Abstract

The root-mean-square (rms) of the residual wavefront, after propagation through atmospheric turbulence and corrected from Zernike polynomials, has been derived for the von Kármán turbulence model. The rms for any location in the telescope pupil and the pupil average rms have been calculated. It is shown that the residual rms on the edge of the pupil can be up to 35% larger than the pupil average residual rms. The results are useful to estimate the required rms stroke of deformable mirror (DM) actuators when they are used as a second stage of correction either in a tip–tilt, single-DM configuration or in a tip–tilt, two-DM (woofer–tweeter) setup.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2007 (1)

2006 (6)

P. L. Wizinowich, J. Chin, E. Johansson, S. Kellner, R. Lafon, D. Le Mignant, C. Neyman, P. Stomski, D. Summers, R. Sumner, and M. Van Dam, "Adaptive optics developments at Keck Observatory," Proc. SPIE 6272, DOI:10.1117/12.672337 (2006).
[CrossRef]

G. Herriot, P. Hickson, B. L. Ellerboek, D. A. Andersen, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," Proc. SPIE 6272, DOI: 10.1117/12.672337 (2006).
[CrossRef]

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, "Adaptive optics simulations for the European Extremely Large Telescope," Proc. SPIE 6272, DOI: 10.1117/12.670187 (2006).
[CrossRef]

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, "Control of a woofer tweeter system of deformable mirrors," Proc. SPIE 6274, DOI: 10.1117/12.672451 (2006).
[CrossRef]

R. Conan, P. Hampton, C. Bradley, and O. Keskin, "The woofer-tweeter experiment," Proc. SPIE 6272, DOI: 10.1117/12.670720 (2006).
[CrossRef]

L. Jolissaint, J.-P. Véran, and R. Conan, "Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach," J. Opt. Soc. Am. A 23, 382-394 (2006).
[CrossRef]

2005 (1)

M. Carbillet, C. Vérinaud, B. Femenía, A. Riccardi, and L. Fini, "Modelling astronomical adaptive optics--I. The software package CAOS," Mon. Not. R. Astron. Soc. 356, 1263-1275 (2005).
[CrossRef]

2004 (2)

M. C. Britton, "Arroyo," Proc. SPIE 5497, 290-300 (2004).
[CrossRef]

Y. Clenet, M. E. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. N. Hubin, "NACO performance: status after 2 years of operation," Proc. SPIE 5490, 107-117 (2004).
[CrossRef]

2003 (2)

V. Kornilov, A. Tokovinin, O. Voziakova, A. Zaitsev, N. Shatsky, S. Potanin, and M. Sarazin, "MASS: a monitor of the vertical turbulence distribution," Proc. SPIE 4839, 837-845 (2003).
[CrossRef]

A. Tokovinin, S. Baumont, and J. Vasquez, "Statistics of turbulence profile at Cerro Tololo," Mon. Not. R. Astron. Soc. 340, 52-58 (2003).
[CrossRef]

2000 (1)

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, "Optical parameters relevant for high angular resolution at paranal from GSM instrument and surface layer contribution," Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

1998 (2)

R. Avila, J. Vernin, and S. Cuevas, "Turbulence profiles with generalized scidar at San Pedro Màrtir Observatory and isoplanatism studies," Publ. Astron. Soc. Pac. 110, 1106-1116 (1998).
[CrossRef]

V. Klückers, N. Wooder, T. Nicholls, M. Adcock, I. Munro, and J. Dainty, "Profiling of atmospheric turbulence strength and velocity using a generalised SCIDAR technique," Astron. Astrophys. Suppl. Ser. 130, 141-155 (1998).
[CrossRef]

1997 (1)

1995 (2)

1994 (1)

1993 (2)

R. Sasiela and J. Shelton, "Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism," J. Opt. Soc. Am. A 10, 646-660 (1993).
[CrossRef]

R. Sasiela and J. Shelton, "Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations," J. Math. Phys. 34, 2572-2617 (1993).
[CrossRef]

1992 (1)

1991 (3)

D. Winker, "Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence," J. Opt. Soc. Am. A 8, 1568-1573 (1991).
[CrossRef]

