Abstract

Traveling waves in two coupled parallel infinite linear point-scatterer arrays are studied analytically for the first time to our knowledge. The two arrays are considered to be generally offset in the axial direction. It is found that slow quasi-even/odd supermodes are supported, as a result of the coupling-induced splitting of the modes of the single array, in direct analogy to standard optical waveguide couplers. Exactly even/odd supermodes are supported when the axial offset is zero. Mode splitting, dispersion curves, and coupling length are numerically investigated versus the inter-element spacing, the inter-array distance, and the axial offset. Potential applications of the concept are in directional optical couplers made of metallic or dielectric nanoparticle chains.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  33. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007). .
  34. We would like to note that, in the Appendix of , energy arguments are provided to show that linear arrays of generalized radiating elements can support only slow modes, provided that the far-field pattern of a single element does not have nulls with respect to the polar angle. These arguments apply to our case too, where the two offset elements of each period can be viewed as a single super-element. However, we believe that a more rigorous derivation of this result is still lacking.

2010 (1)

2009 (2)

L. Xianshu and A. W. Poon, “Many-element coupled-resonator optical waveguides using gapless-coupled microdisk resonators,” Opt. Express 17, 23617–23628 (2009).
[CrossRef]

I. Psarros, G. Fikioris, and M. Vlahoyianni, “Pseudopotentials: a simplified model for certain types of array elements,” IEEE Trans. Antennas Propag. 57, 414–424 (2009).
[CrossRef]

2008 (3)

W. W. Huang, Y. Zhang, and B. J. Li, “Ultracompact wavelength and polarization splitters in periodic dielectric waveguides,” Opt. Express 16, 1600–1609 (2008).
[CrossRef] [PubMed]

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

I. Psarros and I. Chremmos, “Resonance splitting in two coupled circular closed-loop arrays and investigation of analogy to traveling-wave optical resonators,” Prog. Electromagn. Res. 87, 197–214 (2008).
[CrossRef]

2007 (1)

2006 (3)

2005 (1)

R. A. Shore and A. D. Yaghjian, “Traveling electromagnetic waves on linear periodic arrays of lossless penetrable spheres,” IEICE Trans. Commun. E88-B, 2346–2352 (2005).
[CrossRef]

2004 (1)

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

2003 (1)

P. St. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

2002 (3)

S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, “Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects,” Opt. Lett. 27, 2079–2081 (2002).
[CrossRef]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002).
[CrossRef]

A. D. Yaghjian, “Scattering-matrix analysis of linear periodic arrays,” IEEE Trans. Antennas Propag. 50, 1050–1064 (2002).
[CrossRef]

2000 (1)

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356–R16359 (2000).
[CrossRef]

1999 (1)

1998 (2)

1995 (1)

1990 (1)

G. Fikioris, R. W. P. King, and T. T. Wu, “The resonant circular array of electrically short elements,” J. Appl. Phys. 68, 431–439(1990).
[CrossRef]

1978 (1)

1945 (1)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

1928 (1)

F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–600 (1928).
[CrossRef]

Aers, G. C.

Asakawa, K.

Ashili, S. P.

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

Astratov, V. N.

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

Atwater, H. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002).
[CrossRef]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356–R16359 (2000).
[CrossRef]

Aussenegg, F. R.

Bloch, F.

F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–600 (1928).
[CrossRef]

Bock, P. J.

Brongersma, M. L.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356–R16359 (2000).
[CrossRef]

Chang, K.-D.

Cheben, P.

Chen, J. C.

Chremmos, I.

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

I. Psarros and I. Chremmos, “Resonance splitting in two coupled circular closed-loop arrays and investigation of analogy to traveling-wave optical resonators,” Prog. Electromagn. Res. 87, 197–214 (2008).
[CrossRef]

I. Chremmos and G. Fikioris, “Note on acoustic pseudopotentials: energy viewpoint, real scatterers and linear arrays,” Acta Acust. Acust. 97, 364–372 (2011).
[CrossRef]

Delâge, A.

Densmore, A.

DeRose, G. A.

Devenyi, A.

Fan, S.

Fikioris, G.

