Abstract

In wavefront characterization, often the combination of a Shack-Hartmann sensor and a reconstruction method utilizing the Cartesian derivatives of Zernike circle polynomials (the least-squares method, to be called here Method A) is used, which is known to introduce crosstalk. In [J. Opt. Soc. Am. A 31, 1604 (2014) [CrossRef]  ], a crosstalk-free analytic expression of the LMS estimator of the wavefront Zersectnike coefficients is given in terms of wavefront partial derivatives (leading to what we call Method B). Here, we show an implementation of this analytic result where the derivative data are obtained using the Shack-Hartmann sensor and compare it with the conventional least-squares method.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. J. E. M. Janssen, “Zernike expansion of derivatives and laplacians of the zernike circle polynomials,” J. Opt. Soc. Am. A 31, 1604–1613 (2014).
    [Crossref]
  2. The choice for ANSI convention is motivated here by how the Shack-Hartmann sensor deals with the circle polynomials. We would like to point out that using the Born and Wolf convention (that uses exponential instead of cosine/sine functions), renders the results in [1] more transparent and concise.
  3. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
    [PubMed]
  4. G. M. Dai, Wavefront Optics for Vision Correction, vol. 179 (SPIE Press, 2008).
    [Crossref]
  5. V. N. Mahajan and E. Acosta, “Vector polynomials for direct analysis of circular wavefront slope data,” J. Opt. Soc. Am. 34, 1908–1913 (2017).
    [Crossref]
  6. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25, 1642–1651 (2008).
    [Crossref]
  7. R. Fletcher, Practical Methods of Optimization (John Wiley & Sons, 1987).
  8. O. Soloviev and G. Vdovin, “Estimation of the total error of modal wavefront reconstruction with zernike polynomials and hartmann-shack test,” in 5th International Workshop on Adaptive Optics for Industry and Medicine, vol. 6018 (International Society for Optics and Photonics, 2006), p. 60181D.
  9. H. Shengyang, X. Fengjie, L. Changhai, and J. Zongfu, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
    [Crossref]
  10. H. Shengyang, N. Yu, X. Fengjie, and J. Zongfu, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of the Laplacian,” Opt. Commun. 288, 7–12 (2013).
    [Crossref]
  11. A. J. E. M. Janssen and P. Dirksen, “Computing zernike polynomials of arbitrary degree using the discrete fourier transform,” J. Eur. Soc. Rapid Publ. 2, 07012 (2007).
    [Crossref]
  12. B. H. Shakibaei and R. Paramesran, “Recursive formula to compute Zernike radial polynomials,” Opt. Lett. 14, 2487–2489 (2013).
    [Crossref]

2017 (1)

V. N. Mahajan and E. Acosta, “Vector polynomials for direct analysis of circular wavefront slope data,” J. Opt. Soc. Am. 34, 1908–1913 (2017).
[Crossref]

2014 (1)

2013 (2)

H. Shengyang, N. Yu, X. Fengjie, and J. Zongfu, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of the Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

B. H. Shakibaei and R. Paramesran, “Recursive formula to compute Zernike radial polynomials,” Opt. Lett. 14, 2487–2489 (2013).
[Crossref]

2011 (1)

H. Shengyang, X. Fengjie, L. Changhai, and J. Zongfu, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

2008 (1)

2007 (1)

A. J. E. M. Janssen and P. Dirksen, “Computing zernike polynomials of arbitrary degree using the discrete fourier transform,” J. Eur. Soc. Rapid Publ. 2, 07012 (2007).
[Crossref]

2002 (1)

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Acosta, E.

V. N. Mahajan and E. Acosta, “Vector polynomials for direct analysis of circular wavefront slope data,” J. Opt. Soc. Am. 34, 1908–1913 (2017).
[Crossref]

Ando, T.

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Changhai, L.

H. Shengyang, X. Fengjie, L. Changhai, and J. Zongfu, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

Dai, G. M.

