Abstract

A new set of vector solutions to Maxwell’s equations based on solutions to the wave equation in spheroidal coordinates allows laser beams to be described beyond the paraxial approximation. Using these solutions allows us to calculate the complete first-order corrections in the short-wavelength limit to eigenmodes and eigenfrequencies in a Fabry-Perot resonator with perfectly conducting mirrors. Experimentally relevant effects are predicted. Modes which are degenerate according to the paraxial approximation are split according to their total angular momentum. This includes a splitting due to coupling between orbital angular momentum and spin angular momentum.

© 2010 Optical Society of America

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  1. I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
    [CrossRef]
  2. A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon source for distributed quantum networking,” Phys. Rev. Lett. 89, 067901 (2002).
    [CrossRef] [PubMed]
  3. V. F. Lazutkin, “An equation for the natural frequencies of a nonconfocal resonator with cylindrical mirrors which takes mirror aberration into account,” Opt. Spectrosc. 24, 236 (1968).
  4. H. Laabs, and A. T. Friberg, “Nonparaxial Eigenmodes of Stable Resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
    [CrossRef]
  5. J. Visser, and G. Nienhuis, “Spectrum of an optical resonator with spherical aberration,” J. Opt. Soc. Am. A 22, 2490–2497 (2005).
    [CrossRef]
  6. F. Zomer, V. Soskov, and A. Variola, “On the nonparaxial modes of two-dimensional nearly concentric resonators,” Appl. Opt. 46, 6859–6866 (2007).
    [CrossRef] [PubMed]
  7. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  8. R. W. Zimmerer, “Spherical Mirror Fabry-Perot Resonators,” IEEE Trans. Microw. Theory Tech. 11, 371–379 (1963).
    [CrossRef]
  9. W. A. Specht, Jr., “Modes in Spherical-Mirror Resonators,” J. Appl. Phys. 36, 1306–1313 (1965).
    [CrossRef]
  10. C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, Calif., 1957).
  11. M. Zeppenfeld, “Solutions to Maxwell’s equations using spheroidal coordinates,” N. J. Phys. 11, 073007 (2009).
    [CrossRef]
  12. M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

2009 (1)

M. Zeppenfeld, “Solutions to Maxwell’s equations using spheroidal coordinates,” N. J. Phys. 11, 073007 (2009).
[CrossRef]

2008 (1)

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[CrossRef]

2007 (1)

2005 (1)

2002 (1)

A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon source for distributed quantum networking,” Phys. Rev. Lett. 89, 067901 (2002).
[CrossRef] [PubMed]

1999 (1)

H. Laabs, and A. T. Friberg, “Nonparaxial Eigenmodes of Stable Resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1968 (1)

V. F. Lazutkin, “An equation for the natural frequencies of a nonconfocal resonator with cylindrical mirrors which takes mirror aberration into account,” Opt. Spectrosc. 24, 236 (1968).

1965 (1)

W. A. Specht, Jr., “Modes in Spherical-Mirror Resonators,” J. Appl. Phys. 36, 1306–1313 (1965).
[CrossRef]

1963 (1)

R. W. Zimmerer, “Spherical Mirror Fabry-Perot Resonators,” IEEE Trans. Microw. Theory Tech. 11, 371–379 (1963).
[CrossRef]

Friberg, A. T.

H. Laabs, and A. T. Friberg, “Nonparaxial Eigenmodes of Stable Resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

Fuhrmanek, A.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[CrossRef]

Hennrich, M.

A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon source for distributed quantum networking,” Phys. Rev. Lett. 89, 067901 (2002).
[CrossRef] [PubMed]

Kubanek, A.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[CrossRef]

Kuhn, A.

A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon source for distributed quantum networking,” Phys. Rev. Lett. 89, 067901 (2002).
[CrossRef] [PubMed]

Laabs, H.

H. Laabs, and A. T. Friberg, “Nonparaxial Eigenmodes of Stable Resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lazutkin, V. F.

V. F. Lazutkin, “An equation for the natural frequencies of a nonconfocal resonator with cylindrical mirrors which takes mirror aberration into account,” Opt. Spectrosc. 24, 236 (1968).

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Murr, K.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[CrossRef]

Nienhuis, G.

Pinkse, P. W. H.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[CrossRef]

Puppe, T.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[CrossRef]

Rempe, G.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[CrossRef]

A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon source for distributed quantum networking,” Phys. Rev. Lett. 89, 067901 (2002).
[CrossRef] [PubMed]

Schuster, I.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[CrossRef]

Soskov, V.

Specht, W. A.

