Abstract

We investigate near-field imaging of a point dipole by a lossy, nanoscale metamaterial slab. Making use of the electromagnetic angular-spectrum representation, we derive the Green tensor for the field transmission through the metamaterial slab, duly considering multiple reflections, polarizations, and wave-vector signs. With this general formalism, we calculate the point-spread function of the imaging system, which enables us to assess, for instance, resolution and image brightness. Our results demonstrate that with the metamaterial-slab lens one achieves resolution beyond the conventional diffraction limit of half the wavelength. In general, the resolution and image brightness are degraded when the slab thickness and absorption increase, but we show that in some cases the resolution is rather insensitive to the magnitude of the losses in the metamaterial.

© 2009 Optical Society of America

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References

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  1. R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design, and Microwave Applications (Wiley, 2008).
  2. L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford Univ. Press, 2009).
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  4. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and μ,” Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  5. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1981).
  6. D. R. Smith, D. Shurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
    [CrossRef]
  7. D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13, 2127-2134 (2005).
    [CrossRef] [PubMed]
  8. H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
    [CrossRef]
  9. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
    [CrossRef] [PubMed]
  10. S. A. Ramakrishna and O. J. F. Martin, “Resolving the wave vector in negative refractive index media,” Opt. Lett. 30, 2626-2628 (2005). This paper contains a mathematical error in mapping the second Rieman sheet. When calculating the complex square root, one can use either two Rieman sheets or +/- sign in one Rieman sheet, but not both. However, the physical conclusions in Fig. are correct.
    [CrossRef] [PubMed]
  11. R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microwave Opt. Technol. Lett. 41, 315-316 (2004).
    [CrossRef]
  12. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge Univ. Press, 2006).
  13. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481-489 (1987).
    [CrossRef]
  14. G. S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries,” Phys. Rev. A 11, 230-242 (1975).
    [CrossRef]
  15. J. B. Pendry, “Reply to 'Comment on “Negative refraction makes a perfect lens,”'” Phys. Rev. Lett. 87, 249702 (2001).
    [CrossRef]
  16. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).
  17. S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
    [CrossRef]
  18. L. S. Froute-Pérez and R. Carminati,“Controlling the fluorescence lifetime of a single emitter on the nanoscale using a plasmonic superlens,” Phys. Rev. B 78, 125403 (2008).
    [CrossRef]
  19. N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. 82, 161-163 (2003).
    [CrossRef]
  20. A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92, 077401 (2004).
    [CrossRef] [PubMed]

2008 (1)

L. S. Froute-Pérez and R. Carminati,“Controlling the fluorescence lifetime of a single emitter on the nanoscale using a plasmonic superlens,” Phys. Rev. B 78, 125403 (2008).
[CrossRef]

2005 (5)

D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13, 2127-2134 (2005).
[CrossRef] [PubMed]

S. A. Ramakrishna and O. J. F. Martin, “Resolving the wave vector in negative refractive index media,” Opt. Lett. 30, 2626-2628 (2005). This paper contains a mathematical error in mapping the second Rieman sheet. When calculating the complex square root, one can use either two Rieman sheets or +/- sign in one Rieman sheet, but not both. However, the physical conclusions in Fig. are correct.
[CrossRef] [PubMed]

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
[CrossRef]

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

2004 (2)

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microwave Opt. Technol. Lett. 41, 315-316 (2004).
[CrossRef]

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92, 077401 (2004).
[CrossRef] [PubMed]

2003 (2)

N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

D. R. Smith, D. Shurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

2001 (1)

J. B. Pendry, “Reply to 'Comment on “Negative refraction makes a perfect lens,”'” Phys. Rev. Lett. 87, 249702 (2001).
[CrossRef]

2000 (1)

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

1987 (1)

1975 (1)

G. S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries,” Phys. Rev. A 11, 230-242 (1975).
[CrossRef]

1968 (1)

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

Agarwal, G. S.

G. S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries,” Phys. Rev. A 11, 230-242 (1975).
[CrossRef]

Ambati, M.

H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
[CrossRef]

Blaikie, R. J.

Carminati, R.

L. S. Froute-Pérez and R. Carminati,“Controlling the fluorescence lifetime of a single emitter on the nanoscale using a plasmonic superlens,” Phys. Rev. B 78, 125403 (2008).
[CrossRef]

Depine, R. A.