G. M. Dai, Wavefront Optics for Vision Correction, vol. 179 (SPIE Press, 2008).
[Crossref]

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, “Computing zernike polynomials of arbitrary degree using the discrete fourier transform,” J. Eur. Soc. Rapid Publ. 2, 07012 (2007).
[Crossref]

Fengjie, X.

H. Shengyang, N. Yu, X. Fengjie, and J. Zongfu, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of the Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

H. Shengyang, X. Fengjie, L. Changhai, and J. Zongfu, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

Fletcher, R.

R. Fletcher, Practical Methods of Optimization (John Wiley & Sons, 1987).

Fukuchi, N.

Hara, T.

Inoue, T.

Janssen, A. J. E. M.

A. J. E. M. Janssen, “Zernike expansion of derivatives and laplacians of the zernike circle polynomials,” J. Opt. Soc. Am. A 31, 1604–1613 (2014).
[Crossref]

A. J. E. M. Janssen and P. Dirksen, “Computing zernike polynomials of arbitrary degree using the discrete fourier transform,” J. Eur. Soc. Rapid Publ. 2, 07012 (2007).
[Crossref]

Mahajan, V. N.

V. N. Mahajan and E. Acosta, “Vector polynomials for direct analysis of circular wavefront slope data,” J. Opt. Soc. Am. 34, 1908–1913 (2017).
[Crossref]

Matsumoto, N.

Ohtake, Y.

Paramesran, R.

B. H. Shakibaei and R. Paramesran, “Recursive formula to compute Zernike radial polynomials,” Opt. Lett. 14, 2487–2489 (2013).
[Crossref]

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Shakibaei, B. H.

B. H. Shakibaei and R. Paramesran, “Recursive formula to compute Zernike radial polynomials,” Opt. Lett. 14, 2487–2489 (2013).
[Crossref]

Shengyang, H.

H. Shengyang, N. Yu, X. Fengjie, and J. Zongfu, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of the Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

H. Shengyang, X. Fengjie, L. Changhai, and J. Zongfu, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

Soloviev, O.

O. Soloviev and G. Vdovin, “Estimation of the total error of modal wavefront reconstruction with zernike polynomials and hartmann-shack test,” in 5th International Workshop on Adaptive Optics for Industry and Medicine, vol. 6018 (International Society for Optics and Photonics, 2006), p. 60181D.

Thibos, L. N.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Vdovin, G.

O. Soloviev and G. Vdovin, “Estimation of the total error of modal wavefront reconstruction with zernike polynomials and hartmann-shack test,” in 5th International Workshop on Adaptive Optics for Industry and Medicine, vol. 6018 (International Society for Optics and Photonics, 2006), p. 60181D.

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Yu, N.

H. Shengyang, N. Yu, X. Fengjie, and J. Zongfu, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of the Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

Zongfu, J.

H. Shengyang, N. Yu, X. Fengjie, and J. Zongfu, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of the Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

H. Shengyang, X. Fengjie, L. Changhai, and J. Zongfu, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

J. Eur. Soc. Rapid Publ. (1)

A. J. E. M. Janssen and P. Dirksen, “Computing zernike polynomials of arbitrary degree using the discrete fourier transform,” J. Eur. Soc. Rapid Publ. 2, 07012 (2007).
[Crossref]

J. Opt. Soc. Am. (1)

V. N. Mahajan and E. Acosta, “Vector polynomials for direct analysis of circular wavefront slope data,” J. Opt. Soc. Am. 34, 1908–1913 (2017).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Refract. Surg. (1)

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Opt. Commun. (2)

H. Shengyang, X. Fengjie, L. Changhai, and J. Zongfu, “Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing,” Opt. Commun. 284, 2781–2783 (2011).
[Crossref]

H. Shengyang, N. Yu, X. Fengjie, and J. Zongfu, “Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of the Laplacian,” Opt. Commun. 288, 7–12 (2013).
[Crossref]

Opt. Lett. (1)

B. H. Shakibaei and R. Paramesran, “Recursive formula to compute Zernike radial polynomials,” Opt. Lett. 14, 2487–2489 (2013).
[Crossref]

Other (4)

G. M. Dai, Wavefront Optics for Vision Correction, vol. 179 (SPIE Press, 2008).
[Crossref]

The choice for ANSI convention is motivated here by how the Shack-Hartmann sensor deals with the circle polynomials. We would like to point out that using the Born and Wolf convention (that uses exponential instead of cosine/sine functions), renders the results in [1] more transparent and concise.

