Abstract

We derive a nonsingular, polarization-dependent, 3D impulse response that provides unambiguously the wave field scattered by a negative-refractive-index layered lens and distributed in its image volume. By means of a 3D Fourier transform, we introduce the generalized amplitude transfer function in order to gain a deep insight into the resolution power of the optical element. In the near-field regime, fine details containing some depth information may be transmitted through the lens. We show that metamaterials with moderate absorption are appropriate for subwavelength resolution keeping a limited degree of depth discrimination.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2009

R. Kotyński and T. Stefaniuk, “Comparison of imaging with sub-wavelength resolution in the canalization and resonant tunnelling regimes,” J. Opt. A: Pure Appl. Opt. 11, 015001(2009).
[CrossRef]

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Subwavelength electromagnetic near-field imaging of point dipole with metamaterial nanoslab,” J. Opt. Soc. Am. A 26, 2226–2234(2009).
[CrossRef]

2008

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nature Methods 5, 527–529(2008).
[CrossRef] [PubMed]

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. Dorazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

2006

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

R. Marques, M. J. Freire, and J. D. Baena, “Theory of three-dimensional subdiffraction imaging,” Appl. Phys. Lett. 89, 211113 (2006).
[CrossRef]

2005

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

M. W. Feise and Y. S. Kivshar, “Sub-wavelength imaging with a left-handed material flat lens,” Phys. Lett. A 334, 326–330(2005).
[CrossRef]

K. Aydin, I. Bulu, and E. Ozbay, “Focusing of electromagnetic waves by a left-handed metamaterial flat lens,” Opt. Express 13, 8753–8759 (2005).
[CrossRef] [PubMed]

2004

J. B. Brock, A. A. Houck, and I. L. Chuang, “Focusing inside negative index materials,” Appl. Phys. Lett. 85, 2472–2474(2004).
[CrossRef]

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84, 1290–1293 (2004).
[CrossRef]

M. Nieto-Vesperinas, “Problem of image superresolution with a negative-refractive-index slab,” J. Opt. Soc. Am. A 21, 491–498 (2004).
[CrossRef]

2003

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

D. R. Smith, D. Schurig, M. Rosenbluth, and S. Schultz, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

S. Cummer, “Simulated causal subwavelength focusing by a negative refractive index slab,” Appl. Phys. Lett. 82, 1503–1505 (2003).
[CrossRef]

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

2002

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

2001

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Imaging, compression and Poynting vector streamlines with negative permittivity materials,” Electron. Lett. 37, 1243–1244 (2001).
[CrossRef]

2000

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

1999

1996

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
[CrossRef]

1980

1968

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and μ,” Phys. Uspekhi 10, 509–514 (1968).
[CrossRef]

1964

1962

1960

Akozbek, N.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. Dorazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Aydin, K.

Baena, J. D.

R. Marques, M. J. Freire, and J. D. Baena, “Theory of three-dimensional subdiffraction imaging,” Appl. Phys. Lett. 89, 211113 (2006).
[CrossRef]

Bennett, B. T.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nature Methods 5, 527–529(2008).
[CrossRef] [PubMed]

Bewersdorf, J.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nature Methods 5, 527–529(2008).
[CrossRef] [PubMed]

Bloemer, M. J.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. Dorazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, seventh (expanded) ed. (Cambridge, 1999).

Brock, J. B.

J. B. Brock, A. A. Houck, and I. L. Chuang, “Focusing inside negative index materials,” Appl. Phys. Lett. 85, 2472–2474(2004).
[CrossRef]

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Bulu, I.

Cappeddu, M. G.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. Dorazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Centini, M.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. Dorazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Chuang, I. L.

J. B. Brock, A. A. Houck, and I. L. Chuang, “Focusing inside negative index materials,” Appl. Phys. Lett. 85, 2472–2474(2004).
[CrossRef]

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Collet, E.

Cummer, S.

S. Cummer, “Simulated causal subwavelength focusing by a negative refractive index slab,” Appl. Phys. Lett. 82, 1503–1505 (2003).
[CrossRef]

de Ceglia, D.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. Dorazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Dereux, A.

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
[CrossRef]

Dorazio, A.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. Dorazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Fang, N.

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

Feise, M. W.

M. W. Feise and Y. S. Kivshar, “Sub-wavelength imaging with a left-handed material flat lens,” Phys. Lett. A 334, 326–330(2005).
[CrossRef]

Freire, M. J.

R. Marques, M. J. Freire, and J. D. Baena, “Theory of three-dimensional subdiffraction imaging,” Appl. Phys. Lett. 89, 211113 (2006).
[CrossRef]

Friberg, A. T.

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

Girard, C.

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
[CrossRef]

Gould, T. J.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nature Methods 5, 527–529(2008).
[CrossRef] [PubMed]

Grill, S.

