Abstract

The partial wave scattering and interior amplitudes for the interaction of an electromagnetic plane wave with a modified Luneburg lens are derived in terms of the exterior and interior radial functions of the scalar radiation potentials evaluated at the lens surface. A Debye series decomposition of these amplitudes is also performed and discussed. The effective potential inside the lens for the transverse electric polarization is qualitatively examined, and the approximate lens size parameters of morphology-dependent resonances are determined. Finally, the physical optics model is used to calculate wave scattering in the vicinity of the ray theory orbiting condition in order to demonstrate the smoothing of ray theory discontinuities by the diffraction of scattered waves.

© 2008 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]
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  20. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Sec. 2.6.
  21. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, 1992), p. 59.
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  26. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand, 1990), pp. 192-193.
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2008

2006

2005

2002

F. Michel, G. Reidenmeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152704-4 (2002).
[CrossRef]

1996

1994

1993

1992

1991

N. Fiedler-Ferrari, H. M. Nussenzveig, and W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005-1038 (1991).
[CrossRef] [PubMed]

1972

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972), Sec. 6.3.
[CrossRef]

1964

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects (errata),” Phys. Rev. 134, AB1 (1964).
[CrossRef]

1962

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837-1843 (1962).
[CrossRef]

1959

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259-286 (1959).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 505, Eqs. (13.1.31) and (13.1.32).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 504, Eqs. (13.1.2) and (13.1.3).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 756, Eqs. (13.149) and (13.150).

Berry, M. V.

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972), Sec. 6.3.
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand, 1990), pp. 192-193.

Fiedler-Ferrari, N.

N. Fiedler-Ferrari, H. M. Nussenzveig, and W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005-1038 (1991).
[CrossRef] [PubMed]

Ford, K. W.

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259-286 (1959).
[CrossRef]

Griffiths, D. J.

D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed. (Pearson, 2005), pp. 68, 69, and 316.

Han, X.

Jamison, J. M.

Jiang, H.

Johnson, B. R.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), pp. 233-235.

Lam, C. C.

Leung, P. T.

Li, R.

Lin, C.-Y.

Lock, J. A.

Marion, J. B.

J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems, 3rd ed. (Harcourt Brace Jovanovich, 1988), p. 256, Eq. (7.38).

Marston, P. L.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Sec. 2.6.

Michel, F.

F. Michel, G. Reidenmeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152704-4 (2002).
[CrossRef]

Mount, K. E.

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972), Sec. 6.3.
[CrossRef]

Nussenzveig, H. M.

N. Fiedler-Ferrari, H. M. Nussenzveig, and W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005-1038 (1991).
[CrossRef] [PubMed]

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, 1992), p. 59.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, 1992), Chap. 14.
[CrossRef]

Ohkubo, S.

F. Michel, G. Reidenmeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152704-4 (2002).
[CrossRef]

Reidenmeister, G.

F. Michel, G. Reidenmeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152704-4 (2002).
[CrossRef]

Ren, K. F.

Schiller, S.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 505, Eqs. (13.1.31) and (13.1.32).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 504, Eqs. (13.1.2) and (13.1.3).

Tai, C.-T.

C.-T. Tai, Dyadic Green's Functions in Electromagnetic Theory (Intext, 1971), pp. 201-204.

Thornton, S. T.

J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems, 3rd ed. (Harcourt Brace Jovanovich, 1988), p. 256, Eq. (7.38).

Wheeler, J. A.

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259-286 (1959).
[CrossRef]

Wiscombe, W. J.

N. Fiedler-Ferrari, H. M. Nussenzveig, and W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005-1038 (1991).
[CrossRef] [PubMed]

Wyatt, P. J.

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects (errata),” Phys. Rev. 134, AB1 (1964).
[CrossRef]

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837-1843 (1962).
[CrossRef]

Young, K.

Ann. Phys. (N.Y.)

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259-286 (1959).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Phys. Acoust.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1-234 (1992), Sec. 2.6.

Phys. Rev.