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microwave Opt. Technol. Lett. 41, 315-316 (2004).
[CrossRef]

Durant, S.

H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
[CrossRef]

Fang, N.

H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
[CrossRef]

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

N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Froute-Pérez, L. S.

L. S. Froute-Pérez and R. Carminati,“Controlling the fluorescence lifetime of a single emitter on the nanoscale using a plasmonic superlens,” Phys. Rev. B 78, 125403 (2008).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge Univ. Press, 2006).

Kissel, V. N.

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92, 077401 (2004).
[CrossRef] [PubMed]

Lagarkov, A. N.

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92, 077401 (2004).
[CrossRef] [PubMed]

Lakhtakia, A.

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microwave Opt. Technol. Lett. 41, 315-316 (2004).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1981).

Lee, H.

H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
[CrossRef]

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

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1981).

Marques, R.

R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design, and Microwave Applications (Wiley, 2008).

Martin, F.

R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design, and Microwave Applications (Wiley, 2008).

Martin, O. J. F.

Melville, D. O. S.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge Univ. Press, 2006).

Pendry, J. B.

D. R. Smith, D. Shurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

J. B. Pendry, “Reply to 'Comment on “Negative refraction makes a perfect lens,”'” Phys. Rev. Lett. 87, 249702 (2001).
[CrossRef]

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

Ramakrishna, S. A.

S. A. Ramakrishna and O. J. F. Martin, “Resolving the wave vector in negative refractive index media,” Opt. Lett. 30, 2626-2628 (2005). This paper contains a mathematical error in mapping the second Rieman sheet. When calculating the complex square root, one can use either two Rieman sheets or +/- sign in one Rieman sheet, but not both. However, the physical conclusions in Fig. are correct.
[CrossRef] [PubMed]

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

D. R. Smith, D. Shurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Rosenbluth, M.

D. R. Smith, D. Shurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Schultz, S.

D. R. Smith, D. Shurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Shamonina, E.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford Univ. Press, 2009).

Shurig, D.

D. R. Smith, D. Shurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Sipe, J. E.

Smith, D. R.

D. R. Smith, D. Shurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Solymar, L.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford Univ. Press, 2009).

Sorolla, M.

R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design, and Microwave Applications (Wiley, 2008).

Srituravanich, W.

H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
[CrossRef]

Sun, C.

H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
[CrossRef]

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

Veselago, V. G.

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

Xiong, Y.

H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
[CrossRef]

Zhang, X.

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

H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
[CrossRef]

N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

Appl. Phys. Lett. (2)

D. R. Smith, D. Shurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

J. Opt. Soc. Am. B (1)

Microwave Opt. Technol. Lett. (1)

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microwave Opt. Technol. Lett. 41, 315-316 (2004).
[CrossRef]

New J. Phys. (1)

H. Lee, Y. Xiong, N. Fang, W. Srituravanich, S. Durant, M. Ambati, C. Sun, and X. Zhang, “Realization of optical superlens imaging below the diffraction limit,” New J. Phys. 7, 255-270 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

G. S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries,” Phys. Rev. A 11, 230-242 (1975).
[CrossRef]

Phys. Rev. B (1)

L. S. Froute-Pérez and R. Carminati,“Controlling the fluorescence lifetime of a single emitter on the nanoscale using a plasmonic superlens,” Phys. Rev. B 78, 125403 (2008).
[CrossRef]

Phys. Rev. Lett. (3)

J. B. Pendry, “Reply to 'Comment on “Negative refraction makes a perfect lens,”'” Phys. Rev. Lett. 87, 249702 (2001).
[CrossRef]

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed-material plate,” Phys. Rev. Lett. 92, 077401 (2004).
[CrossRef] [PubMed]

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

Rep. Prog. Phys. (1)

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

Science (1)

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

Sov. Phys. Usp. (1)

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

Other (5)

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1981).

R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design, and Microwave Applications (Wiley, 2008).