Hakkarainen, T.

Haus, J. W.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. Dorazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Hess, S. T.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nature Methods 5, 527–529(2008).
[CrossRef] [PubMed]

Houck, A. A.

J. B. Brock, A. A. Houck, and I. L. Chuang, “Focusing inside negative index materials,” Appl. Phys. Lett. 85, 2472–2474(2004).
[CrossRef]

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Juette, M. F.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nature Methods 5, 527–529(2008).
[CrossRef] [PubMed]

Kalinin, V. A.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Imaging, compression and Poynting vector streamlines with negative permittivity materials,” Electron. Lett. 37, 1243–1244 (2001).
[CrossRef]

Kivshar, Y. S.

M. W. Feise and Y. S. Kivshar, “Sub-wavelength imaging with a left-handed material flat lens,” Phys. Lett. A 334, 326–330(2005).
[CrossRef]

Kotynski, R.

R. Kotyński and T. Stefaniuk, “Comparison of imaging with sub-wavelength resolution in the canalization and resonant tunnelling regimes,” J. Opt. A: Pure Appl. Opt. 11, 015001(2009).
[CrossRef]

Lee, H. S.

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

Lessard, M. D.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nature Methods 5, 527–529(2008).
[CrossRef] [PubMed]

Marques, R.

R. Marques, M. J. Freire, and J. D. Baena, “Theory of three-dimensional subdiffraction imaging,” Appl. Phys. Lett. 89, 211113 (2006).
[CrossRef]

McCutchen, C. W.

Merlin, R.

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84, 1290–1293 (2004).
[CrossRef]

Miyamoto, K.

Mlodzianoski, M. J.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nature Methods 5, 527–529(2008).
[CrossRef] [PubMed]

Nagpure, B. S.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100nm resolution fluorescence microscopy of thick samples,” Nature Methods 5, 527–529(2008).
[CrossRef] [PubMed]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, “Problem of image superresolution with a negative-refractive-index slab,” J. Opt. Soc. Am. A 21, 491–498 (2004).
[CrossRef]

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

Ozbay, E.

Pendry, J. B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

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

Ramakrishna, S. A.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

Ringhofer, K. H.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Imaging, compression and Poynting vector streamlines with negative permittivity materials,” Electron. Lett. 37, 1243–1244 (2001).
[CrossRef]

Rosenbluth, M.

D. R. Smith, D. Schurig, M. Rosenbluth, and S. Schultz, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

Scalora, M.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. Dorazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Schultz, S.

D. R. Smith, D. Schurig, M. Rosenbluth, and S. Schultz, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

Schurig, D.

D. R. Smith, D. Schurig, M. Rosenbluth, and S. Schultz, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

Setälä, T.

Shamonina, E.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Imaging, compression and Poynting vector streamlines with negative permittivity materials,” Electron. Lett. 37, 1243–1244 (2001).
[CrossRef]

Smith, D. R.

D. R. Smith, D. Schurig, M. Rosenbluth, and S. Schultz, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

Solymar, L.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Imaging, compression and Poynting vector streamlines with negative permittivity materials,” Electron. Lett. 37, 1243–1244 (2001).
[CrossRef]

Stefaniuk, T.

R. Kotyński and T. Stefaniuk, “Comparison of imaging with sub-wavelength resolution in the canalization and resonant tunnelling regimes,” J. Opt. A: Pure Appl. Opt. 11, 015001(2009).
[CrossRef]

Stelzer, E. H. K.

Stewart, W. J.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

Sun, C.

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

Tsai, D. P.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and μ,” Phys. Uspekhi 10, 509–514 (1968).
[CrossRef]

Vincenti, M. A.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. Dorazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Welford, W. T.

Wiltshire, M. C. K.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

Wolf, E.

Wood, B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

Zhang, X.

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

Appl. Phys. Lett.

D. R. Smith, D. Schurig, M. Rosenbluth, and S. Schultz, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003).
[CrossRef]

S. Cummer, “Simulated causal subwavelength focusing by a negative refractive index slab,” Appl. Phys. Lett. 82, 1503–1505 (2003).
[CrossRef]

J. B. Brock, A. A. Houck, and I. L. Chuang, “Focusing inside negative index materials,” Appl. Phys. Lett. 85, 2472–2474(2004).
[CrossRef]

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84, 1290–1293 (2004).
[CrossRef]

R. Marques, M. J. Freire, and J. D. Baena, “Theory of three-dimensional subdiffraction imaging,” Appl. Phys. Lett. 89, 211113 (2006).
[CrossRef]

Electron. Lett.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Imaging, compression and Poynting vector streamlines with negative permittivity materials,” Electron. Lett. 37, 1243–1244 (2001).
[CrossRef]

J. Mod. Opt.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

J. Opt. A: Pure Appl. Opt.

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Figures (8)

Fig. 1
Fig. 1

Schematic geometry of the planar-layer-based perfect lens.

