 30 dB, (d) sources at x=±0.32λ and adding PN with SNR of 30 dB.

Equations (12)

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

s(x)= x = D x x =+ D x J( x )G(x x )d x
S ˜ ( k x )= J ˜ ( k x ) G ˜ ( k x )
J ^ ˜ ( k x )= S ˜ ( k x )/ G ˜ ( k x )
J ^ ˜ ( k x )=1 for k x m k x k x m .
J ^ (x)=2 k x m sinc( k x m π x).
F ˜ ( k x )= n=N N a n δ( k x nΔk k x m )
f(x)= n=N N (ω ω zn ) = n=N N a n ω n where ω= e jxΔk .
|f(x)|= a 0 + a 1 t+ a 2 (2 t 2 1)++ a n T n (t)++ a N T N (t)
F ˜ ( k x , k y )= n=N N m=N N a m a n δ( k x nΔk k x m , k y mΔk k x m ) .
J ^ ˜ SOF ( k x )= F ˜ ( k x ) k x m k x k x m .
Δx,Δy<λ/2
Δf<c/(2 R max )

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