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford Univ. Press, 2009).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge Univ. Press, 2006).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

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Figures (8)

Fig. 1
Fig. 1

Illustration of the imaging system. A slab of metamaterial of thickness d (medium 2 with ϵ r , 2 and μ r , 2 ) is surrounded by vacuum (medium 1 with ϵ r , 1 = μ r , 1 = 1 ). The slab is used to collect light emitted by a point dipole in the object plane at a distance d 2 in front of the slab. The slab produces an image of the dipole on the opposite side. We consider the image at the distance of d 2 from the slab, which corresponds to the perfect lens condition.

Fig. 2
Fig. 2

Complex k z , 2 2 plane showing the different regions for the fast-decaying and slowly decaying (growing) waves. The parameters a = Re ( k z , 2 2 ) and b = Im ( k z , 2 2 ) are explicitly given in Eqs. (5, 6), respectively. The branch cut for the square root is along the positive real axis.

Fig. 3
Fig. 3

Illustration of the complex k z plane. The eight regions correspond to the regions in the two Riemann sheets of the k z 2 plane. The parameters a = Re ( k z , 2 2 ) and b = Im ( k z , 2 2 ) are defined in Eqs. (5, 6), respectively.

Fig. 4
Fig. 4

Magnitude of the transmission coefficient for a metamaterial slab as a function of the transversal component of the wave vector. In (a) the slab thickness d is varied ( ϵ r , 2 = μ r , 2 = 1 + i 0.1 ) , and in (b) the absorption in the slab is altered. For the chosen material parameters T s = T p in all cases.

Fig. 5
Fig. 5

Logarithmic intensity distributions of the dipole field in free space at the plane z = 80 nm . The dipole is located in the origin and oriented along (a) the x axis and (b) the z axis. The dipole radiates at a wavelength of λ = 633 nm .

Fig. 6
Fig. 6

Intensity distributions of the dipole field in the image plane behind the slab. The object dipole is oriented along (a) the x axis and (b) the z axis, and it radiates at a wavelength of λ = 633 nm . The thickness and material parameters of the slab are d = 40 nm and ϵ r , 2 = μ r , 2 = 1 + i 0.1 , respectively.

Fig. 7
Fig. 7

Point-spread functions for the x-oriented object dipole along the x axis (blue online) and the y axis (red online) with (a) variable thicknesses of the slab ( ϵ r , 2 = μ r , 2 = 1 + i 0.1 ) , and (b) variable imaginary parts of the slab’s material parameters ( d = 40 nm ) . The dipole radiates at a wavelength of λ = 633 nm .

Fig. 8
Fig. 8

Peak value (left arrows, blue online) and FWHM (right arrows, red online) of the psf of z- and x-oriented object dipoles as a function of (a) the thickness of the slab ( ϵ r , 2 = μ r , 2 = 1 + i 0.1 ) and (b) the imaginary parts of ϵ r , 2 and μ r , 2 ( d = 40 nm ) .

Equations (35)

