Abstract

Negative refraction through a triangular prism may be explained without assigning a negative refractive index to the prism by using array theory. For the case of a beam incident upon the wedge, the array theory accurately predicts the beam transmission angle through the prism and provides an estimate of the frequency interval at which negative refraction occurs. The hypotenuse of the prism has a staircase shape because it is built of cubic unit cells. The large phase delay imparted by each unit cell, combined with the staircase shape of the hypotenuse, creates the necessary conditions for negative refraction. Full-wave simulations using the finite-difference time-domain method show that array theory accurately predicts the beam transmission angle.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [Crossref]
  2. 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]
  3. C. G. Parazzoli, R. B. Greegor, B. Koltenbach, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
    [Crossref]
  4. X.-X. Liu and A. Alu, “Homogenization of quasi-isotropic metamaterials composed by dense arrays of magnetodielectric spheres,” Metamaterials 5, 56–63 (2011).
    [Crossref]
  5. F. Y. Meng, Q. Wu, B. S. Jin, H. L. Wang, and J. Wu, “Numerical verification of the NIR features for 2D isotropic LHM,” Acta Phys. Sinica 55, 4514–4519 (2006).
  6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [Crossref]
  7. A. Ishimaru, S.-W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antennas Propag. 51, 2550–2557 (2003).
    [Crossref]
  8. R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
    [Crossref]
  9. Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
    [Crossref]
  10. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961), pp. 106–113.
  11. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 2012), p. 284.
  12. Remcom, XFdtd Reference Manual (2016).
  13. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  14. K. S. Kunz and R. S. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).
  15. P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401 (2002).
    [Crossref]
  16. B. A. Munk, Metamaterials: Critique and Alternatives (Wiley, 2008), pp. 11–14.
  17. S. J. Weiss, G. A. Talalai, and A. Zaghloul, “Negative refraction and array theory,” in Antenna Applications Symposium, Monticello, 2015.
  18. X. Liu and A. Alu, “Limitations and potentials of metamaterial lenses,” J. Nanophoton. 5, 53509 (2011).
    [Crossref]
  19. A. Grbic and G. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
    [Crossref]

2011 (2)

X.-X. Liu and A. Alu, “Homogenization of quasi-isotropic metamaterials composed by dense arrays of magnetodielectric spheres,” Metamaterials 5, 56–63 (2011).
[Crossref]

X. Liu and A. Alu, “Limitations and potentials of metamaterial lenses,” J. Nanophoton. 5, 53509 (2011).
[Crossref]

2007 (1)

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[Crossref]

2006 (2)

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[Crossref]

F. Y. Meng, Q. Wu, B. S. Jin, H. L. Wang, and J. Wu, “Numerical verification of the NIR features for 2D isotropic LHM,” Acta Phys. Sinica 55, 4514–4519 (2006).

2004 (1)

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

2003 (3)

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]

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

A. Ishimaru, S.-W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antennas Propag. 51, 2550–2557 (2003).
[Crossref]

2002 (1)

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

2001 (1)

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

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Alu, A.

X.-X. Liu and A. Alu, “Homogenization of quasi-isotropic metamaterials composed by dense arrays of magnetodielectric spheres,” Metamaterials 5, 56–63 (2011).
[Crossref]

X. Liu and A. Alu, “Limitations and potentials of metamaterial lenses,” J. Nanophoton. 5, 53509 (2011).
[Crossref]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 2012), p. 284.

Brock, J. B.

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]

Chan, C. T.

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[Crossref]

Chuang, I. L.

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]

Cui, T. J.

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[Crossref]

Eleftheriades, G.

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

Grbic, A.

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

Greegor, R. B.

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

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961), pp. 106–113.

Houck, A. A.

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]

Huang, D.

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[Crossref]

Ishimaru, A.

A. Ishimaru, S.-W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antennas Propag. 51, 2550–2557 (2003).
[Crossref]

Jandhyala, V.

A. Ishimaru, S.-W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antennas Propag. 51, 2550–2557 (2003).
[Crossref]

Jin, B. S.