A. Kolmogorov, "Dissipation of energy in the locally isotropic turbulence," Proc. R. Soc. London, Ser. A 434, 15-17 (1991).
[CrossRef]

A. Kolmogorov, "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers," Proc. R. Soc. London, Ser. A 434, 9-13 (1991).
[CrossRef]

1979 (1)

1978 (1)

1976 (3)

1974 (1)

1972 (2)

1971 (1)

1967 (1)

G. Heidbreder, "Image degradation with random wavefront tilt compensation," IEEE Trans. Antennas Propag. AP-15, 90-98 (1967).
[CrossRef]

1966 (1)

1958 (1)

A. Favre, J. Caviglio, and R. Dumas, "Further space-time correlations of velocity in a turbulent boundary layer," J. Fluid Mech. 3, 344-356 (1958).
[CrossRef]

1957 (1)

A. Favre, J. Caviglio, and R. Dumas, "Space-time correlations and spectra in a turbulent boundary layer," J. Fluid Mech. 2, 313-342 (1957).
[CrossRef]

Appl. Opt. (7)

Astron. Astrophys. Suppl. Ser. (2)

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, "Optical parameters relevant for high angular resolution at paranal from GSM instrument and surface layer contribution," Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

V. Klückers, N. Wooder, T. Nicholls, M. Adcock, I. Munro, and J. Dainty, "Profiling of atmospheric turbulence strength and velocity using a generalised SCIDAR technique," Astron. Astrophys. Suppl. Ser. 130, 141-155 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

G. Heidbreder, "Image degradation with random wavefront tilt compensation," IEEE Trans. Antennas Propag. AP-15, 90-98 (1967).
[CrossRef]

J. Fluid Mech. (2)

A. Favre, J. Caviglio, and R. Dumas, "Space-time correlations and spectra in a turbulent boundary layer," J. Fluid Mech. 2, 313-342 (1957).
[CrossRef]

A. Favre, J. Caviglio, and R. Dumas, "Further space-time correlations of velocity in a turbulent boundary layer," J. Fluid Mech. 3, 344-356 (1958).
[CrossRef]

J. Math. Phys. (1)

R. Sasiela and J. Shelton, "Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations," J. Math. Phys. 34, 2572-2617 (1993).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (6)

Mon. Not. R. Astron. Soc. (2)

A. Tokovinin, S. Baumont, and J. Vasquez, "Statistics of turbulence profile at Cerro Tololo," Mon. Not. R. Astron. Soc. 340, 52-58 (2003).
[CrossRef]

M. Carbillet, C. Vérinaud, B. Femenía, A. Riccardi, and L. Fini, "Modelling astronomical adaptive optics--I. The software package CAOS," Mon. Not. R. Astron. Soc. 356, 1263-1275 (2005).
[CrossRef]

Proc. R. Soc. London, Ser. A (2)

A. Kolmogorov, "Dissipation of energy in the locally isotropic turbulence," Proc. R. Soc. London, Ser. A 434, 15-17 (1991).
[CrossRef]

A. Kolmogorov, "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers," Proc. R. Soc. London, Ser. A 434, 9-13 (1991).
[CrossRef]

Proc. SPIE (8)

Y. Clenet, M. E. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. N. Hubin, "NACO performance: status after 2 years of operation," Proc. SPIE 5490, 107-117 (2004).
[CrossRef]

P. L. Wizinowich, J. Chin, E. Johansson, S. Kellner, R. Lafon, D. Le Mignant, C. Neyman, P. Stomski, D. Summers, R. Sumner, and M. Van Dam, "Adaptive optics developments at Keck Observatory," Proc. SPIE 6272, DOI:10.1117/12.672337 (2006).
[CrossRef]

G. Herriot, P. Hickson, B. L. Ellerboek, D. A. Andersen, T. Davidge, D. A. Erickson, I. P. Powell, R. Clare, L. Gilles, C. Boyer, M. Smith, L. Saddlemyer, and J.-P. Véran, "NFIRAOS: TMT narrow field near-infrared facility adaptive optics," Proc. SPIE 6272, DOI: 10.1117/12.672337 (2006).
[CrossRef]

