Abstract

The infinite square lattice of coupled microring optical resonators is studied for what we belive to be the first time. Using the standard matrix formalism and the classical Bloch’s theorem for propagation in periodic optical media, the dispersion equation and the amplitudes of propagating Bloch modes are derived analytically. It is found that the dispersion equation ω(kx,ky) of this 2D microring array is expressed as the sum of two independent dispersion equations of the 1D microring array with wavenumbers kx and ky. As a result, the width of the passband is twice that of a microring coupled-resonator optical waveguide and there are no stop bands for an interresonator power coupling ratio greater than 12. The evanescent modes that are important to the analysis of lattices with interrupted periodicity are also studied. The reported analysis is the prerequisite to the future study of superresonators consisting of large finite microring arrays.

© 2008 Optical Society of America

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References

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  1. V. Van, “Circuit-based method for synthesizing serially coupled microring filters,” J. Lightwave Technol. 24, 2912-2929 (2006).
    [CrossRef]
  2. S. Darmawan, Y. M. Landobasa, and M.-K. Chin, “Pole-zero dynamics of high-order ring resonator filters,” J. Lightwave Technol. 25, 1568-1575 (2007).
    [CrossRef]
  3. C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photonics Technol. Lett. 10, 994-996 (1998).
    [CrossRef]
  4. R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photonics Technol. Lett. 12, 1447-1449 (1995).
    [CrossRef]
  5. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
    [CrossRef]
  6. B. Little, S. Chu, J. Hryniewicz, and P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344-346 (2000).
    [CrossRef]
  7. A. Melloni, “Synthesis of a parallel-coupled ring-resonator filter,” Opt. Lett. 26, 917-919 (2001).
    [CrossRef]
  8. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711-713 (1999).
    [CrossRef]
  9. J. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21, 1665-1673 (2004).
    [CrossRef]
  10. Y. Chen and S. Blair, “Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,” Opt. Express 12, 3353-3366 (2004).
    [CrossRef] [PubMed]
  11. J. Heebner, R. Boyd, and Q. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
    [CrossRef]
  12. D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6 nm free spectral range,” Opt. Lett. 22, 1244-1246 (1997).
    [CrossRef] [PubMed]
  13. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
    [CrossRef]
  14. I. Chremmos and N. Uzunoglu, “Properties of regular polygons of coupled microring resonators,” Appl. Opt. 46, 7730-7738 (2007).
    [CrossRef] [PubMed]
  15. J. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photonics Technol. Lett. 16, 1331-1333 (2004).
    [CrossRef]
  16. V. Van, “Synthesis of elliptic optical filters using mutually coupled microring resonators,” J. Lightwave Technol. 25, 584-590 (2007).
    [CrossRef]
  17. I. D. Chremmos and N. K. Uzunoglu, “Propagation in a directional coupler of parallel microring coupled-resonator optical waveguides,” Opt. Commun. 281, 3381-3389 (2008).
    [CrossRef]
  18. Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-D microresonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410-1418 (2005).
    [CrossRef]
  19. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
    [CrossRef]
  20. S. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. B 23, 1565-1573 (2006).
    [CrossRef]
  21. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90-103 (2004).
    [CrossRef] [PubMed]
  22. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

2008 (1)

I. D. Chremmos and N. K. Uzunoglu, “Propagation in a directional coupler of parallel microring coupled-resonator optical waveguides,” Opt. Commun. 281, 3381-3389 (2008).
[CrossRef]

2007 (3)

2006 (2)

2005 (1)

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-D microresonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

2004 (4)

2002 (1)

J. Heebner, R. Boyd, and Q. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

1998 (2)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photonics Technol. Lett. 10, 994-996 (1998).
[CrossRef]

1997 (3)

D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6 nm free spectral range,” Opt. Lett. 22, 1244-1246 (1997).
[CrossRef] [PubMed]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

1995 (1)

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photonics Technol. Lett. 12, 1447-1449 (1995).
[CrossRef]

Absil, P.

Blair, S.

Boriskina, S.

Boyd, R.

J. Heebner, R. Boyd, and Q. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Chen, Y.

Chin, M.-K.

S. Darmawan, Y. M. Landobasa, and M.-K. Chin, “Pole-zero dynamics of high-order ring resonator filters,” J. Lightwave Technol. 25, 1568-1575 (2007).
[CrossRef]

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-D microresonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

Chremmos, I.

Chremmos, I. D.

I. D. Chremmos and N. K. Uzunoglu, “Propagation in a directional coupler of parallel microring coupled-resonator optical waveguides,” Opt. Commun. 281, 3381-3389 (2008).
[CrossRef]

Chu, S.

Chu, S. T.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Darmawan, S.

S. Darmawan, Y. M. Landobasa, and M.-K. Chin, “Pole-zero dynamics of high-order ring resonator filters,” J. Lightwave Technol. 25, 1568-1575 (2007).
[CrossRef]

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-D microresonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Hagness, S. C.

Haus, H. A.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Heebner, J.

J. Heebner, R. Boyd, and Q. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Ho, S. T.

Hryniewicz, J.

Huang, Y.

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Landobasa, Y. M.