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837-1843 (1962).
[CrossRef]

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects (errata),” Phys. Rev. 134, AB1 (1964).
[CrossRef]

Phys. Rev. A

N. Fiedler-Ferrari, H. M. Nussenzveig, and W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005-1038 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett.

F. Michel, G. Reidenmeister, and S. Ohkubo, “Luneburg lens approach to nuclear rainbow scattering,” Phys. Rev. Lett. 89, 152701-1-152704-4 (2002).
[CrossRef]

Rep. Prog. Phys.

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972), Sec. 6.3.
[CrossRef]

Other

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 756, Eqs. (13.149) and (13.150).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 505, Eqs. (13.1.31) and (13.1.32).

D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed. (Pearson, 2005), pp. 68, 69, and 316.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), pp. 233-235.

C.-T. Tai, Dyadic Green's Functions in Electromagnetic Theory (Intext, 1971), pp. 201-204.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, 1992), Chap. 14.
[CrossRef]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 504, Eqs. (13.1.2) and (13.1.3).

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, 1992), p. 59.

J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems, 3rd ed. (Harcourt Brace Jovanovich, 1988), p. 256, Eq. (7.38).

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand, 1990), pp. 192-193.

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Figures (2)

Fig. 1
Fig. 1

Effective radial potential of Eq. (21) as a function of r a for k a = 50.5 , partial waves n = 45 , 50, and 55, and (a) f = 1.2 , (b) f = 1.0 , and (c) f = 0.8 . The effective energy of the size parameter k a = 50.5 is the horizontal line U eff = 2550 . For f = 0.8 , an internal well potential is formed by partial waves with n slightly larger than 50. For f = 1 , the partial wave n = 50 is at the condition for orbiting.

Fig. 2
Fig. 2

TE scattered intensity as a function of the scattering angle θ for f = 1.0 , a = 28.40 μ m , λ = 0.51 μ m , and k a = 350.0 computed by the method described in [7]. For 30 ° < θ < 100 ° , the intensity resembles that of a Fresnel straight-edge pattern corresponding to Eq. (40).

Equations (104)

