Abstract

We present the Ince–Gaussian modes that constitute the third complete family of exact and orthogonal solutions of the paraxial wave equation in elliptic coordinates and that are transverse eigenmodes of stable resonators. The transverse shape of these modes is described by the Ince polynomials and is structurally stable under propagation. Ince–Gaussian modes constitute the exact and continuous transition modes between Laguerre– and Hermite–Gaussian modes. The expansions between the three families are derived and discussed. As with Laguerre–Gaussian modes, it is possible to construct helical Ince–Gaussian modes that exhibit rotating phase features whose intensity pattern is formed by elliptic rings and whose phase rotates elliptically.

© 2004 Optical Society of America

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 96, 8185–8194 (1992).
    [CrossRef]
  4. G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
    [CrossRef] [PubMed]
  5. E. G. Ince, “A linear differential equation with periodic coefficients,” Proc. London Math. Soc. 23, 56–74 (1923).
  6. F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, UK, 1964).
  7. F. M. Arscott, “The Whittaker-Hill equation and the wave equation in paraboloidal coordinates,” Proc. R. Soc. Edinburgh Sect. A 67, 265–276 (1967).
  8. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 19.
  9. A. G. Fox, T. Li, “Resonant modes in maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
    [CrossRef]
  10. I. Kimel, L. R. Elı́as, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
    [CrossRef]
  11. A. T. O’Neil, J. Courtial, “Mode transformations in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase converter” Opt. Commun. 181, 35–45 (2000).
    [CrossRef]
  12. C. P. Boyer, E. G. Kalnins, W. Miller, “Lie theory and separation of variables. 7. The harmonic oscillator in ellipic coordinates and the Ince polynomials,” J. Math. Phys. 16, 512–523 (1975).
    [CrossRef]
  13. L. Allen, M. J. Padgett, M. Babiker, “The orbital momentum of light,” Prog. Opt. 39, 291–371 (1999).
    [CrossRef]
  14. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
    [CrossRef]
  15. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
    [CrossRef]
  16. S. Chávez-Cerda, J. C. Gutiérrez-Vega, G. H. C. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26, 1803–1805 (2001).
    [CrossRef]
  17. J. Arlt, M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity,” Opt. Lett. 25, 191–193 (2000).
    [CrossRef]
  18. K. T. Gahagan, G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1997).
    [CrossRef]

2001

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

S. Chávez-Cerda, J. C. Gutiérrez-Vega, G. H. C. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26, 1803–1805 (2001).
[CrossRef]

2000

1999

L. Allen, M. J. Padgett, M. Babiker, “The orbital momentum of light,” Prog. Opt. 39, 291–371 (1999).
[CrossRef]

1997

1993

I. Kimel, L. R. Elı́as, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 96, 8185–8194 (1992).
[CrossRef]

1975

C. P. Boyer, E. G. Kalnins, W. Miller, “Lie theory and separation of variables. 7. The harmonic oscillator in ellipic coordinates and the Ince polynomials,” J. Math. Phys. 16, 512–523 (1975).
[CrossRef]

1967

F. M. Arscott, “The Whittaker-Hill equation and the wave equation in paraboloidal coordinates,” Proc. R. Soc. Edinburgh Sect. A 67, 265–276 (1967).

1966

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1961

A. G. Fox, T. Li, “Resonant modes in maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

1923

E. G. Ince, “A linear differential equation with periodic coefficients,” Proc. London Math. Soc. 23, 56–74 (1923).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 19.

Allen, L.

L. Allen, M. J. Padgett, M. Babiker, “The orbital momentum of light,” Prog. Opt. 39, 291–371 (1999).
[CrossRef]

G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 96, 8185–8194 (1992).
[CrossRef]

Arlt, J.

Arscott, F. M.

F. M. Arscott, “The Whittaker-Hill equation and the wave equation in paraboloidal coordinates,” Proc. R. Soc. Edinburgh Sect. A 67, 265–276 (1967).

F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, UK, 1964).

Babiker, M.

