Abstract

Suspended-membrane 19-missing-hole microcavities in triangular lattice photonic crystals are numerically modeled by a three-dimensional finite-difference time-domain method. The resonance frequencies and the quality factors are calculated by interpolation of the discrete Fourier transformation series of the field with a Padé polynomial. The numerical results are compared with the photoluminescent spectra measured on the cavity of a nearly identical dimension. The symmetry properties of the defect modes are analyzed with the group theory, and resonance modes in the photonic-crystal cavities are identified as irreducible representations of the C6v point group. The far-field radiations of the identified modes in the free space are also calculated by use of a vector Green’s function. It is found that the numerical results agree very well with the experimental measurement in various aspects.

© 2005 Optical Society of America

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References

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  1. O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic crystal defect laser," Science 284, 1819-1821 (1999).
    [CrossRef] [PubMed]
  2. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-1-11 (2001).
    [CrossRef]
  3. H.-Y. Ryu, H.-G. Park, and Y.-H. Lee, "Two-dimensional photonic crystal semiconductor laser: computational design, fabrication, and characterization," IEEE J. Sel. Top. Quantum Electron. 8, 891-908 (2002)
    [CrossRef]
  4. P.-T. Lee, J. R. Cao, S.-J. Choi, Z. J. Wei, J. D. O'Brien, and P. D. Dapkus, "Room-temperature operation of VCSEL-pumped photonic crystal lasers," IEEE Photonics Technol. Lett. 4, 435-437 (2002).
  5. K. Srinivasan and O. Painter, "Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals," Opt. Express 11, 579-593 (2003), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  6. K. S. Yee, "Numerical solution to initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
  7. A. Taflove, Computational Electrodynamics--The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).
  8. S. Dey and R. Mittra, "Efficient computation of resonant frequencies and quality factors of cavities via a combination of the finite-difference time-domain technique and the Padé approximation," IEEE Microw. Guid. Wave Lett. 8, 415-417 (1998).
    [CrossRef]
  9. V. Heine, Group Theory in Quantum Mechanics (Dover, New York, 1993).
  10. O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, "Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides," J. Opt. A, Pure Appl. Opt. 3, S161-70 (2001)
    [CrossRef]
  11. M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, San Francisco, Calif., 1964).
  12. A. V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  13. K. Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2001).
    [CrossRef]
  14. J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
    [CrossRef]
  15. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

2005 (1)

J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
[CrossRef]

2003 (1)

2002 (2)

H.-Y. Ryu, H.-G. Park, and Y.-H. Lee, "Two-dimensional photonic crystal semiconductor laser: computational design, fabrication, and characterization," IEEE J. Sel. Top. Quantum Electron. 8, 891-908 (2002)
[CrossRef]

P.-T. Lee, J. R. Cao, S.-J. Choi, Z. J. Wei, J. D. O'Brien, and P. D. Dapkus, "Room-temperature operation of VCSEL-pumped photonic crystal lasers," IEEE Photonics Technol. Lett. 4, 435-437 (2002).

2001 (2)

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-1-11 (2001).
[CrossRef]

O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, "Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides," J. Opt. A, Pure Appl. Opt. 3, S161-70 (2001)
[CrossRef]

1999 (1)

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic crystal defect laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

1998 (1)

S. Dey and R. Mittra, "Efficient computation of resonant frequencies and quality factors of cavities via a combination of the finite-difference time-domain technique and the Padé approximation," IEEE Microw. Guid. Wave Lett. 8, 415-417 (1998).
[CrossRef]

1966 (1)

K. S. Yee, "Numerical solution to initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

Bagheri, M.

J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
[CrossRef]

Cao, J. R.

J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
[CrossRef]

P.-T. Lee, J. R. Cao, S.-J. Choi, Z. J. Wei, J. D. O'Brien, and P. D. Dapkus, "Room-temperature operation of VCSEL-pumped photonic crystal lasers," IEEE Photonics Technol. Lett. 4, 435-437 (2002).

Choi, S.-J.

J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
[CrossRef]

P.-T. Lee, J. R. Cao, S.-J. Choi, Z. J. Wei, J. D. O'Brien, and P. D. Dapkus, "Room-temperature operation of VCSEL-pumped photonic crystal lasers," IEEE Photonics Technol. Lett. 4, 435-437 (2002).

Dapkus, P. D.

J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
[CrossRef]

P.-T. Lee, J. R. Cao, S.-J. Choi, Z. J. Wei, J. D. O'Brien, and P. D. Dapkus, "Room-temperature operation of VCSEL-pumped photonic crystal lasers," IEEE Photonics Technol. Lett. 4, 435-437 (2002).

O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, "Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides," J. Opt. A, Pure Appl. Opt. 3, S161-70 (2001)
[CrossRef]

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic crystal defect laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Dey, S.

S. Dey and R. Mittra, "Efficient computation of resonant frequencies and quality factors of cavities via a combination of the finite-difference time-domain technique and the Padé approximation," IEEE Microw. Guid. Wave Lett. 8, 415-417 (1998).
[CrossRef]

Heine, V.