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ψ n , m ( k r , θ , φ ) = [ F n ( k r ) k r ] P n m [ cos ( θ ) ] exp ( i m φ ) ,
2 ψ n , m + N 2 ( r ) k 2 ψ n , m ( k r ) = 0 ,
d 2 F n d ( k r ) 2 + { N 2 ( r ) n ( n + 1 ) ( k r ) 2 ] F n ( k r ) = 0
E n , m TE ( r , θ , φ ) = r × ψ n , m ,
B n , m TE ( r , θ , φ ) = ( i ω ) × ( r × ψ n , m ) ,
ξ n , m ( k r , θ , φ ) = N ( r ) ψ n , m ( k r , θ , φ ) = [ G n ( k r ) k r ] P n m ( cos θ ) exp ( i m φ )
2 ξ n , m [ 2 ( d N d r ) N ] ξ n , m r [ 2 ( d N d r ) ( N r ) ] ξ n , m + N 2 ( r ) k 2 ξ n , m = 0 .
d 2 G n d ( k r ) 2 2 ( d N d r ) [ d G n d ( k r ) ] N k + [ N 2 ( r ) n ( n + 1 ) ( k r ) 2 ] G n ( k r ) = 0
E n , m TM ( r , θ , φ ) = ( i c N 2 ω ) × ( r × ξ n , m ) ,
B n , m TM ( r , θ , φ ) = ( 1 c ) r × ξ n , m .
Ψ inc ( k r , θ , φ ) = n = 1 { i n ( 2 n + 1 ) [ n ( n + 1 ) ] } ψ n ( k r ) P n 1 ( cos θ ) α ( φ ) ,
α ( φ ) = cos ( φ ) for TM ,
= sin ( φ ) for TE .
Ψ scat ( k r , θ , φ ) = n = 1 { i n ( 2 n + 1 ) [ n ( n + 1 ) ] } ζ n ( 1 ) ( k r ) P n 1 ( cos θ ) β n ( φ ) ,
β n ( φ ) = a n cos ( φ ) for TM ,
= b n sin ( φ ) for TE ,
Ψ int ( k r , θ , φ ) = n = 1 { i n ( 2 n + 1 ) [ n ( n + 1 ) ] } γ n ( k r , φ ) P n 1 ( cos θ ) ,
γ n ( k r , φ ) = G n ( k r ) c n cos ( φ ) for TM ,
= F n ( k r ) d n sin ( φ ) for TE ,
a n = [ G n ( k a ) ψ n ( k a ) N 2 ( a ) G n ( k a ) ψ n ( k a ) ] [ G n ( k a ) ζ n ( 1 ) ( k a ) N 2 ( a ) G n ( k a ) ζ n ( 1 ) ( k a ) ] ,
b n = [ F n ( k a ) ψ n ( k a ) F n ( k a ) ψ n ( k a ) ] [ F n ( k a ) ζ n ( 1 ) ( k a ) F n ( k a ) ζ n ( 1 ) ( k a ) ] ,
c n = i N 2 ( a ) [ G n ( k a ) ζ n ( 1 ) ( k a ) N 2 ( a ) G n ( k a ) ζ n ( 1 ) ( k a ) ] ,
d n = i [ F n ( k a ) ζ n ( 1 ) ( k a ) F n ( k a ) ζ n ( 1 ) ( k a ) ] .
N ( r ) = [ 1 + f 2 ( r a ) 2 ] 1 2 f .
d 2 F n d ( k r ) 2 + [ n ( n + 1 ) ( k r ) 2 + ( f 2 + 1 ) f 2 ( k r ) 2 ( f k a ) 2 ] F n = 0 ,
F n ( k r ) = [ ( k r ) 2 f k a ] ( n + 1 ) 2 exp [ ( k r ) 2 2 f k a ] × M [ ( 2 n + 3 ) 4 ( f 2 + 1 ) k a 4 f , ( 2 n + 3 ) 2 ; ( k r ) 2 f k a ] ,
d 2 G n d ( k r ) 2 + [ 2 k r ( f k a ) 2 ] [ ( f 2 + 1 ) f 2 ( k r f k a ) 2 ] 1 d G n d ( k r ) + [ n ( n + 1 ) ( k r ) 2 + ( f 2 + 1 ) f 2 ( k r ) 2 ( f k a ) 2 ] G n = 0 .
d 2 F n d ( k r ) 2 + U eff ( k r ) F n ( k r ) = F n ( k r ) ,
U eff ( k r ) = n ( n + 1 ) ( k r ) 2 + [ 1 + ( k r ) 2 ( k a ) 2 ] f 2 for r a ,
= n ( n + 1 ) ( k r ) 2 for r > a .
X n ( n + 1 ) ( k a ) 2 .
{ ( f 2 + 1 ) [ ( f 2 + 1 ) 2 4 f 2 X ] 1 2 } 2 ( k r k a ) 2 1 .
X sin 2 ( β )
U eff ( k r ) = X ( k a k r ) 2 + ( k r k a ) 2 1 for r a ,
= X ( k a k r ) 2 for r > a .