I. Psarros, G. Fikioris, and M. Vlahoyianni, “Pseudopotentials: a simplified model for certain types of array elements,” IEEE Trans. Antennas Propag. 57, 414–424 (2009).
[CrossRef]

I. Psarros and G. Fikioris, “Two-term theory for infinite linear array and application to study of resonances,” J. Electromagn. Waves Appl. 20, 623–645 (2006).
[CrossRef]

G. Fikioris, R. W. P. King, and T. T. Wu, “The resonant circular array of electrically short elements,” J. Appl. Phys. 68, 431–439(1990).
[CrossRef]

R. W. P. King, G. Fikioris, and R. B. Mack, Arrays of Cylindrical Dipoles (Cambridge University, 2002).
[CrossRef]

I. Chremmos and G. Fikioris, “Note on acoustic pseudopotentials: energy viewpoint, real scatterers and linear arrays,” Acta Acust. Acust. 97, 364–372 (2011).
[CrossRef]

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Franchak, J. P.

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007). .

Hall, T. J.

Hartman, J. W.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356–R16359 (2000).
[CrossRef]

Huang, W. W.

Ikeda, N.

Ishikawa, H.

Janz, S.

Joannopoulos, J. D.

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals, 2nd ed. (Princeton University Press, 2008).

Khurgin, J. B.

J. B. Khurgin and R. S. Tucker, eds., Slow Light Science and Applications (CRC Press, 2009).

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002).
[CrossRef]

King, R. W. P.

G. Fikioris, R. W. P. King, and T. T. Wu, “The resonant circular array of electrically short elements,” J. Appl. Phys. 68, 431–439(1990).
[CrossRef]

R. W. P. King, G. Fikioris, and R. B. Mack, Arrays of Cylindrical Dipoles (Cambridge University, 2002).
[CrossRef]

Krenn, J. R.

Lan, S.

Lapointe, J.

Lee, R. K.

Leitner, A.

Li, B. J.

Luan, P.-G.

Mack, R. B.

R. W. P. King, G. Fikioris, and R. B. Mack, Arrays of Cylindrical Dipoles (Cambridge University, 2002).
[CrossRef]

Maier, S.

S. Maier, Plasmonics: Fundamentals and applications (Springer, 2002).

Maier, S. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002).
[CrossRef]

Marom, E.

Meade, R. D.

Modinos, A.

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

Nishikawa, S.

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic, 2005).

Poon, A. W.

Poon, J. K. S.

Psarros, I.

I. Psarros, G. Fikioris, and M. Vlahoyianni, “Pseudopotentials: a simplified model for certain types of array elements,” IEEE Trans. Antennas Propag. 57, 414–424 (2009).
[CrossRef]

I. Psarros and I. Chremmos, “Resonance splitting in two coupled circular closed-loop arrays and investigation of analogy to traveling-wave optical resonators,” Prog. Electromagn. Res. 87, 197–214 (2008).
[CrossRef]

I. Psarros and G. Fikioris, “Two-term theory for infinite linear array and application to study of resonances,” J. Electromagn. Waves Appl. 20, 623–645 (2006).
[CrossRef]

Quinten, M.

Russell, P. St. J.

P. St. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007). .

Scherer, A.

Schmid, J. H.

Shore, R. A.

R. A. Shore and A. D. Yaghjian, “Traveling electromagnetic waves on linear periodic arrays of lossless penetrable spheres,” IEICE Trans. Commun. E88-B, 2346–2352 (2005).
[CrossRef]

Stefanou, N.

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

Sugimoto, Y.

Tucker, R. S.

J. B. Khurgin and R. S. Tucker, eds., Slow Light Science and Applications (CRC Press, 2009).

Uzunoglu, N.

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

Vlahoyianni, M.

I. Psarros, G. Fikioris, and M. Vlahoyianni, “Pseudopotentials: a simplified model for certain types of array elements,” IEEE Trans. Antennas Propag. 57, 414–424 (2009).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals, 2nd ed. (Princeton University Press, 2008).

Winn, N.

Wu, T. T.

G. Fikioris, R. W. P. King, and T. T. Wu, “The resonant circular array of electrically short elements,” J. Appl. Phys. 68, 431–439(1990).
[CrossRef]

T. T. Wu, “Fermi pseudopotentials and resonances in arrays,” in Resonances-Models and Phenomena, S.Albeverio, L.S.Ferreira and L.Streit, eds. (Springer, 1984), pp. 293–306.
[CrossRef]

Xianshu, L.

Xu, D.-X.

Xu, Y.

Yaghjian, A. D.

R. A. Shore and A. D. Yaghjian, “Traveling electromagnetic waves on linear periodic arrays of lossless penetrable spheres,” IEICE Trans. Commun. E88-B, 2346–2352 (2005).
[CrossRef]

A. D. Yaghjian, “Scattering-matrix analysis of linear periodic arrays,” IEEE Trans. Antennas Propag. 50, 1050–1064 (2002).
[CrossRef]

Yariv, A.