W. A. Specht, Jr., “Modes in Spherical-Mirror Resonators,” J. Appl. Phys. 36, 1306–1313 (1965).
[CrossRef]

Variola, A.

Visser, J.

Zeppenfeld, M.

M. Zeppenfeld, “Solutions to Maxwell’s equations using spheroidal coordinates,” N. J. Phys. 11, 073007 (2009).
[CrossRef]

Zimmerer, R. W.

R. W. Zimmerer, “Spherical Mirror Fabry-Perot Resonators,” IEEE Trans. Microw. Theory Tech. 11, 371–379 (1963).
[CrossRef]

Zomer, F.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

H. Laabs, and A. T. Friberg, “Nonparaxial Eigenmodes of Stable Resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

R. W. Zimmerer, “Spherical Mirror Fabry-Perot Resonators,” IEEE Trans. Microw. Theory Tech. 11, 371–379 (1963).
[CrossRef]

J. Appl. Phys. (1)

W. A. Specht, Jr., “Modes in Spherical-Mirror Resonators,” J. Appl. Phys. 36, 1306–1313 (1965).
[CrossRef]

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

N. J. Phys. (1)

M. Zeppenfeld, “Solutions to Maxwell’s equations using spheroidal coordinates,” N. J. Phys. 11, 073007 (2009).
[CrossRef]

Nat. Phys. (1)

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008).
[CrossRef]

Opt. Spectrosc. (1)

V. F. Lazutkin, “An equation for the natural frequencies of a nonconfocal resonator with cylindrical mirrors which takes mirror aberration into account,” Opt. Spectrosc. 24, 236 (1968).

Phys. Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Phys. Rev. Lett. (1)

A. Kuhn, M. Hennrich, and G. Rempe, “Deterministic single-photon source for distributed quantum networking,” Phys. Rev. Lett. 89, 067901 (2002).
[CrossRef] [PubMed]

Other (2)

M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, Calif., 1957).

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Figures (1)

Fig. 1.
Fig. 1.

The oblate spheroidal coordinate system. The ξ-coordinate describes the set of ellipses with a common pair of focal points separated by a distance d, the η-coordinate describes the set of hyperbolas with the same two focal points, and the ϕ-coordinate describes the rotational angle around the z-axis. A pair of mirrors forming a resonators can be matched to the coordinate system as indicated, in this case with ξ - = -1.25 and ξ + = 1.75

Equations (80)

