Abstract

We study the effect of gaps between the segments on image quality of the extremely large ground-based telescope. We show that if the width of the gap changes arbitrarily from one intersegment boundary to another, the regular diffraction pattern is distorted and speckles appear. Two mechanisms producing irregularity in the gaps' distribution are considered: a random shift of the segment’s center and an error in the segment’s shape. The analytical approach allows an estimation of ensemble averaged characteristics of the point-spread function, such as a halo and high-order diffraction peaks. We deliver simple formulas for the quantitative estimation of the position and the intensity of the halo as well as the intensity of the diffraction peaks.

© 2007 Optical Society of America

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  1. "Thirty meter telescope passes conceptual design review," TMT press release, June 1, 2006.
  2. G. Monnet and R. Gilmozzi, "Status of the European ELT," The Messenger 123, 4-5 (2006).
  3. J. Nelson, T. Mast, and S. Faber, The Design of the Keck Observatory and Telescope, Keck Observatory Report 90 (W. M. Keck Library, 1985), pp. 5-1-5-44.
  4. P. Alvarez, J. Castro Lopez-Tarruella, and J. P. Rodriguez-Espinoza, "The GTC project: preparing first light," Proc. SPIE 6267, 626708 (2006).
    [CrossRef]
  5. G. Chanan, D. G. MacMartin, J. Nelson, and T. Mast, "Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation," Appl. Opt. 43, 1223-1232 (2004).
    [CrossRef] [PubMed]
  6. N. Yaitskova, K. Dohlen, P. Dierickx, and L. Montoya, "Mach-Zehnder interferometer for piston and tip-tilt sensing in segmented telescopes: theory and analytical treatment," J. Opt. Soc. Am. A 22, 1093-1105 (2005).
    [CrossRef]
  7. I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," Proc. SPIE 6267, 626726 (2006).
    [CrossRef]
  8. O. Lardiere, M. Carbillet, A. Riccardi, and P. Salinari, "High-contrast imaging with ELTs: effect of cophasing and AO residual errors on the PSF contrast," Proc. SPIE 5490, 516-526 (2004).
    [CrossRef]
  9. N. Yaitskova and C. Verinaud, "Adaptive optics correction of the wavefront distortions induced by segments misalignment in extremely large telescope," Proc. SPIE 6267, 62672N (2006).
    [CrossRef]
  10. M. Troy, I. Crossfield, G. Chanan, P. Dumont, J. Green, and B. Macintosh, "Effects of diffraction and static wavefront errors on high-contrast imaging from the Thirty Meter Telescope," in Proc. SPIE 6267, 626722C (2006).
  11. A. Sivaramakrishnan and N. Yaitskova, "Lyot coronagraphy on giant segmented telescopes," Astrophys. J. 626, L65-L68 (2005).
    [CrossRef]
  12. N. Yaitskova, K. Dohlen, and P. Dierickx, "Analytical study of diffraction effects in extremely large segmented telescopes," J. Opt. Soc. Am. A 20, 1563-1575 (2003).
    [CrossRef]
  13. T. Mast, J. Nelson, and G. Sommargren, "Primary mirror segment fabrication for CELT," Proc. SPIE 4003, 43-58 (2000).
    [CrossRef]
  14. M. Troy and G. Chanan, "Diffraction effects from giant segmented-mirror telescopes," Appl. Opt. 42, 3745-3753 (2003).
    [CrossRef] [PubMed]
  15. G. Zeider and E. Montgomery, "Diffraction effect with segmented aperture," Proc. SPIE 3356, 799-809 (1998).
    [CrossRef]
  16. N. Yaitskova and K. Dohlen, "Tip-tilt error for extremely large segmented telescopes: detailed theoretical point-spread-function analysis and numerical simulation results," J. Opt. Soc. Am. A 19, 1274-1285 (2002).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1986).
  18. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).
  19. C. Aime and R. Soummer, "The usefulness and limits of coronagraphy in the presence of pinned speckles," Astrophys. J. 612, L85-L88 (2004).
    [CrossRef]

2006 (5)

G. Monnet and R. Gilmozzi, "Status of the European ELT," The Messenger 123, 4-5 (2006).

P. Alvarez, J. Castro Lopez-Tarruella, and J. P. Rodriguez-Espinoza, "The GTC project: preparing first light," Proc. SPIE 6267, 626708 (2006).
[CrossRef]

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," Proc. SPIE 6267, 626726 (2006).
[CrossRef]

N. Yaitskova and C. Verinaud, "Adaptive optics correction of the wavefront distortions induced by segments misalignment in extremely large telescope," Proc. SPIE 6267, 62672N (2006).
[CrossRef]

M. Troy, I. Crossfield, G. Chanan, P. Dumont, J. Green, and B. Macintosh, "Effects of diffraction and static wavefront errors on high-contrast imaging from the Thirty Meter Telescope," in Proc. SPIE 6267, 626722C (2006).