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, "Adaptive optics simulations for the European Extremely Large Telescope," Proc. SPIE 6272, DOI: 10.1117/12.670187 (2006).
[CrossRef]

M. C. Britton, "Arroyo," Proc. SPIE 5497, 290-300 (2004).
[CrossRef]

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, "Control of a woofer tweeter system of deformable mirrors," Proc. SPIE 6274, DOI: 10.1117/12.672451 (2006).
[CrossRef]

R. Conan, P. Hampton, C. Bradley, and O. Keskin, "The woofer-tweeter experiment," Proc. SPIE 6272, DOI: 10.1117/12.670720 (2006).
[CrossRef]

V. Kornilov, A. Tokovinin, O. Voziakova, A. Zaitsev, N. Shatsky, S. Potanin, and M. Sarazin, "MASS: a monitor of the vertical turbulence distribution," Proc. SPIE 4839, 837-845 (2003).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

R. Avila, J. Vernin, and S. Cuevas, "Turbulence profiles with generalized scidar at San Pedro Màrtir Observatory and isoplanatism studies," Publ. Astron. Soc. Pac. 110, 1106-1116 (1998).
[CrossRef]

Other (12)

A. Tokovinin and V. Kornilov, "Measuring turbulence profile from scintillations of single stars," in Astronomical Site Evaluation in the Visible and Radio Range, J.Vernin, C.Muñóz-Tuñón, and Z.Benkhaldoun, eds. (Astronomical Society of the Pacific, 2002), Vol. 266, pp. 104-112.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

F. Roddier, "The effect of atmospheric turbulence in optical astronomy," in Progress in Optics, E.Wolf, ed. (North-Holland, 1981), Vol. XIX, pp. 281-376.
[CrossRef]

R. Avila and J. Vernin, "Mechanism of formation of atmospheric turbulence relevant for optical astronomy," in Interstellar Turbulence, J.Franco and A.Carramiñna, eds. (Cambridge U. Press, 1998).

J.-P. Véran, F. Rigaut, J. Stoesz, G. Herriot, and B. Ellerbroek, "Preliminary commissioning results of Altair," in Science with Adaptive Optics, W.Brandner and M.E.Kasper, eds. (Springer, 2005), pp. 19-25.
[CrossRef]

J. Mariotti and G. D. Benedetto, "Pathlength stability of synthetic aperture telescopes: the case of the 25 cm CERGA interferometer," in Very Large Telescopes, Their Instrumentation and Programs, M.Ulrich and K.Jear, eds. (International Astronomical Union, 1984), Vol. 79, pp. 257-265.

V. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, 1961).

V. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).

L. Landau and E. Lifchitz, Mécanique Des Fluides (Editions MIR, 1971).

U. Frish, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge U. Press, 1995).

M. Lesieur, La Turbulence (Presses Universitaires de Grenoble, 1994).

R. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (Springer-Verlag, 1994).

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Figures (8)

Fig. 1
Fig. 1

Phase structure function for L 0 = 25 m .

Fig. 2
Fig. 2

Map of normalized residual phase rms when the first eight Zernike polynomials are successively removed from the turbulent phase for D = 30 m and L 0 = 60 m .

Fig. 3
Fig. 3

Radial cuts of residual phase rms for all Zernike polynomials from radial order 3 to 6 fully removed and for D = 30 m and L 0 = 60 m .

Fig. 4
Fig. 4

Plot of σ ϵ ( 1 , 3 ) versus D L 0 . The percentage of stroke increase with respect to Δ n 1 2 is also superimposed for D L 0 = 0 , 0.5, 1, and 2.

Fig. 5
Fig. 5

Radial cuts of σ ϵ ( ρ , 3 ) for infinite L 0 (solid curve), for a diameter twice the length of L 0 (dotted–dashed curve), and for a diameter half the length of L 0 (dashed curve).