V. Heine, Group Theory in Quantum Mechanics (Dover, New York, 1993).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

Kim, I.

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic crystal defect laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Kuang, W.

J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
[CrossRef]

Lee, P.-T.

P.-T. Lee, J. R. Cao, S.-J. Choi, Z. J. Wei, J. D. O'Brien, and P. D. Dapkus, "Room-temperature operation of VCSEL-pumped photonic crystal lasers," IEEE Photonics Technol. Lett. 4, 435-437 (2002).

Lee, R. K.

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic crystal defect laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Lee, Y.-H.

H.-Y. Ryu, H.-G. Park, and Y.-H. Lee, "Two-dimensional photonic crystal semiconductor laser: computational design, fabrication, and characterization," IEEE J. Sel. Top. Quantum Electron. 8, 891-908 (2002)
[CrossRef]

Loncar, M.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-1-11 (2001).
[CrossRef]

Mabuchi, H.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-1-11 (2001).
[CrossRef]

Mittra, R.

S. Dey and R. Mittra, "Efficient computation of resonant frequencies and quality factors of cavities via a combination of the finite-difference time-domain technique and the Padé approximation," IEEE Microw. Guid. Wave Lett. 8, 415-417 (1998).
[CrossRef]

O'Brien, J. D.

J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
[CrossRef]

P.-T. Lee, J. R. Cao, S.-J. Choi, Z. J. Wei, J. D. O'Brien, and P. D. Dapkus, "Room-temperature operation of VCSEL-pumped photonic crystal lasers," IEEE Photonics Technol. Lett. 4, 435-437 (2002).

O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, "Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides," J. Opt. A, Pure Appl. Opt. 3, S161-70 (2001)
[CrossRef]

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic crystal defect laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Painter, O.

K. Srinivasan and O. Painter, "Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals," Opt. Express 11, 579-593 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, "Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides," J. Opt. A, Pure Appl. Opt. 3, S161-70 (2001)
[CrossRef]

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic crystal defect laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Park, H.-G.

H.-Y. Ryu, H.-G. Park, and Y.-H. Lee, "Two-dimensional photonic crystal semiconductor laser: computational design, fabrication, and characterization," IEEE J. Sel. Top. Quantum Electron. 8, 891-908 (2002)
[CrossRef]

Ryu, H.-Y.

H.-Y. Ryu, H.-G. Park, and Y.-H. Lee, "Two-dimensional photonic crystal semiconductor laser: computational design, fabrication, and characterization," IEEE J. Sel. Top. Quantum Electron. 8, 891-908 (2002)
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2001).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Scherer, A.

O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, "Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides," J. Opt. A, Pure Appl. Opt. 3, S161-70 (2001)
[CrossRef]

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-1-11 (2001).
[CrossRef]

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic crystal defect laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Srinivasan, K.

K. Srinivasan and O. Painter, "Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals," Opt. Express 11, 579-593 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, "Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides," J. Opt. A, Pure Appl. Opt. 3, S161-70 (2001)
[CrossRef]

Taflove, A.

A. Taflove, Computational Electrodynamics--The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).

Tinkham, M.

M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, San Francisco, Calif., 1964).

Vuckovic, J.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-1-11 (2001).
[CrossRef]

Wei, Z. J.

J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
[CrossRef]

P.-T. Lee, J. R. Cao, S.-J. Choi, Z. J. Wei, J. D. O'Brien, and P. D. Dapkus, "Room-temperature operation of VCSEL-pumped photonic crystal lasers," IEEE Photonics Technol. Lett. 4, 435-437 (2002).

Yariv, A.

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic crystal defect laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Yee, K. S.

K. S. Yee, "Numerical solution to initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

Yu, H.

J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

H.-Y. Ryu, H.-G. Park, and Y.-H. Lee, "Two-dimensional photonic crystal semiconductor laser: computational design, fabrication, and characterization," IEEE J. Sel. Top. Quantum Electron. 8, 891-908 (2002)
[CrossRef]

IEEE Microw. Guid. Wave Lett. (1)

S. Dey and R. Mittra, "Efficient computation of resonant frequencies and quality factors of cavities via a combination of the finite-difference time-domain technique and the Padé approximation," IEEE Microw. Guid. Wave Lett. 8, 415-417 (1998).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

P.-T. Lee, J. R. Cao, S.-J. Choi, Z. J. Wei, J. D. O'Brien, and P. D. Dapkus, "Room-temperature operation of VCSEL-pumped photonic crystal lasers," IEEE Photonics Technol. Lett. 4, 435-437 (2002).