2005 (2)

2004 (3)

G. Chanan, D. G. MacMartin, J. Nelson, and T. Mast, "Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation," Appl. Opt. 43, 1223-1232 (2004).
[CrossRef] [PubMed]

C. Aime and R. Soummer, "The usefulness and limits of coronagraphy in the presence of pinned speckles," Astrophys. J. 612, L85-L88 (2004).
[CrossRef]

O. Lardiere, M. Carbillet, A. Riccardi, and P. Salinari, "High-contrast imaging with ELTs: effect of cophasing and AO residual errors on the PSF contrast," Proc. SPIE 5490, 516-526 (2004).
[CrossRef]

2003 (2)

2002 (1)

2000 (1)

T. Mast, J. Nelson, and G. Sommargren, "Primary mirror segment fabrication for CELT," Proc. SPIE 4003, 43-58 (2000).
[CrossRef]

1998 (1)

G. Zeider and E. Montgomery, "Diffraction effect with segmented aperture," Proc. SPIE 3356, 799-809 (1998).
[CrossRef]

Aime, C.

C. Aime and R. Soummer, "The usefulness and limits of coronagraphy in the presence of pinned speckles," Astrophys. J. 612, L85-L88 (2004).
[CrossRef]

Alvarez, P.

P. Alvarez, J. Castro Lopez-Tarruella, and J. P. Rodriguez-Espinoza, "The GTC project: preparing first light," Proc. SPIE 6267, 626708 (2006).
[CrossRef]

Carbillet, M.

O. Lardiere, M. Carbillet, A. Riccardi, and P. Salinari, "High-contrast imaging with ELTs: effect of cophasing and AO residual errors on the PSF contrast," Proc. SPIE 5490, 516-526 (2004).
[CrossRef]

Castro Lopez-Tarruella, J.

P. Alvarez, J. Castro Lopez-Tarruella, and J. P. Rodriguez-Espinoza, "The GTC project: preparing first light," Proc. SPIE 6267, 626708 (2006).
[CrossRef]

Chanan, G.

Crossfield, I.

M. Troy, I. Crossfield, G. Chanan, P. Dumont, J. Green, and B. Macintosh, "Effects of diffraction and static wavefront errors on high-contrast imaging from the Thirty Meter Telescope," in Proc. SPIE 6267, 626722C (2006).

Dalton, G.

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," Proc. SPIE 6267, 626726 (2006).
[CrossRef]

Dierickx, P.

Dohlen, K.

Dumont, P.

M. Troy, I. Crossfield, G. Chanan, P. Dumont, J. Green, and B. Macintosh, "Effects of diffraction and static wavefront errors on high-contrast imaging from the Thirty Meter Telescope," in Proc. SPIE 6267, 626722C (2006).

Faber, S.

J. Nelson, T. Mast, and S. Faber, The Design of the Keck Observatory and Telescope, Keck Observatory Report 90 (W. M. Keck Library, 1985), pp. 5-1-5-44.

Gilmozzi, R.

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," Proc. SPIE 6267, 626726 (2006).
[CrossRef]

G. Monnet and R. Gilmozzi, "Status of the European ELT," The Messenger 123, 4-5 (2006).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1986).

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

Green, J.

M. Troy, I. Crossfield, G. Chanan, P. Dumont, J. Green, and B. Macintosh, "Effects of diffraction and static wavefront errors on high-contrast imaging from the Thirty Meter Telescope," in Proc. SPIE 6267, 626722C (2006).

Hook, I.

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," Proc. SPIE 6267, 626726 (2006).
[CrossRef]

Lardiere, O.

O. Lardiere, M. Carbillet, A. Riccardi, and P. Salinari, "High-contrast imaging with ELTs: effect of cophasing and AO residual errors on the PSF contrast," Proc. SPIE 5490, 516-526 (2004).
[CrossRef]

Macintosh, B.