1 X [ ( f 2 + 1 ) 2 f ] 2
{ ( f 2 + 1 ) [ ( f 2 + 1 ) 2 4 f 2 X ] 1 2 } 2 ( k r k a ) 2 { ( f 2 + 1 ) + [ ( f 2 + 1 ) 2 4 f 2 X ] 1 2 } 2 .
{ ( f 2 + 1 ) + [ ( f 2 + 1 ) 2 4 f 2 X ] 1 2 } 2 ( k r k a ) 2 X .
F n ( k a ) F n ( k a ) = χ n ( k a ) χ n ( k a ) .
U eff ( k r ) [ ( 2 X 1 2 f ) ( 1 f 2 ) ] + 4 [ k r ( k a ) f 1 2 X 1 4 ] 2 ( k a f ) 2
X 1 2 = [ ( f 2 + 1 ) 2 f ] [ ( 2 S + 1 ) k a ] ,
n ( n + 1 ) ( n + 1 2 ) 2
n + 1 2 = [ ( f 2 + 1 ) k a 2 f ] ( 2 S + 1 )
k a = ( n + 2 S + 3 2 ) [ 2 f ( f 2 + 1 ) ] .
k a ( n + 1 2 ) N + ( n + 1 2 ) 1 3 w S ( 2 1 3 N ) P ( N 2 1 ) 1 2 + ,
P = 1 for TE ,
= 1 N 2 for TM ,
E ( θ ) = [ i k E 0 exp ( i k r ) 2 π r ] A h A v exp [ i Φ ( θ ) ] d x exp ( i k x 2 2 R h ) d y exp ( i k y 2 2 R v ) = [ E 0 exp ( i k r ) r ] A h A v ( R h R v ) 1 2 exp [ i Φ ( θ ) ] ,
R v = R h = a [ 1 cos ( θ ) ] ,
A h = { cos θ [ 1 cos ( θ ) ] } 1 2 ,
A v = { 1 [ 1 cos ( θ ) ] } 1 2 ,
Φ ( θ ) = k a [ ( π 2 ) cos ( θ ) ] π .
θ = ( π 2 ) + Δ ,
E [ ( π 2 ) + Δ ] = [ i k E 0 exp ( i k r ) 2 π r ] A h ave A v exp [ i Φ ( π 2 ) ] × d x exp ( i k x 2 2 R h ) exp ( i k x Δ ) d y exp ( i k y 2 2 R v ) = ( 1 2 1 2 r ) [ E 0 a A h ave exp ( i k r + i Φ ( π 2 ) i π 4 ) ] × exp ( i k a Δ 2 2 ) { F ( ) F [ Δ ( k a π ) 1 2 ] } ,
F ( w ) = 0 W d v exp ( i π v 2 2 ) .
U eff ( r ) = ( L 2 2 m r 2 ) + V ( r ) ,
E = U eff ( r 0 ) .
θ = π 2 r 0 ( L d r r 2 ) { 2 m [ E U eff ( r ) ] } 1 2 .
( d U eff d r ) r 0 0 .
( d U eff d r ) r 0 = 0 .
( d U eff d r ) r 0 = ( d 2 U eff d r 2 ) r 0 = 0 ,
lim k r 0 F n ( k r ) ( k r ) n + 1 ,
lim k r F n ( k r ) cos [ N ( ) k r φ n ] ,
lim k r 0 U n ( k r ) ( k r ) n ,
lim k r U n ( k r ) sin [ N ( ) k r φ n ] .
X n ( 1 2 ) ( k r ) = F n ( k r ) ± i U n ( k r )
lim k r 0 G n ( k r ) ( k r ) n + 1 ,
lim k r G n ( k r ) cos [ N ( ) k r φ n ] ,
lim k r 0 V n ( k r ) ( k r ) n ,
lim k r V n ( k r ) sin [ N ( ) k r φ n ] ,
Z n ( 1 , 2 ) ( k r ) = G n ( k r ) ± i V n ( k r )
N n = F n ( k a ) ψ n ( k a ) F n ( k a ) ψ n ( k a ) ,
D n = F n ( k a ) χ n ( k a ) F n ( k a ) χ n ( k a ) ,
P n = U n ( k a ) ψ n ( k a ) U n ( k a ) ψ n ( k a ) ,
Q n = U n ( k a ) χ n ( k a ) U n ( k a ) χ n ( k a ) .
W n TE ( k a ) = i [ X n ( 1 ) ( k a ) X n ( 2 ) ( k a ) X n ( 1 ) ( k a ) X n ( 2 ) ( k a ) ] 2 ,
W n TM ( k a ) = i [ Z n ( 1 ) ( k a ) Z n ( 2 ) ( k a ) Z n ( 1 ) ( k a ) Z n ( 2 ) ( k a ) ] 2 .
W n TE = N n Q n D n P n ,
N 2 ( a ) W n TM = N n Q n D n P n .
T n 21 = 2 i [ X n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) X n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) ] ,
