Abstract

Coupling coefficients describe the excitation of eigenfunctions by an incident wave. They are derived here for each of the two systems of eigenfunctions of a spherical resonator, characterized by Hermite or Laguerre polynomials, and for incident waves of any transverse mode, of axes not collinear to the axis of the resonator (three parameters of misalignment), and of structure not matched to the resonator (two parameters of mismatching). Alignment and matching may be controlled by suppressing the excitation of all eigenfunctions except the fundamental mode. An estimate of the residual amounts of misalignment and mismatching is an example of the use of the coupling coefficients.

© 1984 Optical Society of America

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References

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  1. R. L. Fork, D. R. Herriott, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
    [CrossRef]
  2. O. O. Andrade, Appl. Opt. 15, 2800 (1976).
    [CrossRef] [PubMed]
  3. H. Kogelnik, in Proceedings, Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), pp. 333–347.
  4. F. Bayer-Helms, PTB-Bericht Me-43, Physikalisch Technische Bundesanstalt, Braunschweig (Feb.1983).
  5. H. W. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  6. F. Bayer-Helms, PTB-Bericht Me-16, Physikalisch Technische Bundesanstalt, Braunschweig (Jan.1977).
  7. F. Bayer-Helms, PTB-Bericht Me-33, Physikalisch Technische Bundesanstalt, Braunschweig (July1981).
  8. W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, Berlin, 1966).
  9. G. Wencker, “Ein Beitrag zur Theorie Gauβscher Strahlen,” Thesis, Technische Hochschule Aachen (1968).
  10. E. Feldheim, Proc. K. Ned. Akad. Wet. 43, 224 (1940).
  11. R. D. Deslattes, A. Henins, Phys. Rev. Lett. 31, 972 (1973).
    [CrossRef]

1976 (1)

1973 (1)

R. D. Deslattes, A. Henins, Phys. Rev. Lett. 31, 972 (1973).
[CrossRef]

1966 (1)

1964 (1)

1940 (1)

E. Feldheim, Proc. K. Ned. Akad. Wet. 43, 224 (1940).

Andrade, O. O.

Bayer-Helms, F.

F. Bayer-Helms, PTB-Bericht Me-16, Physikalisch Technische Bundesanstalt, Braunschweig (Jan.1977).

F. Bayer-Helms, PTB-Bericht Me-33, Physikalisch Technische Bundesanstalt, Braunschweig (July1981).

F. Bayer-Helms, PTB-Bericht Me-43, Physikalisch Technische Bundesanstalt, Braunschweig (Feb.1983).

Deslattes, R. D.

R. D. Deslattes, A. Henins, Phys. Rev. Lett. 31, 972 (1973).
[CrossRef]

Feldheim, E.

E. Feldheim, Proc. K. Ned. Akad. Wet. 43, 224 (1940).

Fork, R. L.

Henins, A.

R. D. Deslattes, A. Henins, Phys. Rev. Lett. 31, 972 (1973).
[CrossRef]

Herriott, D. R.

Kogelnik, H.

R. L. Fork, D. R. Herriott, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
[CrossRef]

H. Kogelnik, in Proceedings, Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), pp. 333–347.

Kogelnik, H. W.

Li, T.

Magnus, W.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, Berlin, 1966).

Oberhettinger, F.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, Berlin, 1966).

Soni, R. P.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, Berlin, 1966).

Wencker, G.

G. Wencker, “Ein Beitrag zur Theorie Gauβscher Strahlen,” Thesis, Technische Hochschule Aachen (1968).

Appl. Opt. (3)

Phys. Rev. Lett. (1)

R. D. Deslattes, A. Henins, Phys. Rev. Lett. 31, 972 (1973).
[CrossRef]

Proc. K. Ned. Akad. Wet. (1)

E. Feldheim, Proc. K. Ned. Akad. Wet. 43, 224 (1940).

Other (6)

F. Bayer-Helms, PTB-Bericht Me-16, Physikalisch Technische Bundesanstalt, Braunschweig (Jan.1977).

F. Bayer-Helms, PTB-Bericht Me-33, Physikalisch Technische Bundesanstalt, Braunschweig (July1981).

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, Berlin, 1966).

G. Wencker, “Ein Beitrag zur Theorie Gauβscher Strahlen,” Thesis, Technische Hochschule Aachen (1968).

H. Kogelnik, in Proceedings, Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), pp. 333–347.

F. Bayer-Helms, PTB-Bericht Me-43, Physikalisch Technische Bundesanstalt, Braunschweig (Feb.1983).

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Figures (3)

Fig. 1
Fig. 1

Coordinate systems x,y,z of the spherical resonator and x(o),y(o),z(o) of the incident wave. y = 0 or y ( o ) = y 2 ( o ) is the plane of the drawing. r1, r2 and d are the radii and distance of the resonator mirrors. w ¯ 0 and z ¯ 2 are the radius of the waist and its distance from the second mirror for a wave function matched to the resonator. w 0 ( o ) and z 2 ( o ) are the radius of the waist and its distance from the second mirror for the incident wave.