Yeh, P.

Zhang, Y.

Zhu, L.

Acta Acust. Acust. (1)

I. Chremmos and G. Fikioris, “Note on acoustic pseudopotentials: energy viewpoint, real scatterers and linear arrays,” Acta Acust. Acust. 97, 364–372 (2011).
[CrossRef]

Appl. Phys. Lett. (2)

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

A. D. Yaghjian, “Scattering-matrix analysis of linear periodic arrays,” IEEE Trans. Antennas Propag. 50, 1050–1064 (2002).
[CrossRef]

I. Psarros, G. Fikioris, and M. Vlahoyianni, “Pseudopotentials: a simplified model for certain types of array elements,” IEEE Trans. Antennas Propag. 57, 414–424 (2009).
[CrossRef]

IEICE Trans. Commun. (1)

R. A. Shore and A. D. Yaghjian, “Traveling electromagnetic waves on linear periodic arrays of lossless penetrable spheres,” IEICE Trans. Commun. E88-B, 2346–2352 (2005).
[CrossRef]

J. Appl. Phys. (1)

G. Fikioris, R. W. P. King, and T. T. Wu, “The resonant circular array of electrically short elements,” J. Appl. Phys. 68, 431–439(1990).
[CrossRef]

J. Electromagn. Waves Appl. (1)

I. Psarros and G. Fikioris, “Two-term theory for infinite linear array and application to study of resonances,” J. Electromagn. Waves Appl. 20, 623–645 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

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

Opt. Express (5)

Opt. Lett. (4)

Phys. Rev. (1)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Phys. Rev. B (2)

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356–R16359 (2000).
[CrossRef]

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

Prog. Electromagn. Res. (1)

I. Psarros and I. Chremmos, “Resonance splitting in two coupled circular closed-loop arrays and investigation of analogy to traveling-wave optical resonators,” Prog. Electromagn. Res. 87, 197–214 (2008).
[CrossRef]

Science (1)

P. St. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

Z. Phys. (1)

F. Bloch, “Über die quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–600 (1928).
[CrossRef]

Other (8)

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals, 2nd ed. (Princeton University Press, 2008).

J. B. Khurgin and R. S. Tucker, eds., Slow Light Science and Applications (CRC Press, 2009).

K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic, 2005).

S. Maier, Plasmonics: Fundamentals and applications (Springer, 2002).

T. T. Wu, “Fermi pseudopotentials and resonances in arrays,” in Resonances-Models and Phenomena, S.Albeverio, L.S.Ferreira and L.Streit, eds. (Springer, 1984), pp. 293–306.
[CrossRef]

R. W. P. King, G. Fikioris, and R. B. Mack, Arrays of Cylindrical Dipoles (Cambridge University, 2002).
[CrossRef]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007). .

We would like to note that, in the Appendix of , energy arguments are provided to show that linear arrays of generalized radiating elements can support only slow modes, provided that the far-field pattern of a single element does not have nulls with respect to the polar angle. These arguments apply to our case too, where the two offset elements of each period can be viewed as a single super-element. However, we believe that a more rigorous derivation of this result is still lacking.

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Figures (7)

Fig. 1
Fig. 1

Two parallel infinite linear pseudopotential arrays. Both arrays have spacing b x . Array B is offset by b y in direction y and by b 0 in direction x.

Fig. 2
Fig. 2

Dispersion curves θ 0 ( k 0 b x ) of an isolated pseudopotential array for the indicated values of the pseudopotential parameter α.

Fig. 3
Fig. 3

Dispersion curves of the even (thin solid curves) and odd (dotted curves) supermode of two parallel pseudopotential arrays with zero offset ( b 0 = 0 ) for several values of the normalized distance b y / b x = 0.4 , 0.5 , 0.7 , 1.0 , 1.5 . The thick solid curve, which separates the lines of the even and odd modes, corresponds to the isolated array ( b y ) . Pseudopotential parameter α = 1 .

Fig. 4
Fig. 4

Same as Fig. 3 but with pseudopotential parameter α = 1 . The numbers next to the curves indicate the ratio b y / b x .

Fig. 5
Fig. 5

Dispersion curves of the even (lower, solid curves) and odd (upper, dotted curves) supermode of two parallel pseudopotential arrays at distance b y = 0.5 b x and varying axial offset b 0 / b x = 0 , 0.1 , 0.2 , 0.3 , 0.4 , 0.5 . The arrows indicate the direction of increasing b 0 . The thick solid curve that separates the two groups corresponds to the isolated array ( b y ) . Pseudopotential parameter α = 1 .