F. Y. Meng, Q. Wu, B. S. Jin, H. L. Wang, and J. Wu, “Numerical verification of the NIR features for 2D isotropic LHM,” Acta Phys. Sinica 55, 4514–4519 (2006).

Koltenbach, B.

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

Kuga, Y.

A. Ishimaru, S.-W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antennas Propag. 51, 2550–2557 (2003).
[Crossref]

Kunz, K. S.

K. S. Kunz and R. S. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Lee, S.-W.

A. Ishimaru, S.-W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antennas Propag. 51, 2550–2557 (2003).
[Crossref]

Li, J.

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[Crossref]

Liu, R.

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[Crossref]

Liu, X.

X. Liu and A. Alu, “Limitations and potentials of metamaterial lenses,” J. Nanophoton. 5, 53509 (2011).
[Crossref]

Liu, X.-X.

X.-X. Liu and A. Alu, “Homogenization of quasi-isotropic metamaterials composed by dense arrays of magnetodielectric spheres,” Metamaterials 5, 56–63 (2011).
[Crossref]

Luebbers, R. S.

K. S. Kunz and R. S. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Meng, F. Y.

F. Y. Meng, Q. Wu, B. S. Jin, H. L. Wang, and J. Wu, “Numerical verification of the NIR features for 2D isotropic LHM,” Acta Phys. Sinica 55, 4514–4519 (2006).

Munk, B. A.

B. A. Munk, Metamaterials: Critique and Alternatives (Wiley, 2008), pp. 11–14.

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Parazzoli, C. G.

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

Schultz, S.

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Shelby, R. A.

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

Smith, D. R.

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[Crossref]

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Talalai, G. A.

S. J. Weiss, G. A. Talalai, and A. Zaghloul, “Negative refraction and array theory,” in Antenna Applications Symposium, Monticello, 2015.

Tanielian, M.

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

Valanju, A. P.

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

Valanju, P. M.

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

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Walser, R. M.

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

Wang, H. L.

F. Y. Meng, Q. Wu, B. S. Jin, H. L. Wang, and J. Wu, “Numerical verification of the NIR features for 2D isotropic LHM,” Acta Phys. Sinica 55, 4514–4519 (2006).

Weiss, S. J.

S. J. Weiss, G. A. Talalai, and A. Zaghloul, “Negative refraction and array theory,” in Antenna Applications Symposium, Monticello, 2015.

Wu, J.

F. Y. Meng, Q. Wu, B. S. Jin, H. L. Wang, and J. Wu, “Numerical verification of the NIR features for 2D isotropic LHM,” Acta Phys. Sinica 55, 4514–4519 (2006).

Wu, Q.

F. Y. Meng, Q. Wu, B. S. Jin, H. L. Wang, and J. Wu, “Numerical verification of the NIR features for 2D isotropic LHM,” Acta Phys. Sinica 55, 4514–4519 (2006).

Wu, Y.

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[Crossref]

Zaghloul, A.

S. J. Weiss, G. A. Talalai, and A. Zaghloul, “Negative refraction and array theory,” in Antenna Applications Symposium, Monticello, 2015.

Zhang, Z. Q.

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[Crossref]

Zhao, B.

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[Crossref]

Acta Phys. Sinica (1)

F. Y. Meng, Q. Wu, B. S. Jin, H. L. Wang, and J. Wu, “Numerical verification of the NIR features for 2D isotropic LHM,” Acta Phys. Sinica 55, 4514–4519 (2006).

IEEE Trans. Antennas Propag. (1)

A. Ishimaru, S.-W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antennas Propag. 51, 2550–2557 (2003).
[Crossref]

J. Nanophoton. (1)

X. Liu and A. Alu, “Limitations and potentials of metamaterial lenses,” J. Nanophoton. 5, 53509 (2011).
[Crossref]

Metamaterials (1)

X.-X. Liu and A. Alu, “Homogenization of quasi-isotropic metamaterials composed by dense arrays of magnetodielectric spheres,” Metamaterials 5, 56–63 (2011).
[Crossref]

Phys. Rev. B (1)

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[Crossref]

Phys. Rev. E (1)

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[Crossref]

Phys. Rev. Lett. (5)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[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]

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

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

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

Science (1)

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

Other (7)

B. A. Munk, Metamaterials: Critique and Alternatives (Wiley, 2008), pp. 11–14.

S. J. Weiss, G. A. Talalai, and A. Zaghloul, “Negative refraction and array theory,” in Antenna Applications Symposium, Monticello, 2015.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961), pp. 106–113.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 2012), p. 284.

