Abstract

By means of the angular spectrum representation of wave fields, a discussion is given on the propagation and restoration of the wave-front structure in a slab of a left-handed medium (or negative-index medium) whose surface impedance matches that of vacuum, namely, one whose effective optical parameters are n==μ=-1. This restoration was previously discussed [Phys. Rev. Lett. 85, 3866 (2000)] in regard to whether it may yield superresolved images. The divergence of the wave field in the slab, and its equivalence with that of the inverse diffraction propagator in free space, is addressed. Further, the existence of absorption, its regularization of this divergence, and the trade-off of a resulting limited superresolution are analyzed in detail in terms of its effect on the evanescent components of the wave field and hence on the transfer function width.

© 2004 Optical Society of America

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    [CrossRef]
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  4. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbach, M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401–107404 (2003).
    [CrossRef]
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    [CrossRef]
  6. R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625-1–056625-15 (2001).
    [CrossRef]
  7. N. Garcia, M. Nieto-Vesperinas, “Is there an experimental verification of a negative index of refraction yet?” Opt. Lett. 27, 885–887 (2002).
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  8. P. M. Valanju, R. M. Walserand, A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401–187404 (2002).
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    [CrossRef]

2003 (8)

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbach, M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401–107404 (2003).
[CrossRef]

A. A. Houck, J. B. Brock, I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401-1–137401-4 (2003).
[CrossRef]

P. Markos, C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
[CrossRef] [PubMed]

A. Grbic, G. V. Eleftheriades, “Growing evanescent waves in negative-refractive-index transmission-line media,” Appl. Phys. Lett. 82, 1815–1817 (2003).
[CrossRef]

J. B. Pendry, “Comment on ‘Left-handed materials do not make a perfect lens’,” Phys. Rev. Lett. 91, 099701 (2003).
[CrossRef]

M. Nieto-Vesperinas, N. Garcia, “Reply to ‘Comment on “Left-handed materials do not make a perfect lens”’,” Phys. Rev. Lett. 91, 099701 (2003).
[CrossRef]

N. Garcia, M. Nieto-Vesperinas, “Erratum: Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 90, 229903 (2003).
[CrossRef]

A. A. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602(R)-1–025602(R)-4 (2003).
[CrossRef]

2002 (7)

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

N. Garcia, M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403–207406 (2002).
[CrossRef] [PubMed]

N. Garcia, M. Nieto-Vesperinas, “Is there an experimental verification of a negative index of refraction yet?” Opt. Lett. 27, 885–887 (2002).
[CrossRef]

P. M. Valanju, R. M. Walserand, A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401–187404 (2002).
[CrossRef] [PubMed]

A. L. Pokrovski, A. L. Efros, “Electrodynamics of metallic photonic crystals and the problem of left-handed materials,” Phys. Rev. Lett. 89, 093901–093904 (2002).
[CrossRef]

A. Lakhtakia, M. W. McCall, W. S. Weigholfer, “Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),” Int. J. Electron. Commun. 56, 407–410 (2002).
[CrossRef]

E. V. Ponizovskaya, M. Nieto-Vesperinas, N. Garcia, “Losses for microwave transmission in metamaterials for producing left-handed materials,” Appl. Phys. Lett. 81, 4470–4472 (2002).
[CrossRef]

2001 (3)

R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625-1–056625-15 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

D. A. Fletcher, K. E. Goodson, G. S. Kino, “Focusing in microlenses close to a wavelength in diameter,” Opt. Lett. 26, 399–401 (2001).
[CrossRef]

2000 (1)

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

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced non-linear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 195–225 (1999).
[CrossRef]

1997 (1)

J. J. Greffet, R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

1979 (1)

M. Bertero, G. De Mol, G. A. Viano, “On the problems of object restoration and image extrapolation in optics,” J. Math. Phys. 20, 509–521 (1979).
[CrossRef]

1968 (2)

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ∊ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

J. R. Shewell, E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968).
[CrossRef]

Bertero, M.

M. Bertero, G. De Mol, G. A. Viano, “On the problems of object restoration and image extrapolation in optics,” J. Math. Phys. 20, 509–521 (1979).
[CrossRef]

M. Bertero, G. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed., Vol. 20 of Topics in Current Physics, (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

Brock, J. B.

A. A. Houck, J. B. Brock, I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401-1–137401-4 (2003).
[CrossRef]

Carminati, R.

J. J. Greffet, R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Chuang, I. L.

A. A. Houck, J. B. Brock, I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401-1–137401-4 (2003).
[CrossRef]

De Mol, G.