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E ( r , t ) = E 0 ( ω ) e i ( k r ω t ) ,
k x 2 + k y 2 + k z 2 = n 2 k 0 2 ,
k z = [ ( ϵ r μ r ϵ r μ r ) k 0 2 ( k x 2 + k y 2 ) + i ( ϵ r μ r + ϵ r μ r ) k 0 2 ] 1 2 .
k z , 2 2 = a + i b ,
a = ( ϵ r , 2 μ r , 2 ϵ r , 2 μ r , 2 ) k 0 2 ( k x 2 + k y 2 ) ,
b = ( ϵ r , 2 μ r , 2 + ϵ r , 2 μ r , 2 ) k 0 2 .
k z , 2 2 = R e i θ ,
k z , 2 = R 1 2 [ cos ( θ 2 ) + i sin ( θ 2 ) ] .
k z , 2 = [ ± 1 2 ( a + a 2 + b 2 ) ± i 1 2 ( a + a 2 + b 2 ) ] .
E ( r , ω ) = μ r μ 0 ω 2 G ( r , r 0 , ω ) μ .
E ( r , ω ) = i μ r μ 0 ω 2 8 π 2 1 k z ( s ̂ s ̂ + p ̂ ± p ̂ ± ) μ e i [ k ( r r , 0 ) ± k z ( z z 0 ) ] d k x d k y ,
T 1 , 2 = s ̂ t 1 , 2 s s ̂ + p ̂ ± , 2 t 1 , 2 p p ̂ ± , 1 ,
t 1 , 2 s = 2 μ r , 2 k z , 1 μ r , 2 k z , 1 + μ r , 1 k z , 2 ,
t 1 , 2 p = 2 ϵ r , 2 k z , 1 ϵ r , 2 k z , 1 + ϵ r , 1 k z , 2 μ r , 2 ϵ r , 1 μ r , 1 ϵ r , 2
R 1 , 2 = s ̂ r 1 , 2 s s ̂ + p ̂ , 1 r 1 , 2 p p ̂ ± , 1 ,
r 1 , 2 s = μ r , 2 k z , 1 μ r , 1 k z , 2 μ r , 2 k z , 1 + μ r , 1 k z , 2 ,
r 1 , 2 p = ϵ r , 2 k z , 1 ϵ r , 1 k z , 2 ϵ r , 2 k z , 1 + ϵ r , 1 k z , 2 .
T s , p = t 2 , 1 s , p t 1 , 2 s , p e i k z , 2 d 1 ( r 2 , 1 s , p ) 2 e 2 i k z , 2 d ,
E ( r , ω ) = ω 2 μ r , 1 μ 0 G slab ( r , r 0 , ω ) μ ,
G slab ( r , r 0 , ω ) = i 8 π 2 1 k z , 1 e i [ k ( r r , 0 ) + k z , 1 ( z z 0 d ) ] ( T s s ̂ s ̂ + T p p ̂ + , 1 p ̂ + , 1 ) d k x d k y .
G slab ( r , r 0 , ω ) = i 8 π 2 0 k k z , 1 e i k z , 1 ( z z 0 d ) ( T s S + T p P ) d k .
S 11 = π J 0 ( k | r r , 0 | ) + π J 2 ( k | r r , 0 | ) cos ( 2 θ ) ,
S 12 = S 21 = π J 2 ( k | r r , 0 | ) sin ( 2 θ ) ,
S 22 = π J 0 ( k | r r , 0 | ) π J 2 ( k | r r , 0 | ) cos ( 2 θ )
P 11 = k z , 1 2 k 1 2 [ π J 0 ( k | r r , 0 | ) π J 2 ( k | r r , 0 | ) cos ( 2 θ ) ] ,
P 12 = P 21 = k z , 1 2 k 1 2 [ π J 2 ( k | r r , 0 | ) sin ( 2 θ ) ] ,
P 13 = P 31 = k k z , 1 k 1 2 [ 2 π i J 1 ( k | r r , 0 | ) cos θ ] ,
P 22 = k z , 1 2 k 1 2 [ π J 0 ( k | r r , 0 | ) + π J 2 ( k | r r , 0 | ) cos ( 2 θ ) ] ,
P 23 = P 32 = k k z , 1 k 1 2 [ 2 π i J 1 ( k | r r , 0 | ) sin θ ] ,
P 33 = k 2 k 1 2 [ 2 π J 0 ( k | r r , 0 | ) ] .
s ̂ s ̂ = [ sin 2 φ sin φ cos φ 0 sin φ cos φ cos 2 φ 0 0 0 0 ] ,
p ̂ + , 1 p ̂ + , 1 = 1 k 1 2 [ k z 2 cos 2 φ k z 2 sin φ cos φ k k z cos φ k z 2 sin φ cos φ k z 2 sin 2 φ k k z sin φ k k z cos φ k k z sin φ k 2 ] .
k ( r r , 0 ) = k | r r , 0 | cos ( φ θ ) ,
G slab ( r , r 0 , ω ) = i 8 π 2 0 0 2 π k k z , 1 ( T s s ̂ s ̂ + T p p ̂ + , 1 p ̂ + , 1 ) exp { i [ k | r r , 0 | cos ( φ θ ) + k z , 1 ( z z 0 d ) ] } d k d φ .
0 2 π e i q r cos ( φ θ ) { 1 cos φ sin φ cos φ sin φ cos 2 φ sin 2 φ } d φ = { 2 π J 0 ( q r ) 2 π i J 1 ( q r ) cos θ 2 π i J 1 ( q r ) sin θ π J 2 ( q r ) sin 2 θ π J 0 ( q r ) π J 2 ( q r ) cos 2 θ π J 0 ( q r ) + π J 2 ( q r ) cos 2 θ } ,

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