J. R. Cao, W. Kuang, Z. J. Wei, S.-J. Choi, H. Yu, M. Bagheri, J. D. O'Brien, and P. D. Dapkus, "Sapphire-bonded photonic crystal microcavity lasers and their far-field radiation patterns," IEEE Photonics Technol. Lett. 17, 4-6 (2005).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, "Numerical solution to initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

J. Opt. A, Pure Appl. Opt. (1)

O. Painter, K. Srinivasan, J. D. O'Brien, A. Scherer, and P. D. Dapkus, "Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides," J. Opt. A, Pure Appl. Opt. 3, S161-70 (2001)
[CrossRef]

Opt. Express (1)

Phys. Rev. E (1)

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-1-11 (2001).
[CrossRef]

Science (1)

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic crystal defect laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Other (6)

A. Taflove, Computational Electrodynamics--The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).

V. Heine, Group Theory in Quantum Mechanics (Dover, New York, 1993).

M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, San Francisco, Calif., 1964).

A. V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

K. Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2001).
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

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Figures (8)

Fig. 1
Fig. 1

Calculated resonance-mode frequencies and their quality factors for a suspended membrane 19-missing-hole photonic-crystal defect cavity (bars) with a membrane thickness of d a = 0.45 and hole radii of r a = 0.33 except the inner holes around the defect, which have slightly smaller r a = 0.3 . It is compared with the measured photoluminescent spectrum of a cavity with similar dimensions (curve). The left and bottom axes are for the calculation, whereas the right and top axes are for the measurement curve. The difference of frequency is within 0.0035 in normalized scale, or 16 nm in free-space wavelength. Resonance modes with a quality factor over 1000 in the material gain region are also labeled with their corresponding irreducible representation in the C 6 v point symmetry group. The inset shows the top view of the cavity.

Fig. 2
Fig. 2

Magnitude of the H z component at the midplane for the 19-missing-hole suspended-membrane photonic-crystal defect-cavity mode considered (top), with a two-dimensional dielectric contour overlaid. Its projections onto the irreducible representations of the C 6 v group are labeled by A 1 , A 2 , B 1 , B 2 , E 1 , and E 2 .

Fig. 3
Fig. 3

Electrical-field components E x (top left) and E y (top right) at the midplane for the 19-missing-hole suspended-membrane photonic-crystal defect-cavity mode considered (top), with a two-dimensional dielectric contour overlaid. Their projections onto the irreducible representations of the C 6 v group are labeled by A 1 , A 2 , B 1 , B 2 , E 1 , and E 2 .

Fig. 4
Fig. 4

Magnitude of the H z component at the midplane for the 19-missing-hole suspended-membrane photonic-crystal defect-cavity mode considered (top), with a two-dimensional dielectric contour overlaid. Its projections onto the irreducible representations of the C 6 v group are labeled by A 1 , A 2 , B 1 , B 2 , E 1 , and E 2 .

Fig. 5
Fig. 5

Magnitude of the magnetic field H z at the midplane (top) of the 19-missing-hole suspended-membrane photonic-crystal defect cavity for the same mode as in Fig. 5 but excited with different initial condition. Its projections onto the irreducible representations of the C 6 v group are labeled by A 1 , A 2 , B 1 , B 2 , E 1 , and E 2 .

Fig. 6
Fig. 6

Decomposition of the magnetic field H z as shown in Figs. 5, 6, a two-dimensional E 1 mode of the C 6 v group, in one-dimensional representation { B 1 , B 2 } of the C 2 v group.

Fig. 7
Fig. 7

Calculated E x and E y far-field components of the modes corresponding to the B 1 and B 2 representations shown in a polar plot. The dashed circle indicates a numerical aperture of 0.55, identical to the lens system employed for experimental light collection.

Fig. 8
Fig. 8

(a) Measured optical spectrum of the cavity. (b) Polarizations of the two resonances split in frequency during to a rectangular perturbation of a doubly degenerate mode. The insets show the calculated far-field intensities of the respective modes.

Tables (2)

Tables Icon

Table 1 Character Table for the C 6 v Point-Symmetry Group

Tables Icon

Table 2 Character Table for the C 2 v Point-Symmetry Group

Equations (13)

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

g = 2 π n ¯ λ 0 1 Q Γ ,
Δ f f = M N ( a λ 0 ) ,
BW f = π ξ M N ( a λ 0 ) ,
P ( ω k ) = Q I ( ω k ) D J ( ω k ) = i = 0 I α i ( ω k ) i j = 0 J β j ( ω k ) j .
C 6 ν = { E , C 2 , 2 C 3 , 2 C 6 , 3 σ d , 3 σ v } .
P ̂ ( j ) = l j h R χ ( j ) ( R ) P ̂ R ,
E E = E ,
H H = H .
E ( r ) = i ω μ v d 3 r J ( r ) G ̿ ( r , r ) ,
G ̿ ( r , r ) = ( I ̿ + 1 k 0 2 ) exp ( i k 0 r r ) r r .
E ( r ) = s d 2 r × G ̿ ( r , r ) n ̂ × 2 E ( r ) .
( I ̿ + 1 k 0 ) ( I ̿ r ̂ r ̂ ) ,
E ( r ) = i k 0 exp ( i k 0 r ) 4 π r r ̂ 0 × s 2 z ̂ × E s ( r ) exp ( i k 0 r ) d 2 r ,

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