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r = d 2 ( 1 η 2 ) ( 1 + ξ 2 ) and z = d 2 ηξ ,
R mv ( ξ ) = e i c ̄ ξ ( 1 ) v + m / 2 ( 1 + ) v + m / 2 + 1 r mv ( ξ )
S mv ( η ) = c ̄ m / 2 ( 1 η 2 ) m 2 e c ̄ ( 1 η ) s mv ( x ) ,
E = Jσv ( b Jσv + E Jσv + + b Jσv E Jσv ) ,
d = 4 L ( R + + R L ) ( R + L ) ( R L ) ( R + + R 2 L ) 2 .
ξ ± = ± 2 L ( R L ) d ( R + + R 2 L )
z ̅ ± ( r , ϕ ) = z ± r 2 2 R ± c 4 ± r 4 + ( r 6 ) .
ξ S ± = ξ ̄ ± ( x , ϕ ) = ξ ± f 4 ± x 2 c ̅ 2 + ( x 3 c ̄ 3 )
f 4 ± = d ξ ± 2 R ± 2 2 c 4 ± ξ ± 8 ,
S ± ( η , ϕ ) = x ( ξ ̄ ± ( x ( η ) , ϕ ) , η , ϕ )
{ S ± ϕ = h ϕ e ̂ ϕ , S ± η = h η e ̂ η + h ξ e ̂ ξ ξ ̄ ± x x η } ,
h ξ = ( 1 ) , h η = ( c ̄ 1 / 2 ) , h ϕ = ( c ̄ 1 / 2 ) ,
e ̂ ξ · E Jσv ± = ( c ̄ 1 / 2 ) , e ̂ η · E Jσv ± = ( 1 ) ,
ξ · ± x = ( c ̅ 2 ) , x η = ( c ̄ ) ,
e ̂ ϕ · ( x ̂ ± i y ̂ ) = ± i e ±iϕ e ̂ ϕ · z ̂ = 0
e ̂ η · ( x ̂ ± i y ̂ ) = η 1 + ξ 2 η 2 + ξ 2 e ±iϕ e ̂ η · z ̂ = ξ 1 η 2 η 2 + ξ 2
e ̂ ξ · ( x ̂ ± i y ̂ ) = ξ 1 + η 2 η 2 + ξ 2 e ±iϕ e ̂ ξ · z ̂ = η 1 ξ 2 η 2 + ξ 2
e ̂ ϕ · E J σ + v + = i e ψ J 1 , v =
x J 1 2 ( 1 Jx 8 c ̄ ) e x / 2 e iJϕ R J 1 , v ( ξ ± ) { i L v 1 ( J ) + i L v ( J ) + 1 c ̄ [ f 4 ± x 2 L v 1 ( J ) ± f 4 ± x 2 L v ( J )
i ( v + J 1 ) ( 2 v 1 ) 8 L v 2 ( J ) + i ( 2 v 2 + 2 Jv 5 v 2 J + 1 ) 8 + L v 1 ( J ) +
i ( 2 v 2 + 2 Jv 5 v 3 J + 1 ) 8 L v ( J ) i ( v 1 ) ( 2 v + 2 J 1 ) 8 L v + 1 ( J ) ] + ( 1 c ̅ 2 ) } ,
e ̂ ϕ · E J σ - v + = i e ψ J + 1 , v =
x J 1 2 ( 1 Jx 8 c ̄ ) e x / 2 e iJϕ R J + 1 , v ( ξ ± ) { i ( v + J + 1 ) L v ( J ) + i ( v + 1 ) L v + 1 ( J ) +
1 c ̅ [ f 4 ± x 2 ( v + J + 1 ) L v 1 ( J ) ± f 4 ± x 2 ( v + 1 ) L v + 1 ( J ) i ( v + J ) ( v + J + 1 ) ( 2 v + 1 ) 8 L v 1 ( J ) +
i ( v + J + 1 ) ( 2 v 2 + 3 v + J + 2 ) 8 L v ( J ) + i ( v + 1 ) ( 2 v 2 + 4 Jv + 5 v + 2 J 2 + 6 J + 4 ) 8 L v + 1 ( J )
i ( v + 1 ) ( v + 2 ) ( 2 v + 2 J + 3 ) 8 L v + 2 ( J ) ] + ( 1 c ̄ 2 ) } ,
e ̂ η · E J σ + v + =
i e ̂ ϕ · E J σ + v + + x J 1 2 ( 1 Jx 8 c ̄ ) e x / 2 1 c ̄ { i ξ ± 1 + ξ ± 2 [ R J , v 1 ( ξ ± ) ( ( v + J 1 ) L v 2 ( J )
( 2 v + J 1 ) L v 1 ( J ) + v L v ( J ) ) + R Jv ( ξ ± ) ( ( v + J ) L v 1 ( J ) ( 2 v + J + 1 ) L v ( J ) + ( v + 1 ) L v + 1 ( J ) ) ] +
ξ ± 2 2 ( 1 + ξ ± 2 ) R J 1 , v ( ξ ± ) [ ( v + J 1 ) L v 2 ( J ) ( 3 v + 2 J 1 ) L v 1 ( J ) + ( 3 v + J + 1 ) L v ( J ) ( v + 1 ) L v + 1 ( J ) ] } ,
e ̂ η · E J σ v + = 1 i e ̂ ϕ · E J σ - v + + x J 1 2 ( 1 Jx 8 c ̄ ) e x / 2 e iJϕ ×