Fig. 2
Fig. 2

Excitation of the eigenfunctions ψ m ¯ , n ¯ of a spherical resonator with m ¯ + n ¯ = q: by an incident wave of fundamental mode ψ 0 , 0 ( o ); in the asymptotic case of completely isolated interference peaks of heights ϕ,q; depending on the interference order δq/2π = N + (q + 1)ϑ/2π; for discrete values of the parameters of mismatching K ˜ or misalignment (|X|2 + Y2)1/2. The discrete points ϕ,q for integer q are interpolated by curves resulting when q! is replaced by Γ(q + 1). The incident wave is exactly aligned (X = Y = 0) but mismatched ( K ˜ ≠ 0).

Fig. 3
Fig. 3

Excitation of the eigenfunctions ψ m ¯ , n ¯ of a spherical resonator with m ¯ + n ¯ = q: by an incident wave of fundamental mode ψ 0 , 0 ( o ); in the asymptotic case of completely isolated interference peaks of heights ϕ,q; depending on the interference order δq/2π = N + (q + 1)ϑ/2π; for discrete values of the parameters of mismatching K ˜ or misalignment (|X|2 + Y2)1/2. The discrete points ϕ,q for integer q are interpolated by curves resulting when q! is replaced by Γ(q + 1). The incident wave is matched (K = 0) but misaligned (|X|2 + Y2 ≠ 0).

Equations (108)