L. Allen, M. J. Padgett, M. Babiker, “The orbital momentum of light,” Prog. Opt. 39, 291–371 (1999).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 96, 8185–8194 (1992).
[CrossRef]

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, W. Miller, “Lie theory and separation of variables. 7. The harmonic oscillator in ellipic coordinates and the Ince polynomials,” J. Math. Phys. 16, 512–523 (1975).
[CrossRef]

Chávez-Cerda, S.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

S. Chávez-Cerda, J. C. Gutiérrez-Vega, G. H. C. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26, 1803–1805 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

Courtial, J.

A. T. O’Neil, J. Courtial, “Mode transformations in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase converter” Opt. Commun. 181, 35–45 (2000).
[CrossRef]

Eli´as, L. R.

I. Kimel, L. R. Elı́as, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Gahagan, K. T.

Gutiérrez-Vega, J. C.

S. Chávez-Cerda, J. C. Gutiérrez-Vega, G. H. C. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26, 1803–1805 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

Ince, E. G.

E. G. Ince, “A linear differential equation with periodic coefficients,” Proc. London Math. Soc. 23, 56–74 (1923).

Iturbe-Castillo, M. D.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, W. Miller, “Lie theory and separation of variables. 7. The harmonic oscillator in ellipic coordinates and the Ince polynomials,” J. Math. Phys. 16, 512–523 (1975).
[CrossRef]

Kimel, I.

I. Kimel, L. R. Elı́as, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Miller, W.

C. P. Boyer, E. G. Kalnins, W. Miller, “Lie theory and separation of variables. 7. The harmonic oscillator in ellipic coordinates and the Ince polynomials,” J. Math. Phys. 16, 512–523 (1975).
[CrossRef]

New, G. H. C.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

S. Chávez-Cerda, J. C. Gutiérrez-Vega, G. H. C. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26, 1803–1805 (2001).
[CrossRef]

Nienhuis, G.

G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

O’Neil, A. T.

A. T. O’Neil, J. Courtial, “Mode transformations in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase converter” Opt. Commun. 181, 35–45 (2000).
[CrossRef]

Padgett, M. J.

Rami´rez, G. A.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

Rodri´guez-Dagnino, R. M.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 96, 8185–8194 (1992).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 19.

Swartzlander, G. A.

Tepichi´n, E.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 96, 8185–8194 (1992).
[CrossRef]

Bell Syst. Tech. J.

A. G. Fox, T. Li, “Resonant modes in maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

IEEE J. Quantum Electron.

I. Kimel, L. R. Elı́as, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

J. Math. Phys.

C. P. Boyer, E. G. Kalnins, W. Miller, “Lie theory and separation of variables. 7. The harmonic oscillator in ellipic coordinates and the Ince polynomials,” J. Math. Phys. 16, 512–523 (1975).
[CrossRef]

Opt. Commun.

A. T. O’Neil, J. Courtial, “Mode transformations in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase converter” Opt. Commun. 181, 35–45 (2000).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

Opt. Lett.

Phys. Rev. A

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 96, 8185–8194 (1992).
[CrossRef]

G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[CrossRef] [PubMed]

Proc. IEEE

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Proc. London Math. Soc.

E. G. Ince, “A linear differential equation with periodic coefficients,” Proc. London Math. Soc. 23, 56–74 (1923).

Proc. R. Soc. Edinburgh Sect. A

F. M. Arscott, “The Whittaker-Hill equation and the wave equation in paraboloidal coordinates,” Proc. R. Soc. Edinburgh Sect. A 67, 265–276 (1967).

Prog. Opt.

L. Allen, M. J. Padgett, M. Babiker, “The orbital momentum of light,” Prog. Opt. 39, 291–371 (1999).
[CrossRef]

Other

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 19.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, UK, 1964).

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Figures (6)

Fig. 1
Fig. 1

Transverse field distributions of several low-order (a) even, (b) odd IGMs with = 2 . Plots depicted in the bottom row correspond to the phase structure of the modes displayed in the third row.

Fig. 2
Fig. 2

Passive three-dimensional intracavity field distribution of the mode IG 6 , 2 e ( ξ ,   η ,   = 2 ) in a two-mirror stable resonator. The waist is located at z = 0.375   m .