R. Fletcher, Practical Methods of Optimization (John Wiley & Sons, 1987).

O. Soloviev and G. Vdovin, “Estimation of the total error of modal wavefront reconstruction with zernike polynomials and hartmann-shack test,” in 5th International Workshop on Adaptive Optics for Industry and Medicine, vol. 6018 (International Society for Optics and Photonics, 2006), p. 60181D.

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Figures (6)

Fig. 1
Fig. 1 Scheme of the experimental setup. FR is a Faraday isolator, WP is a halfwave plate, SLM is a spatial light modulator, SHS is the Shack Hartmann wavefront sensor. The zeroth order light of the SLM is blocked by the iris.
Fig. 2
Fig. 2 Reference wavefronts (left column) for the first four experiments of Table 1. Reconstructed wavefronts from experimental data using Method A (middle column) and Method B (right column). The RMS errors under each row have been obtained by comparing the reconstructed (middle and right column) with reference wavefront (left column). For the reconstructions, the first 8 degrees in the Zernike expansion has been used.
Fig. 3
Fig. 3 Reference wavefronts (left column) for the second four experiments of Table 1. Reconstructed wavefronts from experimental data using Method A (middle column) and Method B (right column). The RMS errors under each row have been obtained by comparing the reconstructed (middle and right column) with reference wavefront (left column). For the reconstructions, the first 8 degrees in the Zernike expansion has been used.
Fig. 4
Fig. 4 Reference wavefronts (left column) for the last four experiments of Table 1. Reconstructed wavefronts from experimental data using Method A (middle column) and Method B (right column). The RMS errors under each row have been obtained by comparing the reconstructed (middle and right column) with reference wavefront (left column). For the reconstructions, the first 8 degrees in the Zernike expansion has been used.
Fig. 5
Fig. 5 Convergence of coefficients for the sub_zerns_1 experiment.
Fig. 6
Fig. 6 Convergence of coefficients for the sub_zerns_3 experiment.

Tables (2)

Tables Icon

Algorithm 1 Complete measurement and comparison of Shack-Hartmann phase retrieval algorithms

Tables Icon

Table 1 Coefficients Used in Specific Zernike Experiments, Rounded Off to 3 Significant Numbers

Equations (88)