1 c ̄ { i ξ ± 1 + ξ ± 2 [ R Jv ( ξ ± ) ( v + J + 1 ) ( ( v + J ) L v 1 ( J ) + ( 2 v + J + 1 ) L v ( J ) ( v + 1 ) L v + 1 ( J ) ) +
R J , v + 1 ( ξ ± ) ( v + 1 ) ( ( v + J + 1 ) L v ( J ) + ( 2 v + J + 3 ) L v + 1 ( J ) ( v + 2 ) L v + 2 ( J ) ) ] +
ξ ± 2 2 ( 1 + ξ ± 2 ) R J + 1 , v ( ξ ± ) [ ( v + J ) ( v + J + 1 ) L v 1 ( J ) + ( v + J + 1 ) ( 3 v + J + 2 ) L v ( J )
( v + 1 ) ( 3 v + 2 J + 4 ) L v + 1 ( J ) + ( v + 1 ) ( v + 2 ) L v + 2 ( J ) ] } .
v J σ + v = x J 1 2 ( 1 Jx 8 c ̄ ) e x / 2 ( L v 1 ( J ) + L v ( J ) ) ( i e ̂ ϕ e ̂ η ) e iJϕ
v J σ v = x J 1 2 ( 1 Jx 8 c ̄ ) e x / 2 ( ( v + J + 1 ) L v ( J ) ( v + 1 ) L v + 1 ( J ) ) ( i e ̂ ϕ e ̂ η ) e iJϕ ,
u 1 J σ + v = x ( ψ J , v 1 + ψ Jv ) e ̂ η =
1 2 + R J , v 1 ( ξ ) ( v J σ , v 2 v J σ , v 1 ( v + J 1 ) v J σ + , v 1 + v v J σ + v ) +
1 2 + R J v ( ξ ) ( v J σ , v 1 v J σ v ( v + J ) v J σ + v + ( v + 1 ) v J σ + , v + 1 ) ,
u 1 J σ - v = x ( ( v + J + 1 ) ψ J v + ( v + 1 ) ψ J , v + 1 ) e ̂ η =
1 2 R J v ( ξ ) ( v + J + 1 ) ( v J σ , v 1 v J σ v ( v + J ) v J σ + v + ( v + 1 ) v J σ + , v + 1 ) +
1 2 R J , v + 1 ( ξ ) ( v + 1 ) ( v J σ v v J σ , v + 1 ( v + J + 1 ) v J σ + , v + 1 + ( v + 2 ) v J σ + , v + 2 ) ,
u 2 J σ + v = x ψ J 1 , v e e ̂ η =
x 2 R J 1 , v ( ξ ) v J σ + v 1 2 R J 1 , v ( ξ ) ( v J σ , v 2 2 v J σ , v 1 + v J σ v ) + ( 1 c ̄ )
u 2 J σ v = x ψ J + 1 , v e e ̂ η =
x 2 R J + 1 , v ( ξ ) v J σ - v 1 2 R J + 1 , v ( ξ ) [ ( v + J ) ( v + J + 1 ) v J σ + v
2 ( v + 1 ) ( v + J + 1 ) v J σ + v + 1 + ( v + 1 ) ( v + 2 ) v J σ + , v + 2 ] + ( 1 c ̄ ) .
E J σ + v + S ± , = R J 1 , v ( ξ ± ) [ ( 1 i f 4 ± x 2 c ̄ + x 8 c ̄ ) v J σ + v + A J 1 , v v 1 v J σ + , v 1 + A J 1 , v v + 1 v σ + , v + 1 ]
i c ̄ ξ ± 1 + ξ ± 2 u 1 J σ + v ξ = ξ ± + 1 2 c ̄ ξ ± 2 1 + ξ ± 2 u 2 J σ + , v ξ = ξ ± + 𝪪 ( 1 c ̄ 2 )
E J σ - v + S ± , = R J + 1 , v ( ξ ± ) [ ( 1 i f 4 ± x 2 c ̄ + x 8 c ̄ ) v J σ - v + A J + 1 , v v 1 v J σ - , v 1 + A J + 1 , v v + 1 v J σ - , v + 1 ] +
i c ̄ ξ ± 1 + ξ ± 2 u 1 J σ - v ξ = ξ ± + 1 2 c ̄ ξ ± 2 1 + ξ ± 2 u 2 J σ - v ξ = ξ ± + ( 1 c ̄ 2 ) .
E Jσv ± 1 S ± 2 , = σ′v′ v Jσ′v′ a v′v ± 1 , ± 2 , J , σ′ , σ
x v J σ + v = ( v + J 1 ) v J σ + , v 1 + ( 2 v + J ) v J σ + v ( v + 1 ) v J σ + , v + 1
x v J σ - v = ( v + J + 1 ) v J σ - , v 1 + ( 2 v + J + 2 ) v J σ - v ( v + 1 ) v J σ - , v + 1 ,
E S ± , = σv ( b Jσv + E Jσv + S ± , + b Jσv E Jσv S ± , )
= σ′v′ v Jσ′v′ [ σv ( a v′v + , ± , J , σ′ , σ b Jσv + + a v′v , ± , J , σ′ , σ b Jσv ) ] = 0 .