Fig. 3
Fig. 3

Transverse shapes of the subsets L p o , I p o , and H p o for p = 5 . IGMs correspond to = 2 .

Fig. 4
Fig. 4

Graphical representation of the subset p = 5 ( N 5 = 3 ) in a three-dimensional vector space. The transverse modes are shown in Fig. 3. Each gray circle corresponds to a different value of . The model IG 5 , 3 o tends to the negative value of HG 2 , 3 o because for some combinations of indices ( n x ,   n y ) , the HGMs of Eq. (14) do not satisfy the standard parity convention about the positive x axis [i.e., if n = 0 , 1 , 2 , , the even Hermite polynomials H 2 + 4 n ( u ) and the first derivative of the odd Hermite polynomials H 3 + 4 n ( u ) are negative at u = 0 ].

Fig. 5
Fig. 5

Transverse amplitudes and phases of stationary modes IG 10 , 6 σ ( ξ ,   η ,   = 1 ) and the corresponding HIGM at the waist plane. Note that the phase rotates around the line joining the foci of the ellipses. Contiguous rings have a π phase jump.

Fig. 6
Fig. 6

Plots of Ince polynomials (a) C 3 1 ( η ,   ) , { 0 ,   1 ,   2 ,   3 ,   4 } ; (b) C p 4 ( η ,   3 ) , p { 4 ,   6 ,   8 ,   10 ,   12 } ; (c) S 6 2 ( η ,   ) , { 0 ,   1 ,   2 ,   3 ,   4 } ; (d) S p 3 ( η ,   3 ) , p { 3 ,   5 ,   7 ,   9 ,   11 } .

Tables (1)

Tables Icon

Table 1 The Four “Fundamental” Modes

Equations (45)