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z n m ( ρ , θ ) = N n m R n | m | ( ρ ) Θ m ( θ ) ,
N n m = ( 2 δ m 0 ) ( n + 1 ) ,
R n | m | ( ρ ) = s = 0 n | m | 2 ( 1 ) s ( n s ) ! s ! ( n | m | 2 s ) ! ( n + | m | 2 s ) ! ρ n 2 s ,
Θ m ( θ ) = { cos ( m θ ) , if m 0 , sin ( m θ ) , if m < 0 ,
n 0 ,
n | m | is even ,
| m | n ,
1 π 0 1 0 2 π z n m ( ρ , θ ) z n m ( ρ , θ ) ρ d θ d ρ = δ n n δ m m .
W ( ρ , θ ) = m = n η m a n m z n m ( ρ , θ ) ,
Z n m ( ρ , θ ) = R n | m | ( ρ ) e i m θ .
R n | m | ( ρ ) = ρ | m | P n | m | 2 ( 0 , | m | ) ( 2 ρ 2 1 ) ,
0 1 0 2 π Z n m ( ρ , θ ) ( Z n m ( ρ , θ ) ) * ρ d θ d ρ = π n + 1 δ n n δ m m .
W ( ρ , θ ) = m = n η m α n m Z n m ( ρ , θ ) ,
N n m Z n m ( ρ , θ ) = { z n | m | ( ρ , θ ) + i z n | m | ( ρ , θ ) , if m > 0 , z n | m | ( ρ , θ ) i z n | m | ( ρ , θ ) , if m < 0 , z n m , if m = 0 ,
z n m = { N n m 2 ( Z n | m | + Z n | m | ) = N n m ( Z n | m | ) , if m > 0 N n m 2 i ( Z n | m | Z n | m | ) = N n m J ( Z n | m | ) , if m < 0 N n m Z n | m | , if m = 0 .
a n m = { 1 N n m ( α n | m | + α n | m | ) , if m > 0 1 N n m J ( α n | m | α n | m | ) , if m < 0 1 N n m ( α n m ) , if m = 0 .
j = n ( n + 2 ) + m 2 ,
n = 3 + 9 + 8 j 2 ,
m = 2 j n ( n + 2 ) ,
{ 1 A Σ Σ W x d x d y = r Δ x f 1 A Σ Σ W y d x d y = r Δ y f
W ( x , y ) = m = n η m a n m z n m ( x , y ) .
{ W x = m = n η m a n m z n m x W y = m = n η m a n m z n m y .
s = [ W x ¯ | 1 W x ¯ | 2 W x ¯ | n spots W y ¯ | 1 W y ¯ | 2 W y ¯ | n spots ] T .
G = [ Z 1 x ¯ | 1 Z 1 x ¯ | 2 Z 1 x ¯ | n spot Z 1 y ¯ | 1 Z 1 y ¯ | 2 Z 1 y ¯ | n spot Z 2 x ¯ | 1 Z 2 x ¯ | 2 Z 2 x ¯ | n spot Z 2 y ¯ | 1 Z 2 y ¯ | 2 Z 2 y ¯ | n spot Z J x ¯ | 1 Z J x ¯ | 2 Z J x ¯ | n spot Z J y ¯ | 1 Z J y ¯ | 2 Z J y ¯ | n spot ] T .
s G a ,
a G + s ,
α ^ n m = C n m φ n m C n + 2 m φ n + 2 m ,
C n m = 1 + δ n | m | 2 n
φ n m = 1 2 ( β + ) n 1 m + 1 + 1 2 ( β ) n 1 m 1 ,
W x ± i W y = m = n η m ( β ± ) n m Z n m .
a ^ G l + s a ^ G l + G a ,
{ ( n , m , n , m ) | n η m , m = m or m = m ± 2 , n > n , n m + 2 , n η m , m 0 } ,
{ ( n , m , n , m ) | n η m , m = 0 or m = 2 , n > n , n 2 , n η m , m = 0 } ,
Z = [ Z 1 | 1 Z 2 | 1 Z J | 1 Z 1 | 2 Z 2 | 2 Z J | 2 Z 1 | I Z 2 | I Z J | I ] ,