S. Darmawan, Y. M. Landobasa, and M.-K. Chin, “Pole-zero dynamics of high-order ring resonator filters,” J. Lightwave Technol. 25, 1568-1575 (2007).
[CrossRef]

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-D microresonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

Lee, R. K.

Lenz, G.

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photonics Technol. Lett. 10, 994-996 (1998).
[CrossRef]

Little, B.

Little, B. E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Madsen, C. K.

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photonics Technol. Lett. 10, 994-996 (1998).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Melloni, A.

Mookherjea, S.

Orta, R.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photonics Technol. Lett. 12, 1447-1449 (1995).
[CrossRef]

Paloczi, G. T.

Park, Q.

J. Heebner, R. Boyd, and Q. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Poon, J.

J. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21, 1665-1673 (2004).
[CrossRef]

J. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photonics Technol. Lett. 16, 1331-1333 (2004).
[CrossRef]

Poon, J. K. S.

Rafizadeh, D.

Savi, P.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photonics Technol. Lett. 12, 1447-1449 (1995).
[CrossRef]

Scherer, A.

Scheuer, J.

Stair, K. A.

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Taflove, A.

Tascone, R.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photonics Technol. Lett. 12, 1447-1449 (1995).
[CrossRef]

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Tiberio, R. C.

Trinchero, D.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photonics Technol. Lett. 12, 1447-1449 (1995).
[CrossRef]

Uzunoglu, N.

Uzunoglu, N. K.

I. D. Chremmos and N. K. Uzunoglu, “Propagation in a directional coupler of parallel microring coupled-resonator optical waveguides,” Opt. Commun. 281, 3381-3389 (2008).
[CrossRef]

Van, V.

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Xu, Y.

Yariv, A.

Zhang, J. P.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-D microresonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

IEEE Photonics Technol. Lett. (4)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549-551 (1998).
[CrossRef]

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photonics Technol. Lett. 10, 994-996 (1998).
[CrossRef]

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring resonator filters for optical systems,” IEEE Photonics Technol. Lett. 12, 1447-1449 (1995).
[CrossRef]

J. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photonics Technol. Lett. 16, 1331-1333 (2004).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. B (2)

Nature (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

Opt. Commun. (1)

I. D. Chremmos and N. K. Uzunoglu, “Propagation in a directional coupler of parallel microring coupled-resonator optical waveguides,” Opt. Commun. 281, 3381-3389 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. E (1)

J. Heebner, R. Boyd, and Q. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Other (1)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

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Figures (8)

Fig. 1
Fig. 1

Infinite square lattice of coupled microring resonators with lattice constant Λ. The shaded rings form an irreducible supercell for the application of Bloch’s theorem.

Fig. 2
Fig. 2

Map of passbands of three successive microring resonances with azimuthal orders ν 1 , ν, ν + 1 versus the interresonator power coupling ratio κ 2 . The overlap of passbands for κ 2 > 0.5 results in the absence of stop bands.

Fig. 3
Fig. 3

(a) Dispersion diagram corresponding to the (−,−) signs of Eq. (12) for wave vectors in the first Brillouin zone and κ = 0.6 ( θ c = 0.205 π ) . (b) Corresponding isofrequency lines for 20 equally spaced values of δ ϕ in the interval [ 4 θ c , 4 θ c ] . The dotted lines correspond to the negative values of δ ϕ .

Fig. 4
Fig. 4

Dispersion diagrams for wave vectors along the Γ XM triangle of the Brillouin zone for (a) κ = 0.6 ( B c = 1.64 π ) and (b) κ = 0.8 ( B c = 2.36 π ) . In (b), the dashed curves are the bands of the next lower ( δ ϕ = 2 π ) and the next higher ( δ ϕ = 2 π ) microring resonance, which overlap with the band of resonance δ ϕ = 0 .

Fig. 5
Fig. 5

Distribution of mode power inside a 2 Λ × 2 Λ supercell. We have defined P a = a 00 2 , P b = b 00 2 , P c = c 00 2 , and P d = d 00 2 .

Fig. 6
Fig. 6

(a) Mode power b 00 2 in decibels versus k y Λ from Eq. (15) for different values of the coupling coefficient. By reflecting the curve with respect to k y = 0 , we obtain the diagram for d 00 2 versus k x Λ , which is not plotted. (b) Mode power c 00 2 versus k x Λ , k y Λ for κ = 0.6 . Power a 00 2 is normalized to 0 dB .

Fig. 7
Fig. 7

Mapping of regions of complex wavenumbers k x = k x r + j k x i , k y = k y r + j k y i on the θ ( k x ) θ ( k y ) plane. The inner gray square is the region of real wave vectors k = ( k x r , k y r ) . Condition of propagation (14) is mapped on the line with slope 1 .

Fig. 8
Fig. 8

Complete isofrequency diagram for real and complex wavenumbers for 40 equally spaced values of δ ϕ in the interval [ 2 π , 2 π ] and κ = 0.6 .

