Abstract

A detailed study of evanescent waves in a two-dimensional photonic crystal known to be a left-handed medium for propagating modes is presented. We have not found universal amplification of the evanescent modes, connected with negative μ and ϵ of the bulk photonic crystal. An analytical expression for the intensity distribution near the far-field focus is obtained using diffraction theory. This distribution contains some novel features and establishes a new diffraction limit for flat lenses. The distribution is generalized for multiple sources located at different points and is in very good agreement with computer simulations.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. G. Veselago, "Properties of materials having simultaneously negative values of the dielectric (epsilon) and magnetic (µ) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).
  2. J. B. Pendry, "Negative refraction make a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  3. R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  4. N. Garcia and M. Nieto-Vesperinas, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 88, p. 207403 (2002).
    [CrossRef] [PubMed]
  5. A. L. Pokrovsky and A. L. Efros, "Diffraction theory and focusing of light by a slab of left-handed material," cond-mat/0202078, Physica B 338, 333-337 (2003).
    [CrossRef]
  6. F. D. M. Haldane, "Electromagnetic surface modes at interfaces with negative refractive index make a 'not-quite-perfect' lens," cond-mat/0206420.
  7. V. A. Podolskiy and E. E. Narimanov, "Near-sighted superlens," Opt. Lett. 30, 75-77 (2005).
    [CrossRef] [PubMed]
  8. A. L. Efros and A. L. Pokrovsky, "Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability," Solid State Commun. 129, 643-647 (2004).
    [CrossRef]
  9. A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
    [CrossRef]
  10. D. L. Landau and E. M. Lifshitz, The Classical Theory of Fields (Butterworth Heinemann, 2000), p. 156.
  11. M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
    [CrossRef]
  12. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104(R) (2002).
    [CrossRef]
  13. C. Luo, S. G. Johnson and J. D. Joannopoulos and J. B. Pendry, "Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
    [CrossRef]
  14. D. L. Landau and E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth Heinemann, 1984).
  15. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
    [CrossRef] [PubMed]
  16. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, 1984).
  17. See web site http://physics.utah.edu/~efros.
  18. R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 277, 61-64 (2000).
    [CrossRef]
  19. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
    [CrossRef]
  20. F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 44, 15112-15120 (1999).
    [CrossRef]
  21. A. Alu and N. Engheta, "Pairing and epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
    [CrossRef]

2005

2004

A. L. Efros and A. L. Pokrovsky, "Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability," Solid State Commun. 129, 643-647 (2004).
[CrossRef]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

2003

A. L. Pokrovsky and A. L. Efros, "Diffraction theory and focusing of light by a slab of left-handed material," cond-mat/0202078, Physica B 338, 333-337 (2003).
[CrossRef]

C. Luo, S. G. Johnson and J. D. Joannopoulos and J. B. Pendry, "Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[CrossRef]

A. Alu and N. Engheta, "Pairing and epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

2002

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

N. Garcia and M. Nieto-Vesperinas, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 88, p. 207403 (2002).
[CrossRef] [PubMed]

2001

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
[CrossRef]

2000

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

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 277, 61-64 (2000).
[CrossRef]

1999

F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 44, 15112-15120 (1999).
[CrossRef]

1991

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

1967

V. G. Veselago, "Properties of materials having simultaneously negative values of the dielectric (epsilon) and magnetic (µ) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).

Agranovich, V. M.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, 1984).

Alu, A.

A. Alu and N. Engheta, "Pairing and epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Brommer, K. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Derov, J. S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Efros, A. L.

A. L. Efros and A. L. Pokrovsky, "Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability," Solid State Commun. 129, 643-647 (2004).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, "Diffraction theory and focusing of light by a slab of left-handed material," cond-mat/0202078, Physica B 338, 333-337 (2003).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

Engheta, N.

A. Alu and N. Engheta, "Pairing and epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 88, p. 207403 (2002).
[CrossRef] [PubMed]

Ginzburg, V. L.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, 1984).

Haldane, F. D.

F. D. M. Haldane, "Electromagnetic surface modes at interfaces with negative refractive index make a 'not-quite-perfect' lens," cond-mat/0206420.

Halevi, P.

F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 44, 15112-15120 (1999).
[CrossRef]

Heyman, E.

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Joannopoulos, J. D.

C. Luo, S. G. Johnson and J. D. Joannopoulos and J. B. Pendry, "Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Johnson, S. G.

C. Luo, S. G. Johnson and J. D. Joannopoulos and J. B. Pendry, "Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

Landau, D. L.

D. L. Landau and E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth Heinemann, 1984).

D. L. Landau and E. M. Lifshitz, The Classical Theory of Fields (Butterworth Heinemann, 2000), p. 156.

Lifshitz, E. M.