Fig. 2
Fig. 2

Absolute value of the PSF | h 3 | in z d at λ 0 = 600 nm for absorbing slabs of δ = 0.1 and different widths: (a) d = 1.8 μm , (b) d = 600 nm , (c) d = 400 nm , and (d) d = 60 nm . The plot is normalized to unity at r = 0 , and contour lines for a value 1 / 2 (solid line) are drawn in black. The image plane is now shifted to z = 0 (vertical red line).

Fig. 3
Fig. 3

Absolute value of the angular spectrum in the semispace z > 0 associated with the 3D PSF for the superlenses analyzed in Fig. 2.

Fig. 4
Fig. 4

Spherical and hyperboloidal sheets constituting the generalized ATF are shown (a) in 3D and (b) on the meridional plane s x = 0 .

Fig. 5
Fig. 5

Absolute value of the angular spectrum | a | for the numerical simulations of Fig. 2. The component of the far-field is represented in the subfigure on the left and the near-field term of the ATF is shown on the right.

Fig. 6
Fig. 6

Intensity of the wave field in the image vol ume of two point sources centered at R 1 = 60 x ^ nm = R 2 provided under different conditions of depth: (left) both sources are located in the same transverse plane and (right) one source is 40 nm closer to the lens. The layer width is d = 60 nm and material absorption is δ = 10 1 .

Fig. 7
Fig. 7

Same as in Fig. 6, considering a medium of lower loss δ = 10 2 .

Fig. 8
Fig. 8

Same as in Fig. 6, for δ = 10 3 .

Equations (30)

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E out ( R , z ) = E in ( R 0 , z 0 ) h 3 ( R R 0 , z z 1 ) d 2 R 0 ,
h 3 ( R , z ) = 1 ( 2 π ) 2 T ( k ) exp ( i k R + i β 1 z ) d 2 k
T ( k ) = t 12 t 21 exp [ i ( β 1 + β 2 ) d ] 1 r 21 2 exp ( 2 i β 2 d ) ,
β j = σ j k 0 2 ϵ j μ j k · k , for j = { 1 , 2 } .
r j k = μ k β j μ j β k μ k β j + μ j β k
t j k = r j k + 1 .
E out ( R , 2 d z 0 ) = E in ( R 0 , z 0 ) .
h 3 ( R , z < 0 ) = 1 2 π z [ exp ( i k 0 r ) r ] ,
h 3 ( R , z > 0 ) = 1 2 π z [ exp ( i k 0 r ) r ] ,
r = R + z z ^ ,
h 3 ( R , z ) = 1 2 π 0 T ( k ) J 0 ( k R ) exp ( i β 1 z ) k d k ,
k · k = k 2 k 0 2 ,
h 3 ( R , z ) = h N ( R , z ) + h F ( R , z ) .
h F ( R , z ) = i k 0 2 π a ( s ) exp ( i k 0 s r ) d Ω .
k 0 s = k + β 1 z ^ .
a ( s ) = i λ 0 T ( s ) s z ,
h F ( R , 0 ) = J 1 ( k 0 R ) λ 0 R ,
h F ( 0 , z ) = ( 1 i k 0 z ) exp ( i k 0 z ) 1 2 π z 2 .
s · s = 1 ,
h F ( r ) = i k 0 2 π a ( s ) exp ( i k 0 s r ) d 3 s ,
j = 1 , 2 | h 3 ( R R j , z z j ) | 2 .
E in = A ( k ) exp [ i k R + i β 1 ( z + z 0 ) ] d 2 k ,
E r = R ( k ) A ( k ) exp [ i k R i β 1 ( z + z 0 ) ] d 2 k .
A ( k ) = 1 ( 2 π ) 2 E in ( R 0 , z 0 ) exp ( i k R 0 ) d 2 R 0
E t s = T s ( k ) A ( k ) exp [ i k R + i β 2 ( z z 0 ) ] d 2 k ,
E r s = R s ( k ) A ( k ) exp [ i k R i β 2 ( z z 0 ) ] d 2 k .
E out = T ( k ) A ( k ) exp [ i k R + i β 1 ( z z 1 ) ] d 2 k .
R = exp ( 2 i β 1 z 0 ) [ r 12 + t 12 r 21 t 21 exp ( 2 i β 2 d ) ) 1 r 21 2 exp ( 2 i β 2 d ) ] ,
T s = exp [ i ( β 1 + β 2 ) z 0 ] [ t 12 1 r 21 2 exp ( 2 i β 2 d ) ] ,
R s = exp [ i ( β 1 + β 2 ) z 0 ] [ t 12 r 21 exp [ 2 i β 2 ( d z 0 ) ] 1 r 21 2 exp ( 2 i β 2 d ) ] .

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