M. Troy, I. Crossfield, G. Chanan, P. Dumont, J. Green, and B. Macintosh, "Effects of diffraction and static wavefront errors on high-contrast imaging from the Thirty Meter Telescope," in Proc. SPIE 6267, 626722C (2006).

MacMartin, D. G.

Mast, T.

G. Chanan, D. G. MacMartin, J. Nelson, and T. Mast, "Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation," Appl. Opt. 43, 1223-1232 (2004).
[CrossRef] [PubMed]

T. Mast, J. Nelson, and G. Sommargren, "Primary mirror segment fabrication for CELT," Proc. SPIE 4003, 43-58 (2000).
[CrossRef]

J. Nelson, T. Mast, and S. Faber, The Design of the Keck Observatory and Telescope, Keck Observatory Report 90 (W. M. Keck Library, 1985), pp. 5-1-5-44.

Monnet, G.

G. Monnet and R. Gilmozzi, "Status of the European ELT," The Messenger 123, 4-5 (2006).

Montgomery, E.

G. Zeider and E. Montgomery, "Diffraction effect with segmented aperture," Proc. SPIE 3356, 799-809 (1998).
[CrossRef]

Montoya, L.

Nelson, J.

G. Chanan, D. G. MacMartin, J. Nelson, and T. Mast, "Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation," Appl. Opt. 43, 1223-1232 (2004).
[CrossRef] [PubMed]

T. Mast, J. Nelson, and G. Sommargren, "Primary mirror segment fabrication for CELT," Proc. SPIE 4003, 43-58 (2000).
[CrossRef]

J. Nelson, T. Mast, and S. Faber, The Design of the Keck Observatory and Telescope, Keck Observatory Report 90 (W. M. Keck Library, 1985), pp. 5-1-5-44.

Riccardi, A.

O. Lardiere, M. Carbillet, A. Riccardi, and P. Salinari, "High-contrast imaging with ELTs: effect of cophasing and AO residual errors on the PSF contrast," Proc. SPIE 5490, 516-526 (2004).
[CrossRef]

Rodriguez-Espinoza, J. P.

P. Alvarez, J. Castro Lopez-Tarruella, and J. P. Rodriguez-Espinoza, "The GTC project: preparing first light," Proc. SPIE 6267, 626708 (2006).
[CrossRef]

Salinari, P.

O. Lardiere, M. Carbillet, A. Riccardi, and P. Salinari, "High-contrast imaging with ELTs: effect of cophasing and AO residual errors on the PSF contrast," Proc. SPIE 5490, 516-526 (2004).
[CrossRef]

Sivaramakrishnan, A.

A. Sivaramakrishnan and N. Yaitskova, "Lyot coronagraphy on giant segmented telescopes," Astrophys. J. 626, L65-L68 (2005).
[CrossRef]

Sommargren, G.

T. Mast, J. Nelson, and G. Sommargren, "Primary mirror segment fabrication for CELT," Proc. SPIE 4003, 43-58 (2000).
[CrossRef]

Soummer, R.

C. Aime and R. Soummer, "The usefulness and limits of coronagraphy in the presence of pinned speckles," Astrophys. J. 612, L85-L88 (2004).
[CrossRef]

Troy, M.

M. Troy, I. Crossfield, G. Chanan, P. Dumont, J. Green, and B. Macintosh, "Effects of diffraction and static wavefront errors on high-contrast imaging from the Thirty Meter Telescope," in Proc. SPIE 6267, 626722C (2006).

M. Troy and G. Chanan, "Diffraction effects from giant segmented-mirror telescopes," Appl. Opt. 42, 3745-3753 (2003).
[CrossRef] [PubMed]

Verinaud, C.

N. Yaitskova and C. Verinaud, "Adaptive optics correction of the wavefront distortions induced by segments misalignment in extremely large telescope," Proc. SPIE 6267, 62672N (2006).
[CrossRef]

Yaitskova, N.

Zeider, G.