M. Bertero, G. De Mol, G. A. Viano, “On the problems of object restoration and image extrapolation in optics,” J. Math. Phys. 20, 509–521 (1979).
[CrossRef]

M. Bertero, G. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed., Vol. 20 of Topics in Current Physics, (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Efros, A. L.

A. L. Pokrovski, A. L. Efros, “Electrodynamics of metallic photonic crystals and the problem of left-handed materials,” Phys. Rev. Lett. 89, 093901–093904 (2002).
[CrossRef]

Eleftheriades, G. V.

A. Grbic, G. V. Eleftheriades, “Growing evanescent waves in negative-refractive-index transmission-line media,” Appl. Phys. Lett. 82, 1815–1817 (2003).
[CrossRef]

Fletcher, D. A.

Forester, D. W.

A. A. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602(R)-1–025602(R)-4 (2003).
[CrossRef]

Garcia, N.

N. Garcia, M. Nieto-Vesperinas, “Erratum: Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 90, 229903 (2003).
[CrossRef]

M. Nieto-Vesperinas, N. Garcia, “Reply to ‘Comment on “Left-handed materials do not make a perfect lens”’,” Phys. Rev. Lett. 91, 099701 (2003).
[CrossRef]

N. Garcia, M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403–207406 (2002).
[CrossRef] [PubMed]

E. V. Ponizovskaya, M. Nieto-Vesperinas, N. Garcia, “Losses for microwave transmission in metamaterials for producing left-handed materials,” Appl. Phys. Lett. 81, 4470–4472 (2002).
[CrossRef]

N. Garcia, M. Nieto-Vesperinas, “Is there an experimental verification of a negative index of refraction yet?” Opt. Lett. 27, 885–887 (2002).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Goodson, K. E.

Grbic, A.

A. Grbic, G. V. Eleftheriades, “Growing evanescent waves in negative-refractive-index transmission-line media,” Appl. Phys. Lett. 82, 1815–1817 (2003).
[CrossRef]

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbach, M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401–107404 (2003).
[CrossRef]

Greffet, J. J.

J. J. Greffet, R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Heyman, E.

R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625-1–056625-15 (2001).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced non-linear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 195–225 (1999).
[CrossRef]

Houck, A. A.

A. A. Houck, J. B. Brock, I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401-1–137401-4 (2003).
[CrossRef]

Kino, G. S.

Koltenbach, B. E. C.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbach, M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401–107404 (2003).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, M. W. McCall, W. S. Weigholfer, “Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),” Int. J. Electron. Commun. 56, 407–410 (2002).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, UK, 1963).

Li, K.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbach, M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401–107404 (2003).
[CrossRef]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, UK, 1963).

Loschialpo, A. A.

A. A. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602(R)-1–025602(R)-4 (2003).
[CrossRef]

Markos, P.

Maslowski, S. I.

S. A. Tretyakov, I. S. Nefedov, C. R. Simowski, S. I. Maslowski, “Modelling and microwave properties of artificial materials with negative parameters,” in Advances in Electromagnetics of Complex Media and Materials, S. Zouhdi, A. Sihvola, M. Arsalane, eds. (Kluwer, Dordrecht, The Netherlands, 2002), pp. 99–122.

McCall, M. W.

A. Lakhtakia, M. W. McCall, W. S. Weigholfer, “Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),” Int. J. Electron. Commun. 56, 407–410 (2002).
[CrossRef]

Nefedov, I. S.

S. A. Tretyakov, I. S. Nefedov, C. R. Simowski, S. I. Maslowski, “Modelling and microwave properties of artificial materials with negative parameters,” in Advances in Electromagnetics of Complex Media and Materials, S. Zouhdi, A. Sihvola, M. Arsalane, eds. (Kluwer, Dordrecht, The Netherlands, 2002), pp. 99–122.

Nieto-Vesperinas, M.

N. Garcia, M. Nieto-Vesperinas, “Erratum: Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 90, 229903 (2003).
[CrossRef]

M. Nieto-Vesperinas, N. Garcia, “Reply to ‘Comment on “Left-handed materials do not make a perfect lens”’,” Phys. Rev. Lett. 91, 099701 (2003).
[CrossRef]

N. Garcia, M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403–207406 (2002).
[CrossRef] [PubMed]

E. V. Ponizovskaya, M. Nieto-Vesperinas, N. Garcia, “Losses for microwave transmission in metamaterials for producing left-handed materials,” Appl. Phys. Lett. 81, 4470–4472 (2002).
[CrossRef]

N. Garcia, M. Nieto-Vesperinas, “Is there an experimental verification of a negative index of refraction yet?” Opt. Lett. 27, 885–887 (2002).
[CrossRef]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbach, M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401–107404 (2003).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Comment on ‘Left-handed materials do not make a perfect lens’,” Phys. Rev. Lett. 91, 099701 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

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

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced non-linear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 195–225 (1999).
[CrossRef]

Pokrovski, A. L.