Fig. 6
Fig. 6

Plots of σ ϵ ( 1 , N n ) as a function of n for infinite L 0 (circles), for a diameter twice the length of L 0 (diamonds), for a diameter equal to the length of L 0 (dots), and for a diameter half the length of L 0 (squares).

Fig. 7
Fig. 7

Plots of Δ ( n + 1 ) ( n + 2 ) 2 as a function of n for infinite L 0 (circles), for a diameter half the length of L 0 (diamonds), for a diameter equal to the length of L 0 (dots), and for a diameter twice the length of L 0 (squares).

Fig. 8
Fig. 8

Maps of ς ϵ ( r 1 , r 2 , 3 ) in the case of 61 actuators across the pupil diameter, where r 1 is the vector of coordinates of all the actuators and r 2 = r 1 + ( d , 0 ) with d = D 60 .

Equations (60)

Equations on this page are rendered with MathJax. Learn more.

D v ( r ) = C v r 2 3 ,
D T ( r ) = C T 2 r 2 3 , l 1 r L 0 ,
L 0 Δ z 0 α 3 4 ,
C T 2 = α L 0 4 3 ( d T d z ) 2 ,
D n ( r ) = 2 Γ ( 7 6 ) π C n 2 L 0 2 3 [ 1 2 2 3 Γ ( 1 3 ) ( 2 π r L 0 ) 1 3 K 1 3 ( 2 π r L 0 ) ] ,
W n ( f ) = Γ ( 11 6 ) Γ ( 7 6 ) 2 2 3 π 8 3 Γ ( 1 3 ) C n 2 ( f 2 + f 0 2 ) 11 6 ,
Γ ( 11 6 ) Γ ( 7 6 ) 2 2 3 π 8 3 Γ ( 1 3 ) = 9.7 × 10 3 .
W φ ( f ) = k 2 δ z W n ( f ) ,
C φ ( r ) = F 1 [ W φ ( f ) ] ,
C φ ( r ) = 9.7 × 10 3 k 2 δ z C n 2 2 π 0 + d f f ( f 2 + f 0 2 ) 11 6 J 0 ( 2 π r f ) ,
C φ ( r ) = Γ ( 7 6 ) 2 π 5 3 Γ ( 1 3 ) k 2 δ z C n 2 f 0 5 3 ( 2 π r f 0 ) 5 6 K 5 6 ( 2 π r f 0 ) ,
C φ ( r ) = 0.036 k 2 d z C n 2 ( z ) f 0 ( z ) 5 3 ( 2 π r f 0 ( z ) ) 5 6 K 5 6 ( 2 π r f 0 ( z ) ) .
C φ ( r ) = 0.036 f 0 5 3 ( 2 π r f 0 ) 5 6 K 5 6 ( 2 π f 0 r ) k 2 d z C n 2 ( z ) .
C φ ( r ) = Γ ( 11 6 ) 2 5 6 π 8 3 [ 24 5 Γ ( 6 5 ) ] 5 6 ( r 0 f 0 ) 5 3 ( 2 π r f 0 ) 5 6 K 5 6 ( 2 π f 0 r ) ,
k 2 d z C n 2 ( z ) = 5 Γ ( 2 3 ) π Γ ( 1 6 ) [ 24 5 Γ ( 6 5 ) ] 5 6 r 0 5 3 .
W φ ( f ) = Γ ( 11 6 ) Γ ( 7 6 ) 2 2 3 π 8 3 Γ ( 1 3 ) ( f 2 + f 0 2 ) 11 6 k 2 d z C n 2 ( z ) = 0.0229 r 0 5 3 ( f 2 + f 0 2 ) 11 6 ,
0.0229 = Γ 2 ( 11 6 ) 2 π 11 3 [ 24 5 Γ ( 6 5 ) ] .