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z 0 = π w 0 2 / λ ,
γ 0 = w 0 / z 0 = λ / π z 0 = λ / ( π w 0 ) ( rad ) ,
4 π κ = π γ 0 2 = λ / z 0 ( sr ) .
ζ = z - z T z 0 .
z ¯ 0 = { d ( r 1 - d ) ( r 2 - d ) ( r 2 + r 1 - d ) ( r 2 + r 1 - 2 d ) 2 } 1 / 2 ,             z ¯ 2 = d ( r 1 - d ) r 2 + r 1 - 2 d .
ψ m , n ( x , y , z ) = N m , n ( x ) ( 1 + ζ 2 ) - 1 / 2 H m ( a ( x ) ) H n ( a ( y ) ) exp { e m , n } ,
ψ p , l ± ( r , φ , z ) = N p , l ( r ) ( 1 + ζ 2 ) - 1 / 2 a l ( r ) L p ( l ) ( a 2 ( r ) ) exp { e p , l } ,
a ( x ) = x w ¯ 0 ( 2 1 + ζ 2 ) 1 / 2 , e m , n = i ( m + n + 1 ) arctan ζ - x 2 + y 2 w ¯ 0 2 ( 1 - i ζ ) , e p , l = ± i l φ + i ( 2 p + l + 1 ) arctan ζ - r 2 w ¯ 0 2 ( 1 - i ζ ) , arctan ζ < π / 2 ,
- + ψ m , n ψ m , n * d x d y = δ m , m · δ n , n ,
r = 0 φ = 0 2 π ψ p , l ± ψ p , l ± r d r d φ = δ p , p · δ l , l , r = 0 φ = 0 2 π ψ p , l ± ψ p , l * r d r φ = δ p , p · δ l + l , 0 .
x = ( x ( o ) - x 2 ( o ) ) cos γ - z ( o ) sin γ , y = y ( o ) - y 2 ( o ) , z = ( x ( o ) - x 2 ( o ) ) sin γ + z ( o ) cos γ , x ( o ) = x 2 ( o ) + x cos γ + z sin γ , y ( o ) = y 2 ( o ) + y , z ( o ) =             - x sin γ + z cos γ .
K 0 = ( z 0 ( o ) - z ¯ 0 ) / z ¯ 0 > - 1 ,             K 2 = ( z 2 ( o ) - z ¯ 2 ) / z ¯ 0 .
K = ( K 0 + i K 2 ) / 2 ,
K ˜ = K / ( 1 + K ) 2 ,             0 K ˜ < 1.
z 0 ( o ) = ( 1 + K 0 ) z ¯ 0 ,             z 2 ( o ) = K 2 z ¯ 0 + z ¯ 2 ,             w 0 ( o ) = 1 + K 0 w ¯ 0 ,
x ( o ) w 0 ( o ) = x + x 2 ( o ) + z sin γ ( 1 + K 0 ) 1 / 2 w ¯ 0 , y ( o ) w 0 ( o ) = y + y 2 ( o ) ( 1 + K 0 ) 1 / 2 w ¯ 0 , ζ ( o ) = z ( o ) + z 2 ( o ) z 0 ( o ) = ζ + K 2 1 + K 0 ,
sin γ w ¯ 0 / z ¯ 0 = γ γ ¯ 0 ,
ψ m , n ( o ) ( x ( o ) , y ( o ) , z ( o ) ) exp { i ( w t - k z ( o ) ) } = m ¯ = 0 n ¯ = 0 k m , n , m ¯ , n ¯ ψ m ¯ , n ¯ ( x , y , z ) exp { i ( w t - k z ) } .
ψ m , n ( o ) ( x ( o ) , y ( o ) , z ( o ) ) exp { i k ( z - z ( o ) ) } = m ¯ = 0 n ¯ = 0 k m , n , m ¯ , n ¯ ( x . x ) ψ m ¯ , n ¯ ( x , y , z )
= p ¯ = 0 [ k m , n , p ¯ , o ( x . r ) ψ p ¯ , o + l ¯ = 1 ( k m , n , p ¯ , l ¯ ( x , r ) + ψ p ¯ , l ¯ + + k m , n , p ¯ , l ¯ ( x , r ) - ψ p ¯ , l ¯ - ) ] ,
ψ p . l ± ( 0 ) ( r ( o ) , φ ( o ) , z ( o ) ) exp { i k ( z - z ( o ) ) } = m ¯ = 0 n ¯ = 0 k ± p , l , m ¯ , n ¯ ( r . x ) ψ m ¯ , n ¯ ( x , y , z )
= p ¯ = 0 [ k ± p , l . p ¯ , o ( r , r ) ψ p ¯ , o + l ¯ = 1 ( k ± p , l , p ¯ . l ¯ ( r , r ) + ψ p ¯ , l ¯ + + k ± p , l , p ¯ , l ¯ ( r , r ) - ψ p ¯ , l ¯ - ) ] .
- + ψ m , n ( o ) ψ m , n ( o ) * d x d y = m ¯ = 0 n ¯ = 0 k m , n , m ¯ , n ¯ k m , n , m ¯ , n ¯ *
m ¯ = 0 n ¯ = 0 k m , n , m ¯ , n ¯ k m , n , m ¯ , n ¯ * = δ m , m · δ n , n
m ¯ = 0 n ¯ = 0 k m , n , m ¯ , n ¯ 2 = 1 ,
k m , n , m ¯ , n ¯ = exp { i 2 k z sin 2 γ 2 } - + ψ m , n ( o ) exp { i k x sin γ } ψ m ¯ , n ¯ * d x d y ,
