Abstract

Conventional adaptive-optics systems correct the wavefront by adjusting a deformable mirror (DM) based on measurements of the phase aberration taken in a pupil plane. The ability of this technique, known as phase conjugation, to correct aberrations is normally limited by the maximum spatial frequency of the DM. In this paper we show that conventional phase conjugation is not able to achieve the dark nulls needed for high-contrast imaging. Linear combinations of high frequencies in the aberration at the pupil plane “fold” and appear as low-frequency aberrations at the image plane. After describing the frequency-folding phenomenon, we present an alternative optimized solution for the shape of the deformable mirror based on the Fourier decomposition of the effective phase and amplitude aberrations.

© 2006 Optical Society of America

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References

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  1. R. K. Tyson, Introduction to Adaptive Optics (SPIE Press, 2000).
    [Crossref]
  2. C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication, 1999), Vol. 99-3.
  3. L. A. Poyneer and B. Macintosh "Spatially filtered wave-front sensor for high-order adaptive optics," J. Opt. Soc. Am. A 21810-819 (2004).
    [Crossref]
  4. N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, "Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs," Astrophys. J. 582, 1147-1161 (2003).
    [Crossref]
  5. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 2003).
  6. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall, 1989).
  7. N. J. Kasdin, R. J. Vanderbei, N. G. Littman, and D. N. Spergel, "Optimal one-dimensional apodizations and shaped pupils for planet finding coronagraphy," Appl. Opt. 44, 1117-1128 (2005).
    [Crossref] [PubMed]
  8. D. Slepian, "Analytic solution of two apodization problems," J. Opt. Soc. Am. 55, 1110-1115 (1965).
    [Crossref]
  9. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  10. V. N. Mahajan, "Strehl ratio for primary aberrations in terms of their aberration variance," J. Opt. Soc. Am. 73, 860-861 (1983).
    [Crossref]
  11. M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of highStrehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
    [Crossref]
  12. F. Malbet, J. W. Yu, and M. Shao, "High-dynamic-range imaging using a deformable mirror for space coronagraphy," Astron. Soc. Pac. 107, 386-398 (1995).
    [Crossref]
  13. E. E. Bloemhof, R. G. Dekany, M. Troy, and B. R. Oppenheimer, "Behaviour of speckles in an adaptively corrected imaging system," Astrophys. J. 558, L71-L74 (2001).
    [Crossref]
  14. C. Aime and R. Soummer, "The usefullness and limits of coronagraphy in the presence of pinned speckles," Astrophys. J. 612, L85-L88 (2004).
    [Crossref]

2005 (1)

2004 (2)

L. A. Poyneer and B. Macintosh "Spatially filtered wave-front sensor for high-order adaptive optics," J. Opt. Soc. Am. A 21810-819 (2004).
[Crossref]

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

2003 (3)

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of highStrehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[Crossref]

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, "Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs," Astrophys. J. 582, 1147-1161 (2003).
[Crossref]

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 2003).

2001 (1)

E. E. Bloemhof, R. G. Dekany, M. Troy, and B. R. Oppenheimer, "Behaviour of speckles in an adaptively corrected imaging system," Astrophys. J. 558, L71-L74 (2001).
[Crossref]

2000 (1)

R. K. Tyson, Introduction to Adaptive Optics (SPIE Press, 2000).
[Crossref]

1999 (2)

C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication, 1999), Vol. 99-3.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

1995 (1)

F. Malbet, J. W. Yu, and M. Shao, "High-dynamic-range imaging using a deformable mirror for space coronagraphy," Astron. Soc. Pac. 107, 386-398 (1995).
[Crossref]

1989 (1)

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall, 1989).

1983 (1)

1965 (1)

Aime, C.

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

Beichman, C. A.

C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication, 1999), Vol. 99-3.

Bloemhof, E. E.

E. E. Bloemhof, R. G. Dekany, M. Troy, and B. R. Oppenheimer, "Behaviour of speckles in an adaptively corrected imaging system," Astrophys. J. 558, L71-L74 (2001).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 2003).

Dekany, R. G.

E. E. Bloemhof, R. G. Dekany, M. Troy, and B. R. Oppenheimer, "Behaviour of speckles in an adaptively corrected imaging system," Astrophys. J. 558, L71-L74 (2001).
[Crossref]

Graham, J. R.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of highStrehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[Crossref]

Kasdin, N. J.

N. J. Kasdin, R. J. Vanderbei, N. G. Littman, and D. N. Spergel, "Optimal one-dimensional apodizations and shaped pupils for planet finding coronagraphy," Appl. Opt. 44, 1117-1128 (2005).
[Crossref] [PubMed]

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, "Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs," Astrophys. J. 582, 1147-1161 (2003).
[Crossref]

Lindensmith, C. A.