A. L. Pokrovski, A. L. Efros, “Electrodynamics of metallic photonic crystals and the problem of left-handed materials,” Phys. Rev. Lett. 89, 093901–093904 (2002).
[CrossRef]

Ponizovskaya, E. V.

E. V. Ponizovskaya, M. Nieto-Vesperinas, N. Garcia, “Losses for microwave transmission in metamaterials for producing left-handed materials,” Appl. Phys. Lett. 81, 4470–4472 (2002).
[CrossRef]

Rachford, F. J.

A. A. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602(R)-1–025602(R)-4 (2003).
[CrossRef]

Ramakrishna, S. A.

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced non-linear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 195–225 (1999).
[CrossRef]

Schelleng, J.

A. A. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602(R)-1–025602(R)-4 (2003).
[CrossRef]

Schultz, S.

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Schurig, D.

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Shewell, J. R.

Simowski, C. R.

S. A. Tretyakov, I. S. Nefedov, C. R. Simowski, S. I. Maslowski, “Modelling and microwave properties of artificial materials with negative parameters,” in Advances in Electromagnetics of Complex Media and Materials, S. Zouhdi, A. Sihvola, M. Arsalane, eds. (Kluwer, Dordrecht, The Netherlands, 2002), pp. 99–122.

Smith, D. L.

A. A. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602(R)-1–025602(R)-4 (2003).
[CrossRef]

Smith, D. R.

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Soukoulis, C. M.

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced non-linear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 195–225 (1999).
[CrossRef]

Tanielian, M.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbach, M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401–107404 (2003).
[CrossRef]

Tretyakov, S. A.

S. A. Tretyakov, I. S. Nefedov, C. R. Simowski, S. I. Maslowski, “Modelling and microwave properties of artificial materials with negative parameters,” in Advances in Electromagnetics of Complex Media and Materials, S. Zouhdi, A. Sihvola, M. Arsalane, eds. (Kluwer, Dordrecht, The Netherlands, 2002), pp. 99–122.

Valanju, A. P.

P. M. Valanju, R. M. Walserand, A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401–187404 (2002).
[CrossRef] [PubMed]

Valanju, P. M.

P. M. Valanju, R. M. Walserand, A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401–187404 (2002).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ∊ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Viano, G. A.

M. Bertero, G. De Mol, G. A. Viano, “On the problems of object restoration and image extrapolation in optics,” J. Math. Phys. 20, 509–521 (1979).
[CrossRef]

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[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Scheme of propagation: a divergent wave field is created in vacuum at the object plane z=-z0. There is a focus at a plane z=z0 of a LHM with =μ=n=-1. If, as indicated, the latter is a slab of thickness d, there is a second focus in vacuum at the plane z=2d-z0. The ray vector t=S/|S| (S being the Poynting vector) of each plane-wave component points along and opposite to the phase vector k in vacuum and in the LHM, respectively. If d=2z0, then z=2d-z0=z0+d=3z0.

Fig. 2
Fig. 2

(a) Modulus M(s) of the slab transfer function in the evanescent region (ky=k0s>k0). Solid curve: n2=0.1, z0/λ=0.056; dotted–dashed curve: n2=0.1, z0/λ=0.24; dotted curve: n2=0.1, z0/λ=0.7. Curves with symbols correspond to the free-space propagation filter exp(-k0z0s2-1): z0/λ=0.056 (pluses), z0/λ=0.24 (triangles), z0/λ=0.7 (squares). (b) Phase ϕ(s) of the slab transfer function in the evanescent region (ky=k0s>k0).

Fig. 3
Fig. 3

Relationship between n2 and the half-width at half-maximum sc of the transfer function modulus M(s) in the evanescent region: solid curve, z0/λ=0.056; dashed curve, z0/λ=0.09; dotted–dashed curve, z0/λ=0.24.

Fig. 4
Fig. 4

Modulus M(s) of the slab transfer function in the propagating region (ky=k0s<k0).