k m , n , m ¯ , n ¯ = N m , n ( x ) N m ¯ , n ¯ ( x ) ( ( 1 + K 0 ) ( 1 + ζ ( o ) 2 ) ( 1 + ζ 2 ) ( 1 + ζ 2 ) ) - 1 / 2 · exp { i ( m + n + 1 ) arctan ζ ( o ) - i ( m ¯ + n ¯ + 1 ) arctan ζ + i 2 k z sin 2 γ 2 } I m , m ¯ I n , n ¯ , I m , m ¯ = - + H m ( x + x 2 ( o ) + z sin γ w ¯ 0 ( 2 ( 1 + K 0 ) ( 1 + ζ ( o ) 2 ) ) 1 / 2 ) H m ¯ ( a ( x ) ) · exp { - ( x + x 2 ( 0 ) + z sin γ ) 2 ( 1 + K 0 ) w ¯ 0 2 ( 1 - i ζ ( o ) ) - x 2 w ¯ 0 2 ( 1 + i ζ ) + i k x sin γ } d x , I n , n ¯ = - + H n ( y + y 2 ( o ) w ¯ 0 ( 2 ( 1 + K 0 ) ( 1 + ζ ( o ) 2 ) ) 1 / 2 ) H n ¯ ( a ( y ) exp { - ( y + y 2 ( o ) ) 2 ( 1 + K 0 ) w ¯ 0 2 ( 1 - i ζ ( o ) ) - y 2 w ¯ 0 2 ( 1 + i ζ ) } d y .
k m , n , m ¯ , n ¯ = k m , m ¯ ( x ) · k n , n ¯ ( y ) .
- + H s ( ξ ) H s ¯ ( ξ ) exp { - ξ 2 } d ξ = π 2 s s ! δ s , s ¯ .
k m , m ¯ ( x ) = ( - 1 ) m ¯ E ( x ) ( m ! m ¯ ! ( 1 + K 0 ) m + 1 / 2 ( 1 + K * ) - ( m + m ¯ + 1 ) 1 / 2 { S g - S u } , S g = μ = 0 [ m / 2 ] μ ¯ = 0 [ m ¯ / 2 ] ( - 1 ) μ X ¯ m - 2 μ X m ¯ - 2 μ ¯ ( m - 2 μ ) ! ( m ¯ - 2 μ ¯ ) ! σ = 0 min ( μ , μ ¯ ) ( - 1 ) σ F ¯ μ - σ F μ ¯ - σ ( 2 σ ) ! ( μ - σ ) ! ( μ ¯ - σ ) , S u ¯ = μ = 0 [ ( m - 1 ) / 2 ] μ ¯ = 0 [ ( m ¯ - 1 ) / 2 ] ( - 1 ) μ X ¯ m - 2 μ - 1 X m ¯ - 2 μ ¯ 1 ( m - 2 μ - 1 ) ! ( m ¯ - 2 μ ¯ - 1 ) ! σ = 0 min ( μ , μ ¯ ) ( - 1 ) σ F ¯ μ - σ F μ ¯ - σ ( 2 σ + 1 ) ! ( μ - σ ) ! ( μ ¯ - σ ) ! .
X ¯ = ( 1 + K * ) - 1 / 2 ( x 2 ( o ) w ¯ 0 - ( z ¯ 2 z ¯ 0 - i ) γ γ ¯ 0 ) , X = ( 1 + K * ) - 1 / 2 ( x 2 ( o ) w ¯ 0 - ( z ¯ 2 z ¯ 0 + i ( 1 + 2 K * ) ) γ γ ¯ 0 ) , Y = ( 1 + K * ) - 1 / 2 y 2 ( o ) w ¯ 0
F ¯ = K 2 ( 1 + K 0 ) ,             F = K * 2 .
E ( x ) = exp { - X ¯ X 2 - i x 2 ( o ) w ¯ 0 γ γ 0 } ,             E ( y ) = exp { - y 2 2 } ,
E = E ( x ) E ( y ) = exp { - 1 ( 2 + K 0 ) 2 + K 2 2 [ ( 2 + K 0 ) ( ( x 2 ( o ) w ¯ 0 - z ¯ 2 z ¯ 0 γ γ 0 ) 2 + y 2 ( o ) 2 w ¯ 0 2 - γ 2 γ ¯ 0 2 ) - 2 K 2 ( x 2 ( o ) w ¯ 0 - z ¯ 2 z 0 γ γ 0 ) γ γ 0 ] - γ 2 γ ¯ 0 2 } · exp { - i ( 2 + K 0 ) 2 + K 2 2 [ K 2 ( ( x 2 ( o ) w ¯ 0 - z ¯ 2 z ¯ 0 γ γ ¯ 0 ) 2 + y 2 ( o ) 2 w ¯ 0 2 - γ 2 γ ¯ 0 2 ) + 2 ( 2 + K 0 ) ( x 2 ( o ) w ¯ 0 - z ¯ 2 z ¯ 2 γ γ ¯ 0 ) γ γ ¯ 0 ] - i z ¯ 2 z ¯ 0 γ 2 γ ¯ 0 2 } .
σ = 0 min ( μ , μ ¯ ) ( - 1 ) σ F ¯ μ - σ F μ ¯ - σ ( 2 σ ) ! ( μ - σ ) ! ( μ ¯ - σ ) ! = F ¯ μ F μ ¯ μ ! μ ¯ ! F 2 1 ( - μ , - μ ¯ ; 1 2 ; 1 - 1 K ˜ ) = ( 2 μ + 2 μ ¯ ) ! F ¯ μ F μ ¯ ( 2 μ ) ! ( 2 μ ¯ ) ! ( μ + μ ¯ ) ! F 2 1 ( - μ , - μ ¯ ; - μ - μ ¯ + 1 2 ; 1 K ˜ ) = μ ! μ ¯ ! F ¯ μ ( 2 μ ) ! ( 2 μ ) ! σ = 0 min ( μ , μ ¯ ) ( 2 μ + 2 μ ¯ - 2 σ ) ! ( μ + μ ¯ - σ ) ! ( μ - σ ) ! ( μ ¯ - σ ) ! ( - 4 K ¯ ) σ , σ = 0 min ( μ , μ ¯ ) ( - 1 ) F ¯ σ μ - σ F μ ¯ - σ ( 2 σ + 1 ) ! ( μ - σ ) ! ( μ ¯ - σ ) ! = F ¯ μ F μ ¯ μ ! μ ¯ ! F 2 1 ( - μ , - μ ¯ ; 3 2 ; 1 - 1 K ˜ ) = ( 2 μ + 2 μ ¯ + 1 ) ! F ¯ μ F μ ¯ ( 2 μ + 1 ) ! ( 2 μ ¯ + 1 ) ! ( μ + μ ¯ ) ! F 2 1 ( - μ , - μ ¯ ; - μ - μ ¯ + 1 2 ; 1 K ˜ ) = μ ! μ ¯ ! F ¯ μ F μ ¯ ( 2 μ + 1 ) ! ( 2 μ ¯ + 1 ) ! σ = 0 min ( μ , μ ¯ ) ( 2 μ + 2 μ ¯ + 2 - 2 σ ) ! 2 σ ! ( μ + μ ¯ - 1 - σ ) ! ( μ - σ ) ! ( μ ¯ - σ ) ! × ( - 4 K ˜ ) σ .