p = ( Z a ) mod 2 π .
p n e w = ( p o l d γ Z a ^ ) mod 2 π ,
Δ p x = p i , j + 1 p i , j , Δ p y = p i + 1 , j p i , j .
p = Z a ,
ϵ = p ref p rec N 2 ,
f k f k + 1 max { | f k | , | f k + 1 | , 1 } 10 5 ,
max { | proj ( g i ) | i = 1 , , n } 10 5 ,
k 10 3 ,
Z n m ( ν , μ ) Z n m ( ρ , θ ) = R n | m | ( ρ ) e i m θ ,
ν + i μ = ρ e i θ ; ν = ρ cos θ ; μ = ρ sin θ ,
G _ n m = G _ n m ( ρ , θ ) = ( G n , 1 m ( ρ , θ ) , G n , 2 m ( ρ , θ ) ) 2
1 π 0 1 0 2 π Z n m ( ρ , θ ) . G _ n m * ( ρ , θ ) ρ d ρ d θ = δ m , m δ n , n
( z . w ) ( g , h ) = z g + w h ,
( Z n m ) ( ν , μ ) = ( Z n m ν ( ν , μ ) , Z n m μ ( ν , μ ) ) 2 .
α n m = n + 1 π 0 1 0 2 π W ( ρ , θ ) Z n m * ( ρ , θ ) ρ d ρ d θ ,
W = n , m α n m Z n m ,
α n m = 1 π 0 1 0 2 π W ( ρ , θ ) G _ n m * ( ρ , θ ) ρ d ρ d θ .
G _ n m = G n , 1 m ν + G n , 2 m μ = ( n + 1 ) Z n m ,
G n , 1 m ( ρ = 1 , θ ) cos θ + G n , 2 m ( ρ = 1 , θ ) sin θ = 0
G _ n m = U n m .
2 U n m = Δ U n m = ( n + 1 ) Z n m
U n m ν ( ρ = 1 , θ ) cos θ + U n m μ ( ρ = 1 , θ ) sin θ = 0
U n m = [ Z n + 2 m 4 ( n + 2 ) ( n + 1 ) Z n m 2 n ( n + 2 ) + Z n 2 m 4 n ] .
U n m = [ Z n + 2 m 4 ( n + 2 ) ( 3 n + 4 ) Z n m 4 n ( n + 2 ) ] , n = | m | ,
( ν ± i μ ) Z n m ( ν , μ ) = 2 l = 0 n | m | 2 ( n 2 l ) Z n 1 2 l m ± 1 ,
Z n 1 2 l m ± 1 ( 1 , θ ) = e i ( m ± 1 ) θ , l = 0 , 1 , 1 2 ( n | m | 1 ) ,
Z m 1 m 1 ( 1 , θ ) = e i ( m 1 ) θ , Z m 1 m + 1 ( 1 , θ ) = 0 ,
( ν + i μ ) Z n m = 2 l = 0 n m 2 ( n 2 l ) e i ( m + 1 ) θ 2 m e i ( m + 1 ) θ ,
( ν i μ ) Z n m = 2 l = 0 n m 2 ( n 2 l ) e i ( m 1 ) θ .
Z n m ν = 2 e i m θ cos θ l = 0 n m 2 ( n 2 l ) m e i ( m + 1 ) θ ,
Z n m μ = 2 e i m θ sin θ l = 0 n m 2 ( n 2 l ) 1 i m e i ( m + 1 ) θ .
l = 0 n m 2 ( n 2 l ) = n + m 2 ( n + m 2 + 1 ) .
U n m ν ( ρ = 1 , θ ) = = [ 1 4 ( n + 2 ) Z n m ν n + 1 2 n ( n + 2 ) Z n m ν + 1 4 n Z n 2 m ν ] = 2 D n m e i m θ cos θ + m E n e i ( m + 1 ) θ ,
D n m = n + 2 + m 2 ( n + 2 m 2 + 1 ) 4 ( n + 2 ) ( n + 1 ) n + m 2 ( n m 2 + 1 ) 2 n ( n + 2 ) + n 2 + m 2 ( n 2 m 2 + 1 ) 4 n = 0 ,
E n = 1 4 ( n + 2 ) n + 1 2 n ( n + 2 ) + 1 4 n = 0 .
Z n m ( ν , μ ) = ( Z n m ( ν , μ ) ) * .
U n m ν = 1 4 [ Z n + 1 m + 1 + Z n + 1 m 1 Z n 1 m + 1 Z n 1 m 1 ] ,