R n 22 = [ X n ( 2 ) ( k a ) ζ n ( 2 ) ( k a ) X n ( 2 ) ( k a ) ζ n ( 2 ) ( k a ) ] [ X n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) X n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) ]
T n 21 = 2 i N 2 ( a ) [ Z n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) N 2 ( a ) Z n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) ] ,
R n 22 = [ Z n ( 2 ) ( k a ) ζ n ( 2 ) ( k a ) N 2 ( a ) Z n ( 2 ) ( k a ) ζ n ( 2 ) ( k a ) ] [ Z n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) N 2 ( a ) Z n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) ]
T n 21 = 2 i σ [ ( N n + Q n ) + i ( D n P n ) ] ,
R n 22 = [ ( N n + Q n ) + i ( D n + P n ) ] [ ( N n + Q n ) + i ( D n P n ) ] ,
σ = 1 for TE ,
= N 2 ( a ) for TM .
T n 12 = 2 i W n TE [ X n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) X n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) ] ,
R n 11 = [ X n ( 1 ) ( k a ) ζ n ( 1 ) ( k a ) X n ( 1 ) ( k a ) ζ n ( 1 ) ( k a ) ] [ X n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) X n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) ]
T n 12 = 2 i W n TM [ Z n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) N 2 ( a ) Z n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) ] ,
R n 11 = [ Z n ( 1 ) ( k a ) ζ n ( 1 ) ( k a ) N 2 ( a ) Z n ( 1 ) ( k a ) ζ n ( 1 ) ( k a ) ] [ Z n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) N 2 ( a ) Z n ( 2 ) ( k a ) ζ n ( 1 ) ( k a ) ]
T n 12 = 2 i W n [ ( N n + Q n ) + i ( D n P n ) ] ,
R n 11 = [ ( N n + Q n ) i ( D n + P n ) ] [ ( N n + Q n ) + i ( D n P n ) ] .
{ a n b n } = N n ( N n + i D n ) = ( 1 2 ) [ 1 R n 22 T n 21 T n 12 ( 1 R n 11 ) ] = ( 1 2 ) [ 1 R n 22 p = 1 T n 21 ( R n 11 ) p 1 T n 12 ] .
{ c n d n } = i σ ( N n + i D n ) = T n 21 ( 1 R n 11 ) = p = 1 T n 21 ( R n 11 ) p 1 .
{ a 12 M + 1 b 12 M + 1 } = 1 2 [ 1 R M + 1 , Γ , M + 1 T M + 1 , 1 T 1 , M + 1 ( 1 R 1 , Γ , 1 ) ] .
T M + 1 , 1 = t M + 1 , M T M , 1 ( 1 R M , γ , M r M , M + 1 , M ) ,
T 1 , M + 1 = T 1 , M t M , M + 1 ( 1 R M , γ , M r M , M + 1 , M ) ,
R M + 1 , Γ , M + 1 = r M + 1 , M , M + 1 + t M + 1 , M R M , γ , M t M , M + 1 ( 1 R M , γ , M r M , M + 1 , M ) ,
R 1 , Γ , 1 = R 1 , γ , 1 + T 1 , M r M , M + 1 , M T M , 1 ( 1 R M , γ , M r M , M + 1 , M ) ,
I M = R M , γ , M + T M , 1 T 1 , M ( 1 R 1 , γ , 1 )
{ a 12 M b 12 M } = ( 1 2 ) ( 1 I M )
{ a 12 M + 1 b 12 M + 1 } = ( 1 2 ) [ 1 r M + 1 , M , M + 1 t M + 1 , M I M t M , M + 1 ( 1 I M r M , M + 1 , M ) ] .
{ a 12 M + 1 b 12 M + 1 } = ( 1 2 ) ( 1 t M + 1 , M I M t M , M + 1 ) ,

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