k 0 , 0 ( x ) = E ( x ) ( 1 + K 0 ) 1 / 4 ( 1 + K * ) - 1 / 2 , k 1 , 0 ( x ) = E ( x ) ( 1 + K 0 ) 3 / 4 ( 1 + K * ) - 1 X ¯ , k 0 , 1 ( x ) = - E ( x ) ( 1 + K 0 ) 1 / 4 ( 1 + K * ) - 1 X , k 2 , 0 ( x ) = E ( x ) 2 - 1 / 2 ( 1 + K 0 ) 5 / 4 ( 1 + K * ) - 3 / 2 ( X ¯ 2 - 2 F ¯ ) , k 1 , 1 ( x ) = - E ( x ) ( 1 + K 0 ) 3 / 4 ( 1 + K * ) - 3 / 2 ( X ¯ X - 1 ) , k 0 , 2 ( x ) = E ( x ) 2 - 1 / 2 ( 1 + K 0 ) 1 / 4 ( 1 + K * ) - 3 / 2 ( X 2 + 2 F ) .
k 0 , m ¯ ( x ) = E ( x ) ( m ¯ ! ( 1 + K 0 ) 1 / 2 ( 1 + K * ) - ( m ¯ + 1 ) ) 1 / 2 μ ¯ = 0 [ m ¯ / 2 ] ( - X ) m ¯ - 2 μ ¯ F μ ¯ ( m ¯ - 2 μ ¯ ) ! μ ¯ ! .
k m , 0 ( x ) = E ( x ) ( m ! ( 1 + K 0 ) m + 1 / 2 ( 1 + K * ) - ( m + 1 ) ) 1 / 2 × μ = 0 [ m / 2 ] X ¯ m - 2 μ ( - F ¯ ) μ ( m - 2 μ ) ! μ ! .
k m , m ¯ ( x ) = ( - 1 ) m ¯ E ¯ ( x ) ( m ! m ¯ ! ) 1 / 2 μ = 0 min ( m , m ¯ ) ( - 1 ) μ X ¯ m - μ X m ¯ - μ μ ! ( m - μ ) ! ( m ¯ - μ ) ! ,
X ¯ = x 2 ( o ) w ¯ 0 - ( z ¯ 2 z ¯ 0 - i ) γ γ 0 ,             X = X ¯ * ,             Y = y 2 ( o ) w ¯ 0 ,
E ¯ ( x ) = exp { - 1 2 ( ( x 2 ( o ) w ¯ 0 - z ¯ 2 z ¯ 0 γ γ ¯ 0 ) 2 + γ 2 γ ¯ 0 2 ) - i x 2 ( o ) w ¯ 0 γ γ ¯ 0 } , E ¯ ( y ) = exp { - y 2 ( o ) 2 2 w ¯ 0 2 } ,
E ¯ ( x ) E ¯ ( y ) 2 = E E ¯ * = exp { - ( X 2 + y 2 ) } .
k 0 , m ¯ ( x ) = E ¯ ( x ) ( - X ) m ¯ ( m ¯ ! ) 1 / 2 or k m , 0 ( x ) = E ¯ ( x ) X ¯ m ( m ! ) 1 / 2
k 2 m , 2 m ¯ + 1 ( x ) = k 2 m + 1 , 2 m ¯ = 0 ( x ) , k 2 m , 2 m ¯ ( x ) = ( - 1 ) m ( ( 2 m ) ! ( 2 m ¯ ) ! ( 1 - K ˜ ) 1 / 2 ( k ˜ 4 ) m + m ¯ ) 1 / 2 · exp { i ( m + m ¯ + 1 2 ) arc ( 1 + K ) + i ( m - m ¯ ) arc K } σ = 0 min ( m , m ¯ ) ( 4 - 4 / K ˜ ) σ ( 2 σ ) ! ( m - σ ) ! ( m ¯ - σ ) ! , k 2 m + 1 , 2 m ¯ + 1 ( x ) = ( - 1 ) m ( ( 2 m + 1 ) ! ( 2 m ¯ + 1 ) ! ( 1 - K ˜ ) 3 / 2 ( K ˜ 4 ) m + m ¯ ) 1 / 2 · exp { i ( m + m ¯ + 3 2 ) arc ( 1 + K ) + i ( m - m ¯ ) arc K } σ = 0 min ( m , m ¯ ) ( 4 - 4 / K ˜ ) σ ( 2 σ + 1 ) ! ( m - σ ) ! ( m ¯ - σ ) ! .
k 0 , 2 m ¯ ( x ) = 1 m ¯ ! ( ( 2 m ¯ ) ! ( 1 - K ˜ ) 1 / 2 ( K ˜ 4 ) m ¯ ) 1 / 2 · exp { i ( m ¯ + 1 2 ) arc ( 1 + K ) - i m ¯ arc K } ,
k 2 m , 0 ( x ) = ( - 1 ) m m ! ( ( 2 m ) ! ( 1 - K ˜ ) 1 / 2 ( K ˜ 4 ) m ) 1 / 2 · exp { i ( m + 1 2 ) arc ( 1 + K ) + i m arc K } ,
p ! r l L p ( l ) ( r 2 ) exp ( ± i l φ ) = ( - 1 ) p n = 0 q ( ± i ) n 2 - q S p , q - p , n H q - n ( x ) H n ( y ) ,             q = 2 p + l ,
H m ( x ) H n ( y ) = p = 0 [ q / 2 ] ( 1 - 1 2 δ p , q / 2 ) S m , q - m , p p ! r q - 2 p L p ( q - 2 p ) ( r 2 ) · ( i - n exp [ i ( q - 2 p ) φ ] + i n exp [ - i ( q - 2 p ) φ ] ) ,             q = m + n .