U n m μ = 1 4 i [ Z n + 1 m + 1 Z n + 1 m 1 Z n 1 m + 1 + Z n 1 m 1 ] .
G _ n m = ( U n m ν , U n m μ ) .
U n m ν = [ 1 4 ( n + 2 ) l = 0 n + 2 | m | 2 ( n + 2 2 l ) ( Z n + 1 2 l m + 1 + Z n + 1 2 l m 1 ) n + 1 2 n ( n + 2 ) l = 0 n | m | 2 ( n 2 l ) ( Z n 1 2 l m + 1 + Z n 1 2 l m 1 ) + 1 4 n l = 0 n 2 | m | 2 ( n 2 2 l ) ( Z n 3 2 l m + 1 + Z n 3 2 l m 1 ) ]
U n m ν = [ 1 4 ( n + 2 ) ( ( n + 2 ) ( Z n + 1 m + 1 + Z n + 1 m 1 + n ( Z n 1 m + 1 + Z n 1 m 1 ) ) n + 1 2 n ( n + 2 ) n ( Z n 1 m + 1 + Z n 1 m 1 ) ] = [ 1 4 ( Z n + 1 m + 1 + Z n 1 m + 1 ) 1 4 ( Z n 1 m + 1 + Z n 1 m 1 ) ] ,
U = [ Z m + 2 m 4 ( m + 2 ) C Z m m ] .
Z m m ( ρ , θ ) = ρ m e i m θ , Z m + 2 m ( ρ , θ ) = ( ( m + 2 ) ρ m + 2 ( m + 1 ) ρ m ) e i m θ ,
C = 3 m + 4 4 m ( m + 2 ) .
U ν = [ 1 4 ( m + 2 ) Z m + 2 m ν 3 m + 4 4 m ( m + 2 ) Z m m ν ] = [ 1 4 Z m + 1 m + 1 + 1 4 Z m + 1 m 1 + m 4 ( m + 2 ) Z m 1 m 1 3 m + 4 4 m ( m + 2 ) m Z m 1 m 1 ] = [ 1 4 Z m + 1 m + 1 + 1 4 Z m + 1 m 1 1 2 Z m 1 m 1 ] ,
U μ = 1 i [ 1 4 Z m + 1 m + 1 1 4 Z m + 1 m 1 + 1 2 Z m 1 m 1 ] .
W ν ± i W μ = n , m β ±   n m Z n m .
α n m = C n m φ n m C n + 2 m φ n + 2 m , n = | m | , | m | + 2 , ,
φ n m = 1 2 ( β + ) n + 1 m + 1 + ( β ) n 1 m 1 ,
C | m | m = 1 | m | , C n m = 1 2 n , n = | m | + 2 , | m | + 4 , .
α n m = 1 4 n ( β + ) n 1 m + 1 + 1 4 n ( β ) n 1 m 1 1 4 ( n + 2 ) ( β + ) n + 1 m + 1 1 4 ( n + 2 ) ( β ) n + 1 m 1 .
W ν = n , m 1 2 ( ( β + ) n m + ( β ) n m ) Z n m
W μ = n , m 1 2 i ( ( β + ) n m ( β ) n m ) Z n m .
1 π 0 1 0 2 π ( W v U n m * v + W μ U n m * μ ) ρ d ρ d θ = 1 8 π 0 1 0 2 π n , m ( ( β + ) n m + ( β ) n m ) Z n m × [ Z n + 1 m + 1 + Z n + 1 m 1 Z n 1 m + 1 Z n + 1 m 1 ] * ρ d ρ d θ = 1 8 π 0 1 0 2 π n , m ( ( β + ) n m ( β ) n m ) Z n m × [ Z n + 1 m + 1 Z n + 1 m 1 Z n 1 m + 1 + Z n + 1 m 1 ] * ρ d ρ d θ = 1 8 ( n + 2 ) ( ( β + ) n + 1 m + 1 + ( β ) n + 1 m + 1 + ( β + ) n + 1 m 1 + ( β ) n + 1 m 1 ) + 1 8 n ( ( β + ) n 1 m + 1 + ( β ) n 1 m + 1 + ( β + ) n 1 m 1 + ( β ) n 1 m 1 ) 1 8 ( n + 2 ) ( ( β + ) n + 1 m + 1 ( β ) n + 1 m + 1 ( β + ) n + 1 m 1 + ( β ) n + 1 m 1 ) + 1 8 n ( ( β + ) n 1 m + 1 ( β ) n 1 m + 1 ( β + ) n 1 m 1 + ( β ) n 1 m 1 ) ,

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