A σ′σ ± 1 , ± 2 = ( a v′v ± 1 , ± 2 , J , σ′ , σ ) v′v , A ± 1 , ± 2 = ( A σ + σ + ± 1 , ± 2 A σ + σ ± 1 , ± 2 A σ σ + ± 1 , ± 2 A σ σ ± 1 , ± 2 ) , and b σ ± = ( b Jσv ± ) v .
A + , ± ( b σ + + b σ + ) + A , ± ( b σ + b σ ) = 0 .
A = ( A + , ) 1 A , ( A , + ) 1 A + , + .
λ J σ ± v = e 2 i ( c ̅ ( ξ + ξ ) ( 2 v + ( J 1 ) + 1 ) ( arctan ( ξ + ) arctan ( ξ ) ) + ( 1 c ) ) .
a 00 ± 1 , ± 2 = a vv ± 1 , ± 2 , J , σ + , σ + , a 10 ± 1 , ± 2 = a v 1 , v ± 1 , ± 2 , J , σ - , σ + ,
a 01 ± 1 , ± 2 = a v , v 1 ± 1 , ± 2 , J , σ + , σ , a 11 ± 1 , ± 2 = a v 1 . v 1 ± 1 , ± 2 , J , σ - , σ .
a 00 + , ± = R J 1 , v ( ξ ± ) ( 1 i f 4 ± ( 6 v 2 + 6 Jv + J 2 + J ) c ̅ + 2 v + J 8 c ̅ ) +
i c ̅ ξ ± 1 + ξ ± 2 [ v 2 R J , v 1 ( ξ ± ) v + J 2 R Jv ( ξ ± ) ] 2 v + J 4 c ̅ ξ ± 2 1 + ξ ± 2 R J 1 , v ( ξ ± ) + ( 1 c ̅ 2 ) ,
a 10 + , ± = i 2 c ̅ ξ ± 1 + ξ ± 2 [ R J , v 1 ( ξ ± ) R Jv ( ξ ± ) ] + 1 2 c ̅ ξ ± 2 1 + ξ ± 2 R J 1 , v ( ξ ± ) + ( 1 c ̅ 2 ) ,
a 11 + , ± = R J + 1 , v 1 ( ξ ± ) ( 1 i f 4 ± ( 6 v 2 + 6 Jv + J 2 J ) c ̅ 2 v + J 8 c ̅ ) +
i c ̅ ξ ± 1 + ξ ± 2 [ v + J 2 R J , v 1 ( ξ ± ) + v 2 R Jv ( ξ ± ) ] 2 v + J 4 c ̅ ξ ± 2 1 + ξ ± 2 R J + 1 , v 1 ( ξ ± ) + ( 1 c ̅ 2 ) ,
and
a 01 + , ± = iv ( v + J ) 2 c ̅ ξ ± 1 + ξ ± 2 [ R J , v 1 ( ξ ± ) R Jv ( ξ ± ) ] + v ( v + J ) 2 c ̅ ξ ± 2 1 + ξ ± 2 R J + 1 , v 1 ( ξ ± ) + ( 1 c ̄ 2 ) .
( a 00 , ± a 01 , ± a 10 , ± a 11 , ± ) 1 ( a 00 ± , ± a 01 ± , ± a 10 ± , ± a 11 ± , ± ) = 1 a 00 , ± a 11 , ± a 01 , ± a 10 , ± ( a 11 , ± a 00 ± , ± a 01 , ± a 10 ± , ± a 11 , ± a 01 ± , ± a 01 , ± a 11 ± , ± a 10 , ± a 00 ± , ± + a 00 , ± a 10 ± , ± a 10 , ± a 01 ± , ± + a 00 , ± a 11 ± , ± )
arg ( i ( R J , v 1 ( ξ ) R Jv ( ξ ) ) ) = arg ( R J + 1 , v 1 ( ξ ) ) + ( 1 c ̅ ) = arg ( R J 1 , v ( ξ ) ) + ( 1 c ̄ ) .
a 00 , ± a 10 ± , ± a 10 , ± a 00 ± , ± = a 00 , ± a 10 ± , ± ( a 10 ± , ± a 00 , ± ) *
= a 00 , ± a 10 ± , ± a 00 , ± a 10 ± , ± × ( 1 + ( 1 c ̄ ) ) = 𝒪 ( 1 c ̄ 2 )
a 11 , ± a 001 ± , ± a 01 , ± a 11 ± , ± = ( 1 c ̅ 2 ) .
λ J σ + v ± = exp [ ± 2 i ( c ̅ ξ ± ( 2 v + J ) arctan ( ξ ± )
1 c ̅ ( ξ ± 1 + ξ ± 2 v ( v + J ) ± f 4 ± ( 6 v ( v + J ) + J ( J + 1 ) ) ) + ( 1 c ̄ 2 ) ) ]
λ J σ - , v 1 ± = exp [ ± 2 i ( c ̅ ξ ± ( 2 v + J ) arctan ( ξ ± )
1 c ̅ ( ξ ± 1 + ξ ± 2 v ( v + J ) ± f 4 ± ( 6 v ( v + J ) + J ( J 1 ) ) ) + ( 1 c ̅ 2 ) ) ] .
λ m + 1 , σ + ν ± λ m 1 , σ ν ± = exp ( ± 2 i c ̄ ξ ± 1 + ξ ± 2 × m ) = exp ( 2 im k R ± ) .

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