S m , n , p = μ = 0 μ + ν = p m ν = 0 n ( - 1 ) μ ( m μ ) ( n ν )
ψ p , l ± ( γ , φ , z ) = ( - 1 ) p n = 0 q ( ± i ) n K p , q - p , n ψ q - n , n ( x , y , z ) ,             q = 2 p + l ,
ψ m , n ( x , y , z ) = p = 0 [ q / 2 ] ( 1 - 1 2 δ p , q / 2 ) K m , q - m , p ( i - n ψ p , q - 2 p + ( r , φ , z ) + i n ψ p , q - 2 p - ( r , φ , z ) ) ,             q = m + n ,
K m , q - m , p = p ! N m , q - m ( x ) N p , q - 2 p ( r ) S m , q - m , p = ( p ! ( q - p ) ! 2 q m ! ( q - m ) ! ) 1 / 2 S m , q - m , p .
k ± p , l , m , n ( r , x ) = ( - 1 ) p ( ± i ) n K p , p + l , n ,             m = 2 p + l - n ,             0 n 2 p + l
k 2 m , 2 n , p , 0 ( x , r ) = ( - 1 ) n K 2 m , 2 n , p , p = m + n ,
k m , n , p , l ( x , r ) ± = ( i ) n K m , n , p ,             l = m + n - 2 p > 0 ,             0 p [ m + n + 1 2 ]
S m , n , 0 = 1 ,
S 0 , n , p = ( n p ) ,
S m , 0 , p = ( - 1 ) p ( m p ) ,
S m , n , p = ( n p ) F 2 1 ( - m , - p ; n - p + 1 ; - 1 ) , if m n , S n , m , p = ( - 1 ) p S m , n , p , followed by S m , m , 2 p + 1 = 0 ,
S m , m , 2 p = ( - 1 ) p ( m p ) ,
S m , n , m + n - p = ( - 1 ) m S m , n , p , followed by S 2 m + 1 , 2 n + 1 , m + n + 1 = 0 ,
S m + 1 , n , p + 1 = S m , n , p + 1 - S m , n , p ,
S m , n + 1 , p + 1 = S m , n , p + 1 + S m , n , p .
q = 0 q = 1 q = 2 q = 3 ( 1 ) ( 1 1 1 - 1 ) ( 1 2 1 1 0 - 1 1 - 2 1 ) ( 1 3 3 1 1 1 - 1 - 1 1 - 1 - 1 1 1 - 3 3 - 1 )
- + ψ p , l ± ( r , φ , z ) ψ 2 p + l - n , n * ( x , y , z ) d x d y = ( - 1 ) p ( ± i ) n K p , p + l , n ,             0 n 2 p + l ,
r = 0 φ = 0 2 π ψ 2 m , 2 n ( x , y , z ) ψ m + n , 0 ± * ( r , φ , z ) r d r d φ = ( - 1 ) n K 2 m , 2 n , m + n ,
r = 0 φ = 0 2 π ψ m , n ( x , y , z ) ψ p , m + n - 2 p ± * ( r , φ , z ) r d r d φ = ( i ) n K m , n , p ,             0 p [ m + n - 1 2 ] .
S 2 m , 2 n , m + n = ( - 1 ) m ( 2 m ) ! ( 2 n ) ! ( m + n ) ! m ! n ! ,
( q m ) S m , q - m , p = ( q p ) S p , q - p , m or K m , q - m , p = K p , q - p , m ,             0 p q .
K m , q - m , p = sign ( S m , q - m , p ) ( 2 - q S m , q - m , p S p , q - p , m ) 1 / 2 ,
q = 0 q = 1 q = 2 q = 3 ( 1 ) ( a a a - a ) ( a 2 a a 2 a 0 - a a 2 - a a 2 ) ( b c c b c b - b - c c - b - b c b - c c - b ) ,
r = 0 q S m , q - m , r S r , q - r , p = 2 q δ m , p or r = 0 q K m , q - m , r K r , q - r , p = δ m , p
r = 0 q ( - 2 ) r S m , q - m , r S r , q - r , p = 2 q δ q - m , p or r = 0 q ( - 1 ) r K m , q - m , r K r , q - r , p = δ q - m , p ,
r = 0 q S m , q - m , r = 2 q δ q - m , p and r = 0 q ( - 1 ) r S m , q - m , r = 2 q δ q - m , 0 .
k m , n , p ¯ , l ( x , r ) ± = m ¯ = 0 n ¯ = 0 k m , n , m ¯ , n ¯ ( x , x ) r = 0 φ = 0 2 π ψ m ¯ , n ¯ ( x , y , z ) ψ p ¯ , l ¯ ± * ( r , φ , z ) r d r d φ .