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t 2 + 2 ik   z Ψ ( r ) = 0 ,
Ψ G ( r ) = w 0 w ( z ) exp - r 2 w 2 ( z ) + i   kr 2 2 R ( z ) - i ψ GS ( z ) ,
IG ( r ) = E ( ξ ) N ( η ) exp [ iZ ( z ) ] Ψ G ( r ) ,
d 2 E d ξ 2 -   sinh   2 ξ   d E d ξ - ( a - p   cosh   2 ξ ) E = 0 ,
d 2 N d η 2 +   sin   2 η   d N d η + ( a - p   cos   2 η ) N = 0 ,
- z 2 + z R 2 z R d Z d z = p ,
IG p , m e ( r ,   ) = C w 0 w ( z )   C p m ( i ξ ,   ) C p m ( η ,   ) exp - r 2 w 2 ( z ) × exp   i kz + kr 2 2 R ( z ) - ( p + 1 ) ψ GS ( z ) ,
IG p , m o ( r ,   ) = S w 0 w ( z )   S p m ( i ξ ,   ) S p m ( η ,   ) exp - r 2 w 2 ( z ) × exp   i kz + kr 2 2 R ( z ) - ( p + 1 ) ψ GS ( z ) ,
- IG p , m σ IG ¯ p , m σ d S = δ σ σ δ pp δ mm ,
ω sp = c L   [ s π + ( p + 1 ) cos - 1 ( ± g 1 g 2 ) ] ,
q out = Aq in + B Cq in + D .
γ IG ( ξ 2 ,   η 2 ) = K ( ξ 2 ,   η 2 ,   ξ 1 ,   η 1 ) IG ( ξ 1 ,   η 1 ) d S 1 ,
LG n , l e , o ( r ,   ϕ ,   z ) = 4 n ! ( 1 + δ 0 , l ) π ( n + l ) ! 1 / 2 1 w ( z ) cos   l ϕ sin   l ϕ × 2 r w ( z ) l L n l 2 r 2 w ( z ) 2 exp - r 2 w 2 ( z ) × exp   i kz + kr 2 2 R ( z ) - ( 2 n + l + 1 ) ψ GS ( z ) ,
HG n x , n y ( x ,   y ,   z ) = 1 2 n x + n y - 1 π n x ! n y ! 1 / 2 1 w ( z ) × H n x 2 x w ( z ) H n y 2 y w ( z ) exp - r 2 w 2 ( z ) × exp   i kz + kr 2 2 R ( z ) - ( n x + n y + 1 ) ψ GS ( z ) ,
LG n , l σ ( r ,   ϕ ) = m D m IG p = 2 n + l , m σ ( ξ ,   η ,   ) ,
IG p , m σ ( ξ ,   η ,   ) = l , n D l , n LG n , l σ ( r ,   ϕ ) ,
- LG n , l σ IG ¯ p , m σ d S
= δ σ σ δ p , 2 n + l ( - 1 ) n + l + ( p + m ) / 2
× ( 1 + δ 0 , l ) Γ ( n + l + 1 ) n ! A ( l + δ o , σ ) / 2 σ ( a p m ) ,  
N p = ( p + 2 δ σ , e ) / 2 , if p is even , ( p + 1 ) / 2 , if p is odd ,
I p σ = [ LI T p σ ] L p σ ,
H p σ = [ IH T p σ ] I p σ ,
L p σ = [ HL T p σ ] H p σ ,
IG 5 , 1 o IG 5 , 3 o IG 5 , 5 o = 0.938 0.344 0.048 - 0.343 0.901 0.266 0.048 - 0.266 0.963  LG 2 , 1 o LG 1 , 3 o LG 0 , 5 o ,
IG 5 , 1 o IG 5 , 3 o IG 5 , 5 o = 0.101 0.310 0.945 - 0.649 - 0.700 0.298 0.755 - 0.643 0.130  HG 4 , 1 HG 2 , 3 HG 0 , 5 .
HIG p , m ± = IG p , m e ( ξ ,   η ,   ) ± i IG p , m o ( ξ ,   η ,   ) ,
C 2 n 2 k ( η ,   ) = r = 0 n A r cos   2 r η , k = 0 , . . , n ,
C 2 n + 1 2 k + 1 ( η ,   ) = r = 0 n A r cos ( 2 r + 1 ) η , k = 0 , . . , n ,
S 2 n 2 k ( η ,   ) = r = 1 n B r sin   2 r η , k = 1 , . . , n ,
S 2 n + 1 2 k + 1 ( η ,   ) = r = 0 n B r sin ( 2 r + 1 ) η , k = 0 , . . , n .
( p / 2 + 1 ) A 1 = aA 0 ,
( p / 2 + 2 ) A 2 = - p A 0 - ( 4 - a ) A 1 ,
( p / 2 + r + 2 ) A r + 2 = [ a - 4 ( r + 1 ) 2 ] A r + 1 + r - p 2 A r ,
( p / 2 + 2 ) B 2 = ( a - 4 ) B 1 ,
( p / 2 + r + 2 ) B r + 2 = [ a - 4 ( r + 1 ) 2 ] B r + 1 + r - p 2 B r ,
2   ( p + 3 ) A 1 = a - 2   ( p + 1 ) - 1 A 0 ,
2   ( p + 2 r + 3 ) A r + 1 = [ a - ( 2 r + 1 ) 2 ] A r + 2   ( 2 r - p - 1 ) A r - 1 ,
2   ( p + 3 ) B 1 = a + 2   ( p + 1 ) - 1 B 0 ,
2   ( p + 2 r + 3 ) B r + 1 = [ a - ( 2 r + 1 ) 2 ] B r + 2   ( 2 r - p - 1 ) B r - 1 ;
- π π exp - 2       cos   2 η C p m ( η ,   ) C p m ( η ,   ) d η = 0 ,
m m .
C 2 n 2 k ( i ξ ,   ) = r = 0 n A r cosh   2 r η ,
C 2 n + 1 2 k + 1 ( i ξ ,   ) = r = 0 n A r cosh ( 2 r + 1 ) η ,
S 2 n 2 k ( i ξ ,   ) = r = 1 n B r sinh   2 r η ,
S 2 n + 1 2 k + 1 ( i ξ ,   ) = r = 0 n B r sinh ( 2 r + 1 ) η ,

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