D. L. Landau and E. M. Lifshitz, The Classical Theory of Fields (Butterworth Heinemann, 2000), p. 156.

D. L. Landau and E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth Heinemann, 1984).

Lu, W. T.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Luo, C.

C. Luo, S. G. Johnson and J. D. Joannopoulos and J. B. Pendry, "Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

Meade, R. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Narimanov, E. E.

Nieto-Vesperinas, M.

N. Garcia and M. Nieto-Vesperinas, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 88, p. 207403 (2002).
[CrossRef] [PubMed]

Notomi, M.

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

Parimi, P. V.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

C. Luo, S. G. Johnson and J. D. Joannopoulos and J. B. Pendry, "Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

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

Pitaevskii, L. P.

D. L. Landau and E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth Heinemann, 1984).

Podolskiy, V. A.

Pokrovsky, A. L.

A. L. Efros and A. L. Pokrovsky, "Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability," Solid State Commun. 129, 643-647 (2004).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, "Diffraction theory and focusing of light by a slab of left-handed material," cond-mat/0202078, Physica B 338, 333-337 (2003).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

Ramos-Mendieta, F.

F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 44, 15112-15120 (1999).
[CrossRef]

Rappe, A. M.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Ruppin, R.

R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 277, 61-64 (2000).
[CrossRef]

Sokoloff, J.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Sridhar, S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, "Properties of materials having simultaneously negative values of the dielectric (epsilon) and magnetic (µ) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).

Vodo, P.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
[CrossRef]

IEEE Trans. Antennas Propag.

A. Alu and N. Engheta, "Pairing and epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Opt. Lett.

Phys. Lett. A

R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 277, 61-64 (2000).
[CrossRef]

Phys. Rev. B

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 44, 15112-15120 (1999).
[CrossRef]

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

C. Luo, S. G. Johnson and J. D. Joannopoulos and J. B. Pendry, "Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[CrossRef]

Phys. Rev. E

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Phys. Rev. Lett.

N. Garcia and M. Nieto-Vesperinas, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 88, p. 207403 (2002).
[CrossRef] [PubMed]

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

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Physica B

A. L. Pokrovsky and A. L. Efros, "Diffraction theory and focusing of light by a slab of left-handed material," cond-mat/0202078, Physica B 338, 333-337 (2003).
[CrossRef]

Solid State Commun.

A. L. Efros and A. L. Pokrovsky, "Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability," Solid State Commun. 129, 643-647 (2004).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

Sov. Phys. Solid State

V. G. Veselago, "Properties of materials having simultaneously negative values of the dielectric (epsilon) and magnetic (µ) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).

Other

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, 1984).

See web site http://physics.utah.edu/~efros.

D. L. Landau and E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth Heinemann, 1984).

D. L. Landau and E. M. Lifshitz, The Classical Theory of Fields (Butterworth Heinemann, 2000), p. 156.

F. D. M. Haldane, "Electromagnetic surface modes at interfaces with negative refractive index make a 'not-quite-perfect' lens," cond-mat/0206420.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Refraction of light outgoing from a point source at x = a and passing through a slab of a LHM at 0 < x < d described by the anomalous Snell’s law. A second focus (the image) is formed at x = 2 d a . The arrows show the direction of the wave vector.

Fig. 2
Fig. 2

Distribution of dimensionless magnetic energy near the foci of the Veselago lens as a function of ζ and η, as given by Eq. (6).

Fig. 3
Fig. 3

Five lowest bands of the photonic spectrum of a two-dimensional PC with a period d 0 . The unit cell of the PC is shown in the inset. The PC consists of a square lattice of circular cylindrical air holes in a dielectric matrix with ϵ m = 12 , μ m = 1 . The radii of the holes is R = 0.35 d 0 .

Fig. 4
Fig. 4

Contour plot of a magnetic field in a plane wave incident on a PC slab as obtained by computer simulation. Periodic boundary conditions are used in the vertical direction. The width of the slab d = 20 d 0 2 λ . The PC slab is surrounded by a homogeneous medium with parameters given by Eq. (9). The arrows show the direction of the wave vector.

Fig. 5
Fig. 5

Magnetic field of the EW in (a) a HLHM and (b), (c) a PC slab surrounded by a regular homogeneous medium at the following parameters for the EW: k y = ( 5 2 ) k 0 , κ = 0.5 k 0 , k 0 = ω n c , n = ϵ μ = 0.3 . The thickness of the PC slab is (b) 20 d 0 2 λ and (c) 10 d 0 λ . Here λ = 2 π k 0 is the wavelength. The solid curves represent the AH surfaces; the dashed curves represent the BH surfaces. The period of oscillation is d 0 . The insets (b) and (c) demonstrate the difference between AH and BH surfaces.