Equations (34)

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E(r)=-A(kx, ky)exp(ik·r)dkxdky.
E(x, y, z=-z0)=f(x, y).
E(r)=-A(kx, ky)exp{i[kxx+kyy+kz(z+z0)]}dkxdky,  (-z0z0).
E(r)=-A(kx, ky)exp{i[kxx+kyy-kz(z-z0)]}dkxdky(z0),
E(r)=-dxdyf(x, y)KLHM(x-x, y-y, z-z0),
KLHM(x-x, y-y, z-z0)=(1/2π)2-exp{i[kx(x-x)+ky(y-y)-kz(z-z0)]}dkxdky (z0).
E(r)=-A(kx, ky)exp{i[kxx+kyy+kz(z-2d+z0)]}dkxdky(zd).
K1(x-x, y-y, z-2d+z0)=(1/2π)2-exp{i[kx(x-x)+ky(y-y)+kz(z-2d+z0)]}dkxdky(zd).
exp(ikR)R=i2πhdkxdkykzexp{i[kx(x-x0)+ky(y-y0)±kz(z-z0)]}+i2πidkxdkyi|kz|exp{i[kx(x-x0)+ky(y-y0)]|kz|(z-z0)},
exp(-ikR)R=-i2πhdkxdkykzexp{i[kx(x-x0)+ky(y-y0)kz(z-z0)]}+i2πidkxdkyi|kz|exp{i[kx(x-x0)+ky(y-y0)]|kz|(z-z0)}.
KLHM(x-x, y-y, z-z)=-12πz0exp(ikR)R,
Kconv(x-x0, y-y0, z-z0)=-12πzexp(-ikR)R=1(2π)2hdkxdky exp{i[kx(x-x0)+ky(y-y0)-kz(z-z0)]}+1(2π)2idkxdky×exp{i[kx(x-x0)+ky(y-y0)]-|kz|(z-z0)}.
KLHM h(x-x, y-y, z-z0)=12πz0exp(-ikR)Rh=-Kconvh(x-x0, y-y0, z-z0),
Kconv-1 i(x-x0, y-y0, z-z0)=1(2π)2idkxdky exp{i[kx(x-x0)+ky(y-y0)]+|kz|(z-z0)}.
KLHM i(x-x, y-y, z-z0)=Kconv-1 i(x-x, y-y, z-z0).
Ex(zd)=A(ky)exp(ikyy)exp[ikz(z+z0)]×4kzkzμ exp(-ikzd)exp(-ikzd)-(kz-μkz)2+(kz+μkz)2 exp(-2ikzd)  (kz=k02-ky2, ky2k02;  kz=iky2-k02, ky2>k02),  (kz=k02n2-ky2, ky2k02n2;  kz=iky2-k02n2, ky2>k02n2).
Ex(0zd)=A(ky)exp(iky)exp(ikzz0)2kzμ{(kz-μkz)exp(-ikzz)+(kz+μkz)exp[ikz(z-2d)]}-(kz-μkz)2+(kz+μkz)2 exp(-2ikzd)
(kz=k02-ky2, ky2k02;kz=iky2-k02, ky2>k02),
(kz=k02n2-ky2, ky2k02n2;kz=iky2-k02n2, ky2>k02n2).
Ex(zd)=A(ky)exp(ikyy)exp[ikz(z+z0)]4(1-in2)exp(-2ikzd)exp(-kzn2d)(2-in2)2+n22 exp(-2ikzd)exp(-2kzn2d)(ky2k02),
Ex(zd)=A(ky)exp(ikyy)exp[-|kz|(z+z0)]4(1-in2)exp(2|kz|d)exp(-i|kz|n2d)(2-in2)2+n22 exp(2|kz|d)exp(-2i|kz|n2d)
(ky2>k02, |kz|=ky2-k02).
Ex(0zd)=A(ky)exp(iky)exp(ikzz0)×2{(2-in2)exp(-ikzz)exp(-kzn2z)-in2 exp[ikz(z-2d)]exp[kzn2(z-2d)]}(2-in2)2+n22 exp(-2ikzd)exp(-2kzn2d)
(ky2k02),
Ex(0zd)=A(ky)exp(iky)exp(-|kz|z0)×2{(2-in2)exp(|kz|z)exp(-i|kz|n2z)-in2 exp[-|kz|(z-2d)]exp[i|kz|n2(z-2d)]}(2-in2)2+n22 exp(2|kz|d)exp(-2i|kz|n2d)
(ky2>k02, |kz|=ky2-k02).
Ex(z=z0+d)=A(ky)exp(ikyy)T(ky),
T(ky)=4(1-in2)exp(-2i|kz|n2z0)(2-in2)2+n22 exp(4|kz|z0)exp(-4i|kz|n2z0)
(d=2z0, ky2>k02, |kz|=ky2-k02).
Ex(z=d=2z0)=Ex(z=z0+d)exp(|kz|z0)  (d=2z0).
T(ky)=4(1-in2)n22exp(-4|kz|z0)exp(2i|kz|n2z0),
M(ky)=1/[1+n224exp(4|kz|z0)].
2z0ky(c)2-k02=ln(2/n2).
n2<2 exp[-4π(z0/λ)sc2-1].

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