G. Zeider and E. Montgomery, "Diffraction effect with segmented aperture," Proc. SPIE 3356, 799-809 (1998).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (2)

A. Sivaramakrishnan and N. Yaitskova, "Lyot coronagraphy on giant segmented telescopes," Astrophys. J. 626, L65-L68 (2005).
[CrossRef]

C. Aime and R. Soummer, "The usefulness and limits of coronagraphy in the presence of pinned speckles," Astrophys. J. 612, L85-L88 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

Proc. SPIE (7)

I. Hook, G. Dalton, and R. Gilmozzi, "Scientific requirements for a European ELT," Proc. SPIE 6267, 626726 (2006).
[CrossRef]

O. Lardiere, M. Carbillet, A. Riccardi, and P. Salinari, "High-contrast imaging with ELTs: effect of cophasing and AO residual errors on the PSF contrast," Proc. SPIE 5490, 516-526 (2004).
[CrossRef]

N. Yaitskova and C. Verinaud, "Adaptive optics correction of the wavefront distortions induced by segments misalignment in extremely large telescope," Proc. SPIE 6267, 62672N (2006).
[CrossRef]

M. Troy, I. Crossfield, G. Chanan, P. Dumont, J. Green, and B. Macintosh, "Effects of diffraction and static wavefront errors on high-contrast imaging from the Thirty Meter Telescope," in Proc. SPIE 6267, 626722C (2006).

P. Alvarez, J. Castro Lopez-Tarruella, and J. P. Rodriguez-Espinoza, "The GTC project: preparing first light," Proc. SPIE 6267, 626708 (2006).
[CrossRef]

T. Mast, J. Nelson, and G. Sommargren, "Primary mirror segment fabrication for CELT," Proc. SPIE 4003, 43-58 (2000).
[CrossRef]

G. Zeider and E. Montgomery, "Diffraction effect with segmented aperture," Proc. SPIE 3356, 799-809 (1998).
[CrossRef]

The Messenger (1)

G. Monnet and R. Gilmozzi, "Status of the European ELT," The Messenger 123, 4-5 (2006).

Other (4)

J. Nelson, T. Mast, and S. Faber, The Design of the Keck Observatory and Telescope, Keck Observatory Report 90 (W. M. Keck Library, 1985), pp. 5-1-5-44.

"Thirty meter telescope passes conceptual design review," TMT press release, June 1, 2006.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1986).

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

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Figures (11)

Fig. 1
Fig. 1

Telescope shape and the corresponding diffraction pattern. Top, pupil shape; middle, PSF for the filled aperture; bottom, PSF in the presence of the regular gaps. PSF is plotted in a logarithmic scale.

Fig. 2
Fig. 2

(a) Peaks classification and (b) plot of intensity. The angular distances are given for λ = 1.65 μ m , d = 1.21 m . The intensity is normalized by the intensity of the central peak.

Fig. 3
Fig. 3

Any segment can be shifted in x and y directions randomly. The amplitude of the shift is Δ r 0 .

Fig. 4
Fig. 4

Difference between PSF for the mirror with randomly shifted segments and the PSF for the mirror with all segments in nominal positions. The positive halo is accompanied by the drop of the intensity in the peaks. Parameters: segment size 1.21 m ; average gap, 10 mm ; shift amplitude, 50% of average gap size; wavelength, 1.65 μ m ; number of segments, 1000.

Fig. 5
Fig. 5

Averaged halo produced by the random shift of the segments: (a) 2D view on a linear scale, (b) projections on x and y axis. Parameters are the same as in Fig. 4.

Fig. 6
Fig. 6

Maximum intensity of the halo as a function of the average gap size. The amplitude of the random shift is 30%, 50%, and 80%. The rest of parameter is the same as in Fig. 4.

Fig. 7
Fig. 7

Relative drop of the intensity of the high-order peaks as a function of segment shift amplitude for the different groups of peaks.

Fig. 8
Fig. 8

(a) Segment size varies randomly between two concentric hexagons. (b) Average transmission function for the uniform distribution of the sizes.

Fig. 9
Fig. 9

Averaged halo produced by the irregularity in segment size. Dispersion of the size is 10 mm ; other parameters are the same as in Fig. 4.

Fig. 10
Fig. 10

Change of the peaks’ intensity due to fluctuations of the segment size as a function of normalized dispersion. The average gap is 10 mm . Other parameters are the same as in Fig. 4.

Fig. 11
Fig. 11

Explanatory schemes for the calculation of the correlation function: (a) 1D dimensional case, (b) 2D case of a hexagon.