Fig. 6
Fig. 6

A magnetic field of EW propagating in a direction that is not parallel to the surface of the PC slab exhibits negative refraction at the interface. The arrows show the directions of the propagating wave vector k. The thickness of the PC slab is 20 d 0 2 λ . The parameters of the EW are the same as in Fig. 5.

Fig. 7
Fig. 7

Magnetic field of the EW with k y = ( 5 2 ) k 0 , k = 0.5 k 0 , k 0 = ω n c is shown with BH surface for the PC slab. (a) The PC slab of the width 10 d 0 is surrounded by HLHM. (b) The same PC slab is surrounded by regular media. The dashed curve is a plot of the function exp ( κ x ) .

Fig. 8
Fig. 8

Distribution of magnetic energy near the focus for both the thin lens and the thick lens with AH surfaces in the lateral direction. The dashed curve is the distribution of magnetic energy in the lateral direction for a thin lens; the solid curve is the same for a thick lens. Analytical results as obtained from Eq. (13) for a thick lens are shown by circles. The dotted curve shows the distribution of magnetic energy of the point source. The inset depicts the geometric meaning of k m .

Fig. 9
Fig. 9

Distribution of magnetic energy near the focus for the thick lens with AH surfaces in lateral and perpendicular directions. The solid curve is the distribution of magnetic energy in the perpendicular direction; the dashed curve is the same for the lateral direction. Circles show the analytical result for perpendicular direction, triangles show the same for lateral direction. The dotted curve shows the distribution of magnetic energy of the point source. Analytical results are obtained from Eq. (13).

Fig. 10
Fig. 10

Simulation results for the distribution of magnetic energy near the focus in the lateral direction for two lenses with AH surfaces with different heights. The solid curve is the distribution for the lens with h = 80 d 0 , and the dashed curve is for the lens with h = 120 d 0 . The dotted curve is the plot of J 0 2 ( 2 π y λ ) . The analytical results are almost indistinguishable from the simulation results.

Fig. 11
Fig. 11

Distribution of magnetic energy along the lateral direction near the focus of three point sources with interval 0.3 λ , with the point sources located at y = 0 , y = 0.3 λ , and y = 0.3 λ (solid curve) and the same for the interval 0.5 λ with the three point sources located at y = 0 , y = 0.5 λ and y = 0.5 λ (dashed curve). The analytical results are almost indistinguishable from the simulation results.

Fig. 12
Fig. 12

Distribution of magnetic energy of three point sources located along the y axis with interval 0.3 λ . The distance between the sources and the PC slab is half of the slab thickness. We use the low-reflecting boundary condition (LRBC) surrounding the system, which means that the transmitted and reflected waves are almost perfectly absorbed by the surface with LRBC.

Equations (14)

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

H ( x , y ) = b r l H ( y ) G L ( k 0 [ x 2 + ( y y ) 2 ] 1 2 ) cos ( θ ) d y .
H 0 ( 1 ) ( k 0 ρ ) ( 2 π k 0 ρ ) 1 2 exp i ( k 0 ρ π 4 ) .
H ( x , y ) = h 0 i a π exp i k 0 { ( a 2 + y 2 ) 1 2 [ x 2 + ( y y ) 2 ] 1 2 } ( a 2 + y 2 ) 3 4 [ x 2 + ( y y ) 2 ] 1 4 d y .
H ( x , y ) = h 0 i π 1 1 exp i ( ζ t η 1 1 t 2 ) 1 t 2 d t .
H ( x , y ) = b l r H ( y ) G R { k 0 [ x 2 + ( y y ) 2 ] 1 2 } cos ( θ ) d y ,
H ( x , y ) = h 0 i π 1 1 exp i ( ζ t + η 1 t 2 ) 1 t 2 d t ,
H ( 2 d a , ζ ) = h 0 2 i π 0 1 c o s ( ζ t ) 1 t 2 d t = i h 0 J 0 ( ζ ) ,
H ( η , 0 ) = h 0 2 i π 0 1 [ c o s ( η q ) + i sin ( η q ) ] 1 q 2 d q = h 0 [ H 0 ( η ) + i J 0 ( η ) ] ,
ϵ = 1.125 , μ = 0.08 .
H p = i ( h 0 π ) k 0 k 0 exp i ( k y + x k 0 2 k 2 ω t ) k 0 2 k 2 d k
H ev = ( h 0 π ) k > k 0 exp ( i k y x k 2 k 0 2 i ω t ) k 2 k 0 2 d k
H f = i ( h 0 π ) k 0 k 0 exp i ( k y + x k 0 2 k 2 ω t ) k 0 2 k 2 d k .
H p f = i ( h 0 π ) k m k m exp i ( k y + x k 0 2 k 2 ω t ) k 0 2 k 2 d k .
H ( x , y ) = k 0 k 0 F ( k ) exp i ( k y + x k 0 2 k 2 ω t ) dk ,

Metrics