σ φ 2 = C φ ( 0 ) = Γ ( 11 6 ) Γ ( 5 6 ) 2 π 8 3 [ 24 5 Γ ( 6 5 ) ] 5 6 ( r 0 f 0 ) 5 3 ,
D φ ( r ) = 2 [ σ φ 2 C φ ( r ) ] = 2 Γ ( 11 6 ) 2 5 6 π 8 3 [ 24 5 Γ ( 6 5 ) ] 5 6 ( r 0 f 0 ) 5 3 × [ Γ ( 5 6 ) 2 1 6 ( 2 π r f 0 ) 5 6 K 5 6 ( 2 π r f 0 ) ] .
K 5 6 ( 2 π r f 0 ) = Γ ( 5 6 ) 2 11 6 ( 2 π r f 0 ) 5 6 + Γ ( 5 6 ) 2 1 6 ( 2 π r f 0 ) 5 6 .
D φ ( r ) = 2 [ 24 5 Γ ( 6 5 ) ] 5 6 ( r r 0 ) 5 3 = 6.88 ( r r 0 ) 5 3 ,
{ Z i even ( r R , γ ) = Z i odd ( r R , γ ) = Z i ( r R , γ ) = } n + 1 R n m ( r R ) { { 2 cos ( m γ ) 2 sin ( m γ ) } m 0 1 m = 0 } ,
R n m ( r R ) = s = 0 ( n m ) 2 ( 1 ) s ( n s ) ! s ! [ ( n + m ) 2 s ] ! [ ( n m ) 2 s ] ! ( r R ) n 2 s ,
d r ( r R ) Z i ( r R , γ ) Z j ( r R , γ ) = δ i j ,
δ i j = 1 if i = j = 0 elsewhere ,
( r R ) = 1 π R 2 if ( r R ) 1 = 0 elsewhere .
Q i ( f , ϕ ) = d r ( r R ) Z i ( r R , γ ) exp ( 2 j π f r ) ,
{ Q i even ( f , ϕ ) = Q i odd ( f , ϕ ) = Q i ( f , ϕ ) = } n + 1 J n + 1 ( 2 π f R ) π f R × { { ( 1 ) ( n + m ) 2 j m 2 cos ( m ϕ ) ( 1 ) ( n + m ) 2 j m 2 sin ( m ϕ ) } m 0 ( 1 ) n 2 m = 0 } ,
φ ( r , γ ) = i = 1 a i Z i ( r R , γ ) .
a i = d r ( r R ) Z i ( r R , γ ) φ ( r , γ ) .
δ N ( r ) = φ ϵ ( r ) 2 = φ ( r ) i = 1 N a i Z i ( r R ) 2 = φ ( r ) 2 + i = 1 N a i Z i ( r R ) 2 2 i = 1 N Z i ( r R ) φ ( r ) a i = φ ( r ) 2 + i = 1 N i = 1 N a i a i Z i ( r R ) Z i ( r R ) 2 i = 1 N Z i ( r R ) C φ a i ( r ) ,
a i a i = d f W φ ( f ) Q i ( f , ϕ ) Q i ( f , ϕ ) .
C φ a i ( r ) = d r 1 ( r 1 R ) Z i ( r 1 R , γ 1 ) φ ( r 1 ) φ ( r ) ,
C φ a i ( r ) = d f W φ ( f ) Q i ( f , ϕ ) exp ( 2 j π f r ) .
Δ N = d r ( r R ) φ ϵ ( r ) 2 = φ ( r ) 2 i = 1 N a i 2 ,
a i 2 = d f W φ ( f ) Q i ( f , ϕ ) 2 ,
Δ N = 2 Γ ( 11 6 ) π 3 2 [ 24 5 Γ ( 6 5 ) ] 5 6 ( D r 0 ) 5 3 i = 1 N ( n i + 1 ) × { Γ [ 5 6 + n i , 7 3 , 11 6 23 6 + n i , 17 6 ] F 3 2 [ 11 6 , 7 3 ; ( π D f 0 ) 2 23 6 + n i , 17 6 , 11 6 n i ] + ( π D f 0 ) 2 n i 5 3 Γ [ 1 + n i , 5 6 n i , 3 2 + n i 3 + 2 n i , 2 + n i ] × [ F 3 2 [ 1 + n i , 3 2 + n i ; ( π D f 0 ) 2 3 + 2 n i , 2 + n i , 1 6 + n i ] δ i 1 ] } ,
D φ ϵ ( r 1 , r 2 ) = φ ϵ ( r 1 ) φ ϵ ( r 2 ) 2 = [ φ ( r 1 ) i = 1 N a i Z i ( r 1 R ) ] [ φ ( r 2 ) i = 1 N a i Z i ( r 2 R ) ] 2 = D φ ( r 2 r 1 ) + i = 1 N i = 1 N a i a i [ Z i ( r 1 R ) Z i ( r 2 R ) ] [ Z i ( r 1 R ) Z i ( r 2 R ) ] 2 i = 1 N [ Z i ( r 1 R ) Z i ( r 2 R ) ] [ C φ a i ( r 1 ) C φ a i ( r 2 ) ] .
σ ϵ ( ρ , N ) = [ δ N ( r ) Δ N ] 1 2 = [ φ ϵ ( r ) 2 d r ( r R ) φ ϵ ( r ) 2 ] 1 2 ,
ς ϵ ( r 1 , r 2 , N ) = [ D φ ϵ ( r 1 , r 2 ) Δ N ] 1 2 .
a i a i = Γ 2 ( 11 6 ) π 14 3 [ 24 5 Γ ( 6 5 ) ] 5 6 r 0 5 3 f 0 11 3 R 2 × ( n + 1 ) ( n + 1 ) ( 1 ) ( n + n m m ) 2 δ m m × 0 + d x x [ 1 + ( x π D f 0 ) 2 ] 11 6 J n + 1 ( x ) J n + 1 ( x ) , if i i is even or m = m = 0 = 0 if i i is odd .
a i a i = 1.16 ( D r 0 ) 5 3 ( n + 1 ) ( n + 1 ) ( 1 ) ( n + n 2 m ) 2 δ m m × { Γ [ 3 + n + n 2 , 2 + n + n 2 , 1 + n + n 2 , 5 6 n + n 2 3 + n + n , 2 + n , 2 + n ] × ( π D f 0 ) n + n 5 3 F 4 3 [ 3 + n + n 2 , 2 + n + n 2 , 1 + n + n 2 ; ( π D f 0 ) 2 3 + n + n , 2 + n , 2 + n , 1 6 + n + n 2 ] + Γ [ n + n 2 5 6 , 7 3 , 17 6 , 11 6 n + n 2 + 23 6 , n n 2 + 17 6 , n n 2 + 17 6 ] × F 4 3 [ 7 3 , 17 6 , 11 6 ; ( π D f 0 ) 2 n + n 2 + 23 6 , n n 2 + 17 6 , n n 2 + 17 6 , 11 6 n + n 2 ] } , if i i is even or m = m = 0 = 0 if i i is odd ,
1.16 = 2 Γ ( 11 6 ) π 3 2 [ 24 5 Γ ( 6 5 ) ] 5 6 ,
Γ [ a 1 , , a p b 1 , , b q ] = Γ [ ( a ) ( b ) ] = n = 1 p Γ ( a n ) m = 1 q Γ ( b m ) .
F q p [ ( a ) ; z ( b ) ] = Γ [ ( b ) ( a ) ] k = 0 + Γ [ ( a ) + k ( b ) + k ] z k k ! .
a i 2 = 1.16 ( D r 0 ) 5 3 ( n + 1 ) × { Γ [ 5 6 + n , 7 3 , 11 6 23 6 + n , 17 6 ] F 3 2 [ 11 6 , 7 3 ; ( π D f 0 ) 2 23 6 + n , 17 6 , 11 6 n ] + ( π D f 0 ) 2 n 5 3 Γ [ 1 + n , 5 6 n , 3 2 + n 3 + 2 n , 2 + n ] × F 3 2 [ 1 + n , 3 2 + n ; ( π D f 0 ) 2 3 + 2 n , 2 + n , 1 6 + n ] } .
a i a i = 2.25 ( D r 0 ) 5 3 ( n + 1 ) ( n + 1 ) ( 1 ) ( n + n 2 m ) 2 δ m m × Γ [ 5 6 + n + n 2 23 6 + n + n 2 , 17 6 + n n 2 , 17 6 + n n 2 ] , if i i is even or m = m = 0 = 0 if i i is odd ,
2.25 = Γ ( 14 3 ) Γ 2 ( 11 6 ) 2 8 3 π [ 24 5 Γ ( 6 5 ) ] 5 6 .
{ C φ a i even ( r ) = C φ a i odd ( r ) = C φ a i ( r ) = } 2 n + 1 R I n , m ( r ) × { { ( 1 ) ( n m ) 2 2 cos ( m θ ) ( 1 ) ( n m ) 2 2 sin ( m θ ) } m 0 ( 1 ) n 2 m = 0 , }
I n , m ( r ) = 0 + d f W φ ( f ) J n + 1 ( 2 π f R ) J m ( 2 π f r ) ,
I n , m ( r ) = 0.0229 r 0 5 3 ( π R ) 8 3 2 Γ ( 11 6 ) l = 0 + p = 0 + ( 1 ) l l ! ( 1 ) p p ! × { Γ [ l + p + n + m 2 + 1 , 5 6 l p n + m 2 l + n + 2 , p + m + 1 ] × ( π R f 0 ) 2 l + n 5 3 ( π r f 0 ) 2 p + m + Γ [ p l + n + m 2 5 6 , l + 11 6 l p + n m 2 + 17 6 , p + m + 1 ] × ( π R f 0 ) 2 l ( r R ) 2 p + m } .
I n , m ( r ) = 0.0229 r 0 5 3 ( π R ) 8 3 2 ( r R ) m Γ [ n + m 2 5 6 n m 2 + 17 6 , m + 1 ] × F 1 2 [ n + m 2 5 6 , m n 2 11 6 ; ( r R ) 2 m + 1 ] .
δ N ( r ) = φ ( r ) 2 P + i = 1 N i = 1 + δ i 1 N a i a i Z i ( r R ) Z i ( r R ) 2 i = 2 N Z i ( r R ) C φ a i ( r ) ,
φ ( r ) 2 P = φ ( r ) 2 + a 1 2 2 C φ a 1 ( r ) = a 1 * 2 2 C φ a 1 * ( r ) ,
a 1 * 2 = a 1 2 φ ( r ) 2 ,
C φ a 1 * ( r ) = C φ a 1 ( r ) φ ( r ) 2 .
a 1 * 2 = 2 π 0 + d f f W φ ( f ) { [ 2 J 1 ( 2 π R f ) 2 π R f ] 2 1 } = 4 Γ 2 ( 11 6 ) π [ 24 5 Γ ( 6 5 ) ] 5 6 ( 2 R r 0 ) 5 3 × 0 + d x x x 11 3 [ J 1 2 ( x ) x 2 4 ] ,
a 1 * 2 = 512 2057 Γ [ 5 6 , 7 3 ] π 3 2 [ 24 5 Γ ( 6 5 ) ] 5 6 ( 2 R r 0 ) 5 3 = 1.0324 ( 2 R r 0 ) 5 3 ,
C φ a 1 * ( r ) = 2 π 0 + d f f W φ ( f ) [ 2 J 1 ( 2 π R f ) 2 π R f J 0 ( 2 π r f ) 1 ] = 2 π Γ 2 ( 11 6 ) [ 24 5 Γ ( 6 5 ) ] 5 6 ( 2 R r 0 ) 5 3 × 0 + d x x x 8 3 [ J 1 ( x ) J 0 ( r R x ) x 2 ] ,
C φ a 1 * ( r ) = Γ 2 ( 11 6 ) 2 8 3 π [ 24 5 Γ ( 6 5 ) ] 5 6 ( 2 R r 0 ) 5 3 × m = 0 ( 1 ) m m ! Γ [ m 5 6 17 6 m , 1 + m ] ( r R ) 2 m = Γ 2 ( 11 6 ) 4 [ 24 5 Γ ( 6 5 ) ] 5 6 ( 2 R r 0 ) 5 3 × Γ [ 5 6 17 6 ] F 1 2 [ 5 6 , 11 6 ; ( r R ) 2 1 ] .

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