Equations (53)

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F ( x ) = j = 1 N θ j ( x r j ) .
PSF ( w ) = 1 N j = 1 N [ exp ( i 2 π λ w r j ) A j A 1 A j θ j ( ξ ) exp ( i 2 π λ w ξ ) d 2 ξ ] 2 .
PSF ( w ) = ( A s A ) 2 1 A s θ s ( ξ ) exp ( i 2 π λ w ξ ) d 2 ξ 2 1 N j = 1 N exp ( i 2 π λ w r j ) 2 = ( 1 ω ) 4 PSF S ( w ) GF ( w ) .
I A 1 0.684 ω 2 , I B 1 0.444 ω 2 , I A 2 0.171 ω 2 ,
I C 1 0.335 ω 2 ,
I A 3 ω 4 , I B 2 0.444 ω 2 , I D 1 0.0944 ω 2 ,
I A 4 0.0427 ω 2 .
PSF seg shift ( w ) = ( A s A ) 2 1 A s θ s ( ξ ) exp ( i 2 π λ w ξ ) d 2 ξ 2 1 N j = 1 N exp [ i 2 π λ w ( r j + Δ r j ) ] 2 .
PSF seg shift ( w ) = ( 1 ω ) 4 PSF S ( w ) × 1 N 2 j = 1 N l = 1 N exp [ i 2 π λ w ( r j r l ) ] exp [ i 2 π λ w ( Δ r j Δ r l ) ] .
exp [ i 2 π λ w ( Δ r j Δ r l ) ] = { 1 , l = j exp ( i 2 π λ w Δ r ) 2 , l j . }
PSF seg shift ( w ) = ( 1 ω ) 4 PSF S ( w ) G F ( w ) exp ( i Δ r w 2 π λ ) 2 + 1 N ( 1 ω ) 4 PSF S ( w ) [ 1 exp ( i Δ r w 2 π λ ) 2 ] .
exp ( i Δ r w 2 π λ ) = exp ( i Δ r w 2 π λ ) = sinc ( Δ r 0 π w x λ ) sinc ( Δ r 0 π w y λ ) .
w peak 0.556 ( 1 + ω ) λ d ,
FWHM 0.568 ( 1 + ω ) λ d ,
I peak 0.102 η 2 ( 1 ω ) 2 N .
I A 1 0.684 ω 2 0.75 η 2 ω 2 , I B 1 0.444 ω 2 1.46 η 2 ω 2 ,
I A 2 0.171 ω 2 0.75 η 2 ω 2 , I C 1 0.335 ω 2 2.57 η 2 ω 2 ,
I A 3 ω 4 9.87 η 2 ω 4 , I B 2 0.444 ω 2 5.84 η 2 ω 2 ,
I D 1 0.094 ω 2 1.35 η 2 ω 2 , I A 4 0.0427 ω 2 0.75 η 2 ω 2 .
PSF segcut ( w ) = 1 N 2 j = 1 N l = 1 N exp [ i 2 π λ w ( r j r l ) ] × 1 A 2 θ j ( ξ ) θ l ( ξ ) exp [ i 2 π λ w ( ξ ξ ) ] d 2 ξ d 2 ξ .
θ j ( ξ ) θ l ( ξ ) = { θ ( ξ ) θ ( ξ ) , l = j θ ( ξ ) θ ( ξ ) , l j } ,
PSF segcut ( w ) = GF ( w ) 1 A θ ( ξ ) exp ( i 2 π λ w ξ ) d 2 ξ 2 + 1 N { 1 A 2 θ ( ξ ) θ ( ξ ) exp [ i 2 π λ w ( ξ ξ ) ] d 2 ξ d 2 ξ 1 A θ ( ξ ) exp ( i 2 π λ w ξ ) d 2 ξ 2 } .
t fl ( w , h ) = 1 A h θ ( ξ ) θ h ( ξ ) exp ( i 2 π λ w ξ ) d 2 ξ .
1 A h f ( ξ ) θ h ( ξ ) exp ( i 2 π λ w ξ ) d 2 ξ = 0 h F ( x ) x [ x 2 h 2 t ( w , x ) ] d x ,
t ( w , x )
= sin ( 3 α β ) sinc ( α 3 + β ) + sin ( 3 α + β ) sinc ( α 3 β ) 2 3 α
F fl ( x ) = { 1 , x d min x d max d min d max , x > d min 0 , x d max } ,