k m , n , p ¯ , 0 ( x , r ) = ν ¯ = 0 p ( - 1 ) ν ¯ K 2 p - 2 ν ¯ , 2 ν ¯ , p ¯ k m , n , 2 p ¯ , 2 ν ¯ , 2 ν ¯ ( x , x ) , k m , n , p ¯ , l ¯ ( x , r ) + = ν ¯ = 0 2 p ¯ + l ¯ ( i ) ν ¯ K 2 p ¯ + l ¯ - ν ¯ , ν ¯ , p ¯ k m , n , 2 p ¯ + l ¯ - ν ¯ , ν ¯ ( x , x ) ,             l ¯ > 0.
k ± p , l , m ¯ , n ¯ ( r , x ) = ( - 1 ) p ν = 0 2 p + l ( ± i ) ν K p , p + l , ν k 2 p + l - ν , ν , m ¯ , n ¯ ( x , x ) .
k ± p , l , p ¯ , 0 ( r , r ) = ( - 1 ) p ν = 0 2 p + l ν ¯ = 0 p ¯ ( ± i ) ν ¯ K p , p + l , ν K 2 p ¯ - 2 ν ¯ , 2 ν ¯ , p ¯ k 2 p + l - ν , ν , 2 p ¯ - 2 ν ¯ , 2 ν ¯ ( x , x ) , k ± p , l , p ¯ , l ( r , r ) + = ( - 1 ) p ν = 0 2 p + l ν ¯ = 0 2 p ¯ + l ¯ ( ± i ) ν ( - i ) ν ¯ K p , p + l , ν K 2 p ¯ + l ¯ - ν ¯ , ν ¯ , p ¯ k 2 p + l - ν , ν , 2 p ¯ + l ¯ - ν ¯ , ν ¯ ( x , x ) ,             l ¯ > 0 , k ± p , l , p ¯ , l ¯ ( r , r ) - =             ( + i ) ν ¯ .
k 0 , 0 , 0 , 0 ( r , r ) = E ( 1 + K 0 ) 1 / 2 ( 1 + K * ) - 1 , k 0 , 0 , 0 , 1 ( r , r ) ± = - E ( 1 + K 0 ) 1 / 2 ( 1 + K * ) - 3 / 2 2 - 1 / 2 ( X i Y ) , k ± 0 , 1 , 0 , 0 ( r , r ) = E ( 1 + K 0 ) ( 1 + K * ) - 3 / 2 2 - 1 / 2 ( X ¯ ± i Y ) , k 0 , 0 , 1 , 0 ( r , r ) = - E ( 1 + K 0 ) 1 / 2 ( 1 + K * ) - 2 { K * + ( X 2 + Y 2 ) / 2 } , k 1 , 0 , 0 , 0 ( r , r ) = E ( 1 + K 0 ) 3 / 2 ( 1 + K * ) - 2 { K / ( 1 + K 0 ) - ( X ¯ 2 + Y 2 ) / 2 } , k ± 0 , 1 , 0 , 1 ( r , r ) = E ( 1 + K 0 ) ( 1 + K * ) - 2 { 1 - ( X ¯ ± i Y ) ( X i Y ) / 2 } , k ± 0 , 1 , 0 , 1 ( r , r ) = - E ( 1 + K 0 ) ( 1 + K * ) - 2 ( X ¯ ± i Y ) ( X ± i Y ) / 2 , k 0 , 0 , 0 , 2 ( r , r ) ± = E ( 1 + K 0 ) 1 / 2 ( 1 + K * ) - 2 2 - 3 / 2 ( X i Y ) 2 , k ± 0 , 2 , 0 , 0 ( r , r ) = E ( 1 + K 0 ) 3 / 2 ( 1 + K * ) - 2 2 - 3 / 2 ( X ¯ ± i Y ) 2 .
k ± p , l , p ¯ , l ( r , r ) + = r = 0 φ = 0 2 π ψ p , l ± ( o ) ψ p ¯ , l ¯ ± * r d r d φ , k ± p , l , p ¯ , l ( r , r ) - = ψ p ¯ , l ¯ - * .
k ± p , l , p ¯ , l ( r , r ) ± = ( p ! ( p + l ) ! p ¯ ! ( p ¯ + l ) ! ) 1 / 2 ( ( 1 + K 0 ) 1 / 2 1 + K * ) l + 1 ( 1 + i ζ ( o ) 1 - i ζ ( o ) ) p ( 1 - i ζ 1 + i ζ ) p ¯ · ρ = 0 ρ ( l ) L p ( l ) ( α 2 ρ ) L p ¯ ( 1 ) ( α ¯ ρ 2 ) exp ( - ρ ) d ρ , α 2 = 1 1 + K * distance 1 + i ζ 1 + i ζ ( o ) ,             α ¯ 2 = 1 + K 0 1 + K * distance 1 - i ζ ( o ) 1 - i ζ
± k p , l , p ¯ , l ( r , r ) ± = ( p ! p ¯ ! ( p + l ) ! ( p ¯ + l ) ! ) 1 / 2 ( ( 1 + K 0 ) 1 / 2 1 + K * ) l + 1 ( K 1 + K * ) p ( - K * 1 + K * ) p ¯ · σ = 0 min ( p , p ¯ ) ( 1 - 1 / K ˜ ) σ σ ! ( l + σ ) ! ( p - σ ) ! ( p ¯ - σ ) ! ,             l 0.
σ = 0 min ( p , p ¯ ) ( 1 - 1 / K ˜ ) σ σ ! ( l + σ ) ! ( p - σ ) ! ( p ¯ - σ ) ! = 1 l ! p ! p ¯ ! F 2 1 ( - p , p ¯ ; l + 1 ; 1 - 1 K ˜ ) = ( p + p ¯ + l ) ! p ! p ¯ ! ( p + l ) ! ( p ¯ + l ) ! F 2 1 ( - p , - p ¯ ; - p - p ¯ - l ; 1 K ˜ ) = 1 ( p + l ) ! ( p ¯ + l ) ! σ = 0 min ( p , p ¯ ) ( p + p ¯ + l - σ ) ! σ ! ( p - σ ) ! ( p ¯ - σ ) ! ( - 1 K ˜ ) σ .