Remcom, XFdtd Reference Manual (2016).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

K. S. Kunz and R. S. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

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Figures (6)

Fig. 1.
Fig. 1. Prism geometry.
Fig. 2.
Fig. 2. Array theory geometry.
Fig. 3.
Fig. 3. Geometry of unit cell and prism. (A) Cubic unit cell with edge length d containing a magneto-dielectric sphere of radius r = 0.45 d . (B) 14° prism of magneto-dielectric spheres with waveguide source for simulation. Both panels show the coordinate axes of their respective geometries.
Fig. 4.
Fig. 4. Excess phase delay of unit cell with magneto-dielectric sphere versus frequency and expected negative refraction region (demarcated by dotted lines).
Fig. 5.
Fig. 5. Instantaneous electric fields refracted through 14° prism of magneto-dielectric spheres. All panels are scaled relative to the peak amplitude in that panel. (A)  k 0 d = 0.514 with peak amplitude of 5.1 kV/m. (B)  k 0 d = 0.660 with peak amplitude of 13.2 kV/m. (C)  k 0 d = 0.734 with peak amplitude of 10.5 kV/m. (D)  k 0 d = 0.807 with peak amplitude of 6.9 kV/m.
Fig. 6.
Fig. 6. Beam transmission angle through 14° prism of magneto-dielectric spheres: Array theory versus xFDTD simulation.

Equations (18)

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

α = cot 1 R = 45 ° , 26.6 ° , 18 ° , ; R = 1 , 2 , 3 , .
ψ = ( k k 0 ) d ,
E = z ^ E ( x ) ,
E ( x ) = E 0 e j ( n 1 ) ψ ; ( n 1 ) R d < x < n R d ,
M s , pec = y ^ × E = x ^ E ( x ) ,
M s = 2 y ^ × E = x ^ 2 E ( x ) ,
g ( r , r ) = 1 4 j H 0 ( 2 ) ( k 0 | r r | ) 1 4 j 2 j π k 0 ρ e j k 0 ρ e j k 0 x cos ϕ ,
F = ϵ 0 0 N R d M s ( x ) g ( r , r ) d x .
F = ϵ 0 1 8 π j k 0 e j k 0 ρ ρ 0 N R d M s ( x ) e j k 0 x cos ϕ d x .
F = x ^ ϵ 0 E 0 2 π j k 0 e j k 0 ρ ρ × n = 1 N [ e j ( n 1 ) ψ e j ( n 1 ) R k 0 d cos ϕ 0 R d e j k 0 x cos ϕ d x ] .
F = x ^ ϵ 0 R d E 0 2 π j k 0 e j k 0 ρ ρ sinc ( R k 0 d 2 cos ϕ ) × n = 1 N [ e j ( n 1 ) ψ e j ( n 1 ) R k 0 d cos ϕ ] .
E = j k 0 ϵ 0 ρ ^ × F ,
E = z ^    P ( ϕ ) · AF ( ϕ ) · e j k 0 ρ ρ ,
P ( ϕ ) = R d E 0 2 π j k 0 sinc ( R k 0 d 2 cos ϕ ) sin ϕ ,
AF ( ϕ ) = n = 1 N e j ( n 1 ) ψ e j ( n 1 ) R k 0 d cos ϕ .
R k 0 d cos ϕ ψ = 2 m π ; m = 0 , ± 1 , ± 2 , .
ϕ = cos 1 ( ψ 2 π R k 0 d ) .
2 π ( 1 d / λ 0 cot α ) < ψ < 2 π ( 1 d / λ 0 cos α ) .

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