t fl ( w , h ) = d min 2 h 2 t ( w , d min ) + d min h x d max d min d max x [ x 2 h 2 t ( w , x ) ] d x .
F halo ( w ) = 1 A max 2 θ ( ξ ) θ ( ξ ) exp [ i 2 π λ w ( ξ ξ ) ] d 2 ξ d 2 ξ .
F halo ( w ) = 1 d max 4 0 d max { x 2 t ( w , x ) x [ x 2 t fl ( w , x ) ] + [ d max 2 t fl ( w , d max ) x 2 t fl ( w , x ) ] x [ x 2 t ( w , x ) ] } d x .
PSF segcut ( w ) = GF ( w ) ( d max d ) 4 t fl 2 ( w , d max ) + 1 N ( d max d ) 4 [ F halo ( w ) t fl 2 ( w , d max ) ] .
ω max = 1 d max d , ω = 1 ( d min + d max ) 2 d ,
η = ( d max d min ) d .
St ( ω , η ) St ( ω ) St ( ω ) η 2 6 10 7 .
FWHM 0.68 ( 1 + ω ) λ d ,
I peak η 2 ( 1 ω ) 2 3 N .
I A 1 0.684 ω 2 0.069 η 2 ω , I B 1 0.444 ω 2 0.111 η 2 ω ,
I A 2 0.171 ω 2 + 0.034 η 2 ω , I C 1 0.335 ω 2 0.048 η 2 ω ,
I A 3 ω 4 + 0.167 η 2 ω 2 , I B 2 0.444 ω 2 0.111 η 2 ω ,
I D 1 0.094 ω 2 + 0.026 η 2 ω , I A 4 0.0427 ω 2 0.017 η 2 ω .
PSF ( w ) = GF ( w ) B ( w ) + D ( w ) B ( w ) N .
St = ( N 1 N ) B ( 0 ) + D ( 0 ) B ( 0 ) N B ( 0 ) , for N 1 .
B ( w ) = ( 1 ω ) 4 PSF S ( w ) exp ( i Δ r j w 2 π λ ) 2 ,
D ( w ) = ( 1 ω ) 4 PSF S ( w ) .
B ( w ) = ( 1 ω max ) 4 t fl 2 ( w , d max ) ,
D ( w ) = ( 1 ω max ) 4 F halo ( w ) .
θ ( ξ ) θ ( ξ ) = { θ ( ξ ) , ξ < ξ θ ( ξ ) , ξ > ξ θ ( ξ ) = θ ( ξ ) , ξ = ξ } .
F halo ( w ) = 1 A max 2 θ ( ξ ) θ ( ξ ) exp [ i 2 π λ w ( ξ ξ ) ] d 2 ξ d 2 ξ
F halo ( w ) = 1 A max 2 θ ( ξ ) θ ( ξ ) θ max ( ξ ) θ max ( ξ ) exp [ i 2 π λ w ( ξ ξ ) ] d 2 ξ d 2 ξ .
F halo ( w ) = 1 A max 2 n = 1 K [ θ n ( ξ ) θ n 1 ( ξ ) ] exp ( i 2 π λ w ξ ) θ ( ξ ) θ ( ξ ) θ max ( ξ ) exp ( i 2 π λ w ξ ) d 2 ξ ,
F halo ( w ) = 1 A max 2 n = 1 K [ θ n ( ξ ) θ n 1 ( ξ ) ] exp ( i 2 π λ w ξ ) × { θ ( ξ ) θ n 1 ( ξ ) exp ( i 2 π λ w ξ ) d 2 ξ + θ ( ξ ) [ θ max ( ξ ) θ n 1 ( ξ ) ] exp ( i 2 π λ w ξ ) d 2 ξ } .
F halo ( w ) = 1 A max 2 n = 1 K [ A n t f l ( w , d n ) A n 1 t f l ( w , d n 1 ) ] A n 1 t ( w , d n 1 ) + [ A K t f l ( w , d K ) A n 1 t f l ( w , d n 1 ) ] [ A n t ( w , d n ) A n 1 t ( w , d n 1 ) ] ,
F halo ( w ) = 1 d max 4 0 d max { x 2 t ( w , x ) x [ x 2 t f l ( w , x ) ] + [ d max 2 t f l ( w , d max ) x 2 t f l ( w , x ) ] x [ x 2 t ( w , x ) ] } d x .

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