Fig. 6
Fig. 6

Coupling number of periods versus the normalized inter-array distance for the various indicated values of the pseudopotential parameter α = 0.7 , 1 , 2 , 5 , 50 . The normalized frequency is fixed at k 0 b x = 0.8 .

Fig. 7
Fig. 7

(a) Directional coupler concept. The two straight parts, with 101 elements each, are at a distance b y = 2 b x . Each of the curved terminal parts has 20 elements and a constant radius of curvature R = 40 b x . Only the first input element is excited, while the last elements at the two outputs are absorbing with τ = 0.5 . The size of the elements has been exaggerated for visualization purposes. (b) Squared amplitude of elements (1 to 141) along the upper (black circles) and lower (white circles) coupler arm at k 0 b x = 0.8 . The scattering coefficient of all but the absorbing elements is τ = 2 j / ( 1 2 j ) .

Equations (23)

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[ ( m n ) b x b 0 ] 2 + b y 2 b m n A B .
n = + S m n I n A + n = + M m n A B I n B = 0 , m = 0 , ± 1 , ± 2 , ... n = + M m n B A I n A + n = + S m n I n B = 0 , m = 0 , ± 1 , ± 2 , ... ,
S n = S n = { 1 j α , n = 0 4 π α k 0 G 0 ( | n | b x ) , n = ± 1 , ± 2 , ...
M n A B = M n B A = 4 π α k 0 G 0 ( b n A B )
I n A = I 0 e j n θ , I n B = u I 0 e j n θ ,
S ( θ ) + u M A B ( θ , b 0 ) = 0 , M B A ( θ , b 0 ) + u S ( θ ) = 0 ,
S ( θ ) = S ( θ ) = n = + S n e j n θ ,
M A B ( θ , b 0 ) = M A B ( θ , b 0 ) = n = + M n A B e j n θ , M B A ( θ , b 0 ) = M B A ( θ , b 0 ) = M A B ( θ , b 0 ) = n = + M n B A e j n θ .
S 2 ( θ ) M A B ( θ , b 0 ) M A B ( θ , b 0 ) = 0 ,
S ( θ ) = 1 j α + α k 0 b x [ T * ( k 0 b x + θ 2 π ) + T * ( k 0 b x θ 2 π ) ] ,
T ( x ) = n = 1 + e j 2 π n x n = ln [ 2 | sin ( π x ) | ] j π ( x x 1 2 )
S ( θ ) = 1 α k 0 b x ln [ 2 | cos ( k 0 b x ) cos θ | ] j π α k 0 b x ( 1 + k 0 b x + θ 2 π + k 0 b x θ 2 π ) .
M A B ( θ , b 0 ) = 4 π α k 0 n = + G 0 ( ( n b x b 0 ) 2 + b y 2 ) e j n θ .
M A B ( θ , b 0 ) = 4 π α k 0 p = + + e j ( 2 p π θ ) t G 0 ( ( t b x b 0 ) 2 + b y 2 ) d t ,
M A B ( θ , b 0 ) = 2 α k 0 b x p = + e j ( 2 p π θ ) b 0 b x + cos ( 2 p π θ k 0 b x w ) e j w 2 + ( k 0 b y ) 2 w 2 + ( k 0 b y ) 2 d w .
M A B ( θ , b 0 ) = 2 α k 0 b x p = + e j ( 2 p π θ ) b 0 b x { j π 2 H 0 ( 2 ) ( b y b x ( k 0 b x ) 2 ( θ 2 p π ) 2 ) , | θ 2 p π | < k 0 b x K 0 ( b y b x ( θ 2 p π ) 2 ( k 0 b x ) 2 ) , | θ 2 p π | > k 0 b x .
M B A ( θ , b 0 ) = M A B ( θ , b 0 ) = [ M A B ( θ , b 0 ) ] * ,
S ( θ ) = ± | M A B ( θ , b 0 ) | .
M A B ( θ , b 0 ) | b y b x 2 α k 0 b x e j θ b 0 b x K 0 ( b y b x θ 2 ( k 0 b x ) 2 ) .
θ 0 = cos 1 [ cos ( k 0 b x ) 1 2 e k 0 b x / α ] .
S ( θ 0 ) = 2 α k 0 b x sin ( θ 0 ) e k 0 b x / α ,
θ o / e θ 0 ± | M A B ( θ 0 , b 0 ) | S ( θ 0 ) ,
A m j 4 π τ m k 0 n = 1 N A n G 0 ( r m , r n ) = δ m , 1 , m = 1 , 2 , ... , N ,

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