Equations (25)

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

( a 00 c 10 ) e j ϕ 8 = ( t j κ j κ t ) ( d 00 b 10 ) e j ϕ 8 ,
( b 10 c 10 ) = C ( d 00 a 00 ) , where C = 1 j κ ( t e j ϕ 4 e j ϕ 4 t ) ,
( a l x + 2 p , l y + 2 q b l x + 2 p , l y + 2 q c l x + 2 p , l y + 2 q d l x + 2 p , l y + 2 q ) = ( a l x , l y b l x , l y c l x , l y d l x , l y ) e j ( k x p + j k y q ) 2 Λ ,
( b 10 c 10 ) = C ( d 00 a 00 ) , ( 4.1 ) ( c 01 d 01 ) = C ( a 00 b 00 ) , ( 4.3 ) ( d 11 a 11 ) = C ( b 01 c 01 ) , ( 4.5 ) ( a 11 b 11 ) = C ( c 10 d 10 ) , ( 4.7 ) ( d 10 a 10 ) = C ( b 00 c 00 ) e j 2 k x Λ , ( 4.2 ) ( a 01 b 01 ) = C ( c 00 d 00 ) e j 2 k y Λ , ( 4.4 ) ( b 11 c 11 ) = C ( d 01 a 01 ) e j 2 k x Λ , ( 4.6 ) ( c 11 d 11 ) = C ( a 10 b 10 ) e j 2 k y Λ . ( 4.8 )
T x = 1 j κ ( 0 e j ϕ 4 t 0 e j ϕ 4 0 0 t t 0 0 e j ϕ 4 0 t e j ϕ 4 0 ) , T y = ( 0 C C 0 ) .
w 10 = T x P x w 00 , ( 6.1 ) w 11 = T y P y w 10 , ( 6.3 ) w 01 = T y P y 1 e j 2 k y Λ w 00 , ( 6.2 ) w 11 = T x P x 1 e j 2 k x Λ w 01 . ( 6.4 )
P x = ( 1 0 0 0 0 e j 2 k x Λ 0 0 0 0 e j 2 k x Λ 0 0 0 0 1 ) ,
P y = ( e j 2 k y Λ 0 0 0 0 e j 2 k y Λ 0 0 0 0 1 0 0 0 0 1 ) .
M ( k x , k y , ϕ ) w 00 = 0 .
M ( k x , k y , ϕ ) = T y P y T x P x T x P x 1 T y P y 1 e j 2 ( k x + k y ) Λ = ( 0 P k x ϕ Q k x , k y ϕ P k y ϕ P k x ϕ 0 P k y ϕ e j 2 k x Λ Q k x , k y ϕ e j 2 k x Λ Q k x , k y ϕ P k y ϕ e j 2 k x Λ 0 P k x ϕ e j 2 k y Λ P k y ϕ Q k x , k y ϕ e j 2 k x Λ P k x ϕ e j 2 k y Λ 0 ) ,
P k x , y ϕ = t e j ϕ 4 ( 1 + e j 2 k x , y Λ ) ,
Q k x , k y ϕ = e j ( ϕ 2 2 k x Λ ) e j ( ϕ 2 + 2 k y Λ ) .
M = 16 κ 8 e j ( k x + k y ) 2 Λ { sin 4 ( ϕ 2 ) 2 κ 2 sin 2 ( ϕ 2 ) [ cos 2 ( k x Λ ) + cos 2 ( k y Λ ) 2 κ 2 cos 2 ( k x Λ ) cos 2 ( k y Λ ) ] + κ 4 [ cos 2 ( k x Λ ) cos 2 ( k y Λ ) ] 2 } ,
sin ( ϕ 2 ) = ± κ cos ( k x Λ ) 1 κ 2 cos 2 ( k y Λ ) ± κ cos ( k y Λ ) 1 κ 2 cos 2 ( k x Λ ) .
sin [ θ ( K ) 2 ] = κ cos ( K Λ ) ,
δ ϕ ( ω ) = θ ( k x ) + θ ( k y ) ,
a 00 = 1 ,
b 00 = j ν t 1 { cos [ θ ( k y ) 2 ] κ sin ( k y Λ ) } exp [ j ξ ( k x , k y ) ] ,
c 00 = ( 1 ) ν cos [ θ ( k y ) 2 ] κ sin ( k y Λ ) cos [ θ ( k x ) 2 ] κ sin ( k x Λ ) ,
d 00 = j ν t 1 { cos [ θ ( k x ) 2 ] + κ sin ( k x Λ ) } exp [ j ξ ( k x , k y ) ] ,
ξ ( k x , k y ) = [ θ ( k x ) θ ( k y ) ] 4 .
w 10 = [ Ξ ( k x , k y ) 0 0 Ξ ( k x , k y ) ] w 00 exp ( j k x Λ ) ,
w 01 = [ 0 Ξ + ( k x , k y ) Ξ + ( k x , k y ) 0 ] w 00 exp ( j k y Λ ) ,
w 11 = ( 1 ) ν [ 0 I I 0 ] w 00 exp [ j ( k x + k y ) Λ ] ,
Ξ ± ( k x , k y ) = [ 0 exp [ + j ξ ( k x , k y ) ± j ν π 2 ] exp [ j ξ ( k x , k y ) j ν π 2 ] 0 ]

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