C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication, 1999), Vol. 99-3.

Littman, M. G.

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, "Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs," Astrophys. J. 582, 1147-1161 (2003).
[Crossref]

Littman, N. G.

Macintosh, B.

Mahajan, V. N.

Makidon, R. B.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of highStrehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[Crossref]

Malbet, F.

F. Malbet, J. W. Yu, and M. Shao, "High-dynamic-range imaging using a deformable mirror for space coronagraphy," Astron. Soc. Pac. 107, 386-398 (1995).
[Crossref]

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall, 1989).

Oppenheimer, B. R.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of highStrehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[Crossref]

E. E. Bloemhof, R. G. Dekany, M. Troy, and B. R. Oppenheimer, "Behaviour of speckles in an adaptively corrected imaging system," Astrophys. J. 558, L71-L74 (2001).
[Crossref]

Perrin, M. D.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of highStrehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[Crossref]

Poyneer, L. A.

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall, 1989).

Shao, M.

F. Malbet, J. W. Yu, and M. Shao, "High-dynamic-range imaging using a deformable mirror for space coronagraphy," Astron. Soc. Pac. 107, 386-398 (1995).
[Crossref]

Sivaramakrishnan, A.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of highStrehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[Crossref]

Slepian, D.

Soummer, R.

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

Spergel, D. N.

N. J. Kasdin, R. J. Vanderbei, N. G. Littman, and D. N. Spergel, "Optimal one-dimensional apodizations and shaped pupils for planet finding coronagraphy," Appl. Opt. 44, 1117-1128 (2005).
[Crossref] [PubMed]

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, "Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs," Astrophys. J. 582, 1147-1161 (2003).
[Crossref]

Troy, M.

E. E. Bloemhof, R. G. Dekany, M. Troy, and B. R. Oppenheimer, "Behaviour of speckles in an adaptively corrected imaging system," Astrophys. J. 558, L71-L74 (2001).
[Crossref]

Tyson, R. K.

R. K. Tyson, Introduction to Adaptive Optics (SPIE Press, 2000).
[Crossref]

Vanderbei, R. J.

N. J. Kasdin, R. J. Vanderbei, N. G. Littman, and D. N. Spergel, "Optimal one-dimensional apodizations and shaped pupils for planet finding coronagraphy," Appl. Opt. 44, 1117-1128 (2005).
[Crossref] [PubMed]

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, "Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs," Astrophys. J. 582, 1147-1161 (2003).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Woolf, N. J.

C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication, 1999), Vol. 99-3.

Yu, J. W.

F. Malbet, J. W. Yu, and M. Shao, "High-dynamic-range imaging using a deformable mirror for space coronagraphy," Astron. Soc. Pac. 107, 386-398 (1995).
[Crossref]

Appl. Opt. (1)

Astron. Soc. Pac. (1)

F. Malbet, J. W. Yu, and M. Shao, "High-dynamic-range imaging using a deformable mirror for space coronagraphy," Astron. Soc. Pac. 107, 386-398 (1995).
[Crossref]

Astrophys. J. (4)

E. E. Bloemhof, R. G. Dekany, M. Troy, and B. R. Oppenheimer, "Behaviour of speckles in an adaptively corrected imaging system," Astrophys. J. 558, L71-L74 (2001).
[Crossref]

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

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of highStrehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[Crossref]

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, "Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs," Astrophys. J. 582, 1147-1161 (2003).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Other (5)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 2003).

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall, 1989).

R. K. Tyson, Introduction to Adaptive Optics (SPIE Press, 2000).
[Crossref]

C. A. Beichman, N. J. Woolf, and C. A. Lindensmith, The Terrestrial Planet Finder (JPL Publication, 1999), Vol. 99-3.

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Figures (9)

Fig. 1
Fig. 1

Left, one-dimensional-Kaiser-window-based apodization function. Right, corresponding PSF.

Fig. 2
Fig. 2

Simulating an ideally corrected system.

Fig. 3
Fig. 3

Demonstration of the frequency-folding effect using the polar-coordinate decomposition.

Fig. 4
Fig. 4

PSF (log scaled) of the ideal system. Left, two-dimensional image. Right, horizontal slice through the middle of the image.

Fig. 5
Fig. 5

Phase and amplitude aberrations used in this simulation. Left, phase aberration [ ϕ ( x , y ) in Eq. (2)] in units of wavelength. Right, amplitude aberration [ e α ( x , y ) in Eq. (2).]

Fig. 6
Fig. 6

Simulation of the PSF of the system with phase aberrations only. Left, aberrated system. Right, simulation of conventