Abstract

The pyramid wave-front sensor in its original form works with a mechanical modulation that adapts the linear range of the sensor to seeing and sensing conditions. For adaptive optics systems working in an astronomical context, the way in which the aberrations produced by the atmospheric turbulence, which are not seen by the sensor owing to its limited temporal bandwidth, act as modulators is shown. These aberrations have the same effect of increasing the linear range and localizing the measurement as does mechanical modulation. The effect of residual wave-front aberrations is estimated for some example conditions of telescope diameter, system bandwidth, wind velocity, and Fried parameter.

© 2005 Optical Society of America

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References

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  1. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
    [CrossRef]
  2. S. Esposito, O. Feeney, A. Riccardi, “Laboratory test of a pyramid wavefront sensor,” in Adaptive Optical System Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 416–422 (2000).
    [CrossRef]
  3. A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
    [CrossRef]
  4. R. Ragazzoni, E. Diolati, E. Viard, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
    [CrossRef]
  5. S. Esposito, A. Riccardi, “Pyramid wavefront sensor behaviour in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
    [CrossRef]
  6. R. Ragazzoni, J. Farinato, “Sensitivity of a pyramidic wavefront sensor in closed loop adaptive optics,” Astron. Astrophys. 305, L23–L26 (1999).
  7. C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27–38 (2004).
    [CrossRef]
  8. J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
    [CrossRef]
  9. O. Feeney, “Theory and laboratory characterisation of novel wavefront sensor for adaptive optics systems,” Ph.D. dissertation (National University of Ireland, Galway, Ireland, 2001).
  10. R. G. Wilson, “Wavefront-error evaluation by mathematical analysis of experimental Foucault-test data,” Appl. Opt. 14, 2286–2297 (1975).
    [CrossRef] [PubMed]
  11. E. H. Linfoot, “On the theory of the zonal Foucault test,” Mon. Not. R. Astron. Soc. 108, 428–445 (1948).
  12. J. M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
    [CrossRef]
  13. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. A 55, 1427–1435 (1965).
    [CrossRef]
  14. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976). fault fault
    [CrossRef]

2004

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27–38 (2004).
[CrossRef]

2002

R. Ragazzoni, E. Diolati, E. Viard, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

2001

S. Esposito, A. Riccardi, “Pyramid wavefront sensor behaviour in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

1999

R. Ragazzoni, J. Farinato, “Sensitivity of a pyramidic wavefront sensor in closed loop adaptive optics,” Astron. Astrophys. 305, L23–L26 (1999).

1996

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

1995

1976

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976). fault fault
[CrossRef]

1975

1965

D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. A 55, 1427–1435 (1965).
[CrossRef]

1948

E. H. Linfoot, “On the theory of the zonal Foucault test,” Mon. Not. R. Astron. Soc. 108, 428–445 (1948).

Baruffolo, A.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Carbillet, M.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
[CrossRef]

Cecconi, M.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Conan, J. M.

Costa, J. B.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
[CrossRef]

Crimi, G.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Diolati, E.

R. Ragazzoni, E. Diolati, E. Viard, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Esposito, S.

S. Esposito, A. Riccardi, “Pyramid wavefront sensor behaviour in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
[CrossRef]

S. Esposito, O. Feeney, A. Riccardi, “Laboratory test of a pyramid wavefront sensor,” in Adaptive Optical System Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 416–422 (2000).
[CrossRef]

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Farinato, J.

R. Ragazzoni, J. Farinato, “Sensitivity of a pyramidic wavefront sensor in closed loop adaptive optics,” Astron. Astrophys. 305, L23–L26 (1999).

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
[CrossRef]

Feeney, O.

O. Feeney, “Theory and laboratory characterisation of novel wavefront sensor for adaptive optics systems,” Ph.D. dissertation (National University of Ireland, Galway, Ireland, 2001).

S. Esposito, O. Feeney, A. Riccardi, “Laboratory test of a pyramid wavefront sensor,” in Adaptive Optical System Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 416–422 (2000).
[CrossRef]

Feldt, M.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
[CrossRef]

Fini, L.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Fried, D. L.

D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. A 55, 1427–1435 (1965).
[CrossRef]

Ghedina, A.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
[CrossRef]

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Ghigo, M.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot, “On the theory of the zonal Foucault test,” Mon. Not. R. Astron. Soc. 108, 428–445 (1948).

Madec, P.-Y.

Marchetti, E.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Niero, T.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976). fault fault
[CrossRef]

Puga, E.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
[CrossRef]

Puglisi, A.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

Ragazzoni, R.

R. Ragazzoni, E. Diolati, E. Viard, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

R. Ragazzoni, J. Farinato, “Sensitivity of a pyramidic wavefront sensor in closed loop adaptive optics,” Astron. Astrophys. 305, L23–L26 (1999).

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
[CrossRef]

Riccardi, A.

S. Esposito, A. Riccardi, “Pyramid wavefront sensor behaviour in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

S. Esposito, O. Feeney, A. Riccardi, “Laboratory test of a pyramid wavefront sensor,” in Adaptive Optical System Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 416–422 (2000).
[CrossRef]

Rousset, G.

Verinaud, C.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
[CrossRef]

Vérinaud, C.

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27–38 (2004).
[CrossRef]

Viard, E.

R. Ragazzoni, E. Diolati, E. Viard, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

Wilson, R. G.

Appl. Opt.

Astron. Astrophys.

S. Esposito, A. Riccardi, “Pyramid wavefront sensor behaviour in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

R. Ragazzoni, J. Farinato, “Sensitivity of a pyramidic wavefront sensor in closed loop adaptive optics,” Astron. Astrophys. 305, L23–L26 (1999).

J. Mod. Opt.

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

J. Opt. Soc. Am. A

J. M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. A 55, 1427–1435 (1965).
[CrossRef]

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976). fault fault
[CrossRef]

Mon. Not. R. Astron. Soc.

E. H. Linfoot, “On the theory of the zonal Foucault test,” Mon. Not. R. Astron. Soc. 108, 428–445 (1948).

Opt. Commun.

R. Ragazzoni, E. Diolati, E. Viard, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27–38 (2004).
[CrossRef]

Other

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 288–298 (2003).
[CrossRef]

O. Feeney, “Theory and laboratory characterisation of novel wavefront sensor for adaptive optics systems,” Ph.D. dissertation (National University of Ireland, Galway, Ireland, 2001).

S. Esposito, O. Feeney, A. Riccardi, “Laboratory test of a pyramid wavefront sensor,” in Adaptive Optical System Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 416–422 (2000).
[CrossRef]

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolati, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, A. Puglisi, “On sky test of pyramid wavefront sensor,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 869–877 (2003).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Signal computation for the knife-edge wave-front sensor.

Fig. 2
Fig. 2

Square modulus of the error transfer function for a system with bandwidth νc.

Fig. 3
Fig. 3

Normalized residual variance σj/(Δj − Δj+1) for some Zernike polynomials as a function of a, the ratio of the wind speed over the system’s bandwidth multiplied by the telescope’s diameter [a = V/(νcDπ)].

Fig. 4
Fig. 4

Modulation function across the telescope aperture [top, in the pupil center and bottom, at point (x = 1.5 m, y = 0 m)] for several system bandwidths νc, for a telescope diameter of 4 m, a wind velocity of 30 m/s, and an r0 of 1 m.

Fig. 5
Fig. 5

Integrated modulation function across the telescope aperture for several points along the y = 0 chord, normalized with the integral at (x = 0 m, y = 0 m) for several system bandwidths νc, for a telescope diameter of 4 m, a wind velocity of 30 m/s, and an r0 of 1 m.

Fig. 6
Fig. 6

Modulation function for a telescope with diameter of 4 m with a wind velocity of 30 m/s, r0 = 1 m, and a system bandwidth of 15 Hz (for x = 0, y = 0) (solid curve). For comparison we have also plotted the mechanical modulation function for three modulation amplitudes.

Equations (32)

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

S x ( x , y ) = 1 π - B ( y ) B ( y ) d x ( p υ ) sin [ ϕ ( x , y ) - ϕ ( x , y ) ] ( x - x ) ,
S x ( x , y ) = 1 π - B ( y ) B ( y ) d x ( p υ ) sin [ ϕ ( x , y ) - ϕ ( x , y ) ] ( x - x ) × Δ t sin [ a 0 ( x - x ) ] a 0 ( x - x ) modulation .
ϕ ( x , y ; t ) = n c n ( t ) A n ( x , y ) ,
P ( c n ) = 1 σ n 2 π exp ( - c n 2 2 σ n ) .
ϕ ( x , y ; t ) = ϕ s ( x , y ) + j c j ( t ) A j ( x , y ) .
S x ( x , y ) = 1 T 0 T d t S x ( x , y ; t ) .
S x ( x , y ) = 1 C + - j d c j 1 π - B ( y ) B ( y ) d x × ( p υ ) sin [ ϕ s ( x , y ) - ϕ s ( x , y ) ] x - x × cos { j c j [ A j ( x , y ) - A j ( x , y ) ] } × exp ( - j c j 2 2 σ j ) .
S x ( x , y ) = 1 π - B ( y ) B ( y ) d x ( p υ ) sin [ ϕ s ( x , y ) - ϕ s ( x , y ) ] x - x × M ( x , x , y ) ,
M ( x , x , y ) Π j C - + d c j exp { i c j [ A j ( x , y ) - A j ( x , y ) ] } exp ( - c j 2 2 σ j ) .
M ( x , x , y ) = exp { - j σ j 2 [ A j ( x , y ) - A j ( x , y ) ] 2 } .
S x ( x , y ) C d ϕ s ( x , y ) d x
w Z j ( ν ) = 0.033 C N 2 d h V ( 2 π ) - 2 / 3 ( 2 π λ ) 2 × - + d f y | Z ˜ j ( ν V , f y ) | 2 [ ( ν V ) 2 + f y 2 ] - 11 / 6 ,
Z ˜ f ( f x , f x ) = ( n + 1 ) 1 / 2 2 J n + 1 ( π D f ) π D f × { 2 cos ( m θ ) m 0 2 sin ( m θ ) m 0 1 m = 0 .
σ j = - + T ( ν ) 2 w Z j ( ν ) d ν ,
| T ( ν ν c ) | 2 = ( ν ν c ) 2 θ [ 1 - ( ν ν c ) 2 ] + θ [ ( ν ν c ) 2 - 1 ] .
σ j = - + T ( ν ) 2 w Z j ( ν ) d ν - + w Z j ( ν ) d ν ( Δ j - Δ j + 1 ) ,
Δ j 0.2944 j - 3 / 2 ( D r 0 ) 5 / 3 .
u i ( x , y ) = 1 2 u 0 ( x , y ) ± i 2 π - B ( y ) B ( y ) d x ( p υ ) u 0 ( x , y ) ( x - x ) ,
u i + ( x , y ) 2 = [ 1 2 u 0 ( x , y ) - i 2 π - B ( y ) B ( y ) d x ( p υ ) u 0 ( x , y ) ( x - x ) ] × [ 1 2 u 0 * ( x , y ) + i 2 π - B ( y ) B ( y ) d x × ( p υ ) u 0 * ( x , y ) ( x - x ) ] ,
S x ( x , y ) = u i - 2 - u i + 2 u 0 2 .
S x = 2 i 4 π [ u 0 * - B ( y ) B ( y ) d x ( p υ ) u 0 ( x , y ) ( x - x ) - u 0 - B ( y ) B ( y ) d x ( p v ) u 0 * ( x , y ) ( x - x ) ] ,
S x ( x , y ) = 1 π - B ( y ) B ( y ) d x ( p v ) sin [ ϕ ( x , y ) - ϕ ( x , y ) ] ( x - x ) ,
ϕ ˜ ( x , y ; t ) = ϕ ( x , y ) + a 0 2 t Δ t x
w Z j ( ν ) = 4 ( n + 1 ) C 1 - + d f y J n + 1 ( π D f ) 2 π D f [ ( ν V ) 2 + f y 2 ] - 11 / 6 { 2 cos 2 ( m θ ) m 0 2 sin 2 ( m θ ) m 0 1 m = 0 ,
w Z j ( ν ) = 4 ( n + 1 ) C 1 V 14 / 3 π 2 D 2 C 2 - + d μ × J n + 1 [ ( π D / V ) ( ν 2 + μ 2 ) 1 / 2 ] 2 ( ν 2 + μ 2 ) 17 / 6 × { 2 cos 2 { m arcsin [ μ / ( ν 2 + μ 2 ) 1 / 2 ] } m 0 2 sin 2 { m arcsin [ μ / ( ν 2 + μ 2 ) 1 / 2 ] } m 0 1 m = 0 .
- + d ν w Z j ( ν ) = C 2 0 2 π d θ 0 + d r J n + 1 ( π D r / V ) 2 r 14 / 3 × { 2 cos 2 ( m θ ) m 0 2 sin 2 ( m θ ) m 0 1 m = 0
= 2 π C 2 0 + d r J n + 1 ( π D r / V ) 2 r 14 / 3 .
- + d ν w Z j ( ν ) = 2 π C 2 ( V π D ) - 11 / 3 0 + d ρ J n + 1 ( ρ ) 2 ρ 14 / 3 .
- + d ν | T ( ν ν c ) | 2 w Z j ( ν ) = C 2 ( π D V ) 11 / 3 × 0 2 π d θ 0 + d ρ T ( a ρ cos θ ) 2 J n + 1 ( ρ ) 2 ρ 14 / 3 × { 2 cos 2 ( m θ ) m 0 2 sin 2 ( m θ ) m 0 1 m = 0.
M ( x , x , y ) = exp { - j 2 σ j D 2 [ P j ( x , y ) - P j ( x , y ) ] 2 } .
σ j         f j ( a ) j - 3 / 2 ( D / r 0 ) 5 / 3 .
M ( x , x , y ) [ exp ( - 1 D 2 ) ( D r 0 ) 5 / 3 ] ( exp { - j f j ( a ) j - 3 / 2 × [ P j ( x , y ) - P j ( x , y ) ] 2 } ) .

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