k 0 , 0 , 0 , 0 ( r , r ) = ( 1 - K ˜ ) 1 / 2 exp { i arc ( 1 + K ) } , k 0 , 0 , 1 , 0 ( r , r ) = ( ( 1 - K ˜ ) K ˜ ) 1 / 2 exp { i 2 arc ( 1 + K ) - i arc K } , k 1 , 0 , 0 , 0 ( r , r ) = ( ( 1 - K ˜ ) K ˜ ) 1 / 2 exp { i 2 arc ( 1 + K ) + i arc K } , k ± 0 , 1 , 0 , 1 ( r , r ) ± = ( 1 - K ˜ ) exp { i 2 arc ( 1 + K ) } .
τ 1 τ 2 ρ 1 ρ 2 t exp { - i ( δ - ( m ¯ - n ¯ + 1 ) ϑ ) t } ;
ϑ = 2 arccos ( ( 1 - d / r 1 ) ( 1 - d / r 2 ) ) 1 / 2
ψ ˜ m ¯ , n ¯ = t = 0 ψ m ¯ , n ¯ τ 1 τ 2 ρ 1 ρ 2 2 exp { - i ( δ - ( m ¯ + n ¯ + 1 ϑ ) t ) } = τ 1 τ 2 ψ m ¯ , n ¯ 1 - ρ 1 ρ 2 exp { - i ( δ - ( m ¯ + n ¯ + 1 ) ϑ ) } ,
ψ ˜ m , n ( o ) = m ¯ = 0 n ¯ = 0 τ 1 τ 2 k m , n , m ¯ , n ¯ ψ m ¯ , n ¯ 1 - ρ 1 ρ 2 exp { - i ( δ - ( m ¯ + n ¯ + 1 ) ϑ ) } .
ϕ ( ) = - + ψ ˜ m , n ( o ) ψ ˜ m , n ( o ) * d x d y = m ¯ = 0 n ¯ = 0 τ 1 τ 2 2 k m , n , m ¯ , n ¯ k m , n , m ¯ , n ¯ * 1 + ρ 1 ρ 2 2 - 2 ρ 1 ρ 2 cos ( δ - ( m ¯ + n ¯ + 1 ) ϑ ) ,
F = π ρ 1 ρ 2 1 / 2 1 - ρ 1 ρ 2 = π δ H ,
δ q = 2 π N + ( q + 1 ) ϑ ,
ϕ ( ) = q = 0 τ 1 τ 2 2 H ϕ , q 1 + ρ 1 ρ 2 2 - 2 ρ 1 ρ 2 cos ( δ - ( q + 1 ) ϑ ) ,
H ϕ , q = m ¯ + n ¯ = 1 k m , n , m ¯ , n ¯ 2 ,
q = 0 H ϕ , q = 1 ,
H ϕ , q = m ¯ + n ¯ = q k 0 , m ¯ ( x ) k 0 , n ¯ ( y ) 2
H ϕ , 0 = E E * ( 1 - K ˜ ) ,
H ϕ , 1 / H ϕ , 0 = 1 - K ˜ 1 + K 0 1 + K { X 2 + Y 2 } ,
H ϕ , 2 / H ϕ , 0 = ( 1 - K ˜ ) 2 2 ( 1 + K 0 ) 2 1 + K 2 { X 2 + K * 2 + 2 X Y 2 + Y 2 + K * 2 } .
H ϕ , q = ( X 2 + Y 2 ) q q ! exp { - ( X 2 + Y 2 ) } .
H ϕ , 2 q = ( 1 - K ˜ ) K ˜ q .
r = 0 q ( 2 r ) ! ( 2 q - 2 r ) ! 2 2 q ( r ! ( q - r ) ! ) 2 = 1.
H ϕ , 1 / H ϕ , 0 X 2 + Y 2 ,             H ϕ , 2 / H ϕ , 0 K ˜
X 2 + Y 2 ( x 2 ( o ) - z ¯ 2 sin γ w ¯ 0 ) 2 + ( γ γ ¯ 0 ) 2 + ( y 2 ( o ) w ¯ 0 ) N S , K ˜ K 0 2 + K 2 2 4 N S .
| x 2 ( o ) - z ¯ 2 sin γ w ¯ 0 | ( N 3 S ) 1 / 2 ,             | γ γ ¯ 0 | ( N 3 S ) 1 / 2 ,             | y 2 ( o ) w ¯ 0 | ( N 3 S ) 1 / 2 , K 0 = | z 0 ( o ) - z ¯ 0 z ¯ 0 | ( 2 N S ) 1 / 2 ,             K 2 = | z 2 ( o ) - z ¯ 2 z ¯ 0 | ( 2 N S ) 1 / 2 ;
q = i π λ ( 1 q ( o ) - 1 q ¯ * ) ,             α = i π λ ( 1 q ( o ) - 1 q ( o ) * ) ,             β = i π λ ( 1 q ¯ - 1 q ¯ * ) .
q - α = i π λ ( - 1 q * - 1 q ¯ ( o ) * ) ,             q - β = - ( q - α ) * ,             q - α - β = - q * , q - α q · exp { i 2 arctan ζ ( o ) } = z 0 ( o ) - z ¯ 0 - i ( z T ( o ) - z ¯ T ) z 0 ( o ) + z ¯ 0 + i ( z T ( o ) - z ¯ T ) = K * 1 + K ,
q - β q · exp { i 2 arctan ζ ¯ } = - z 0 ( o ) - z ¯ 0 + i ( z T ( o ) - z ¯ T ) z 0 ( o ) + z ¯ 0 + i ( z T ( o ) - z ¯ T ) = - K 1 + K
q ( q - α - β ) ( q - α ) ( q - β ) = ( q q - α ) ( q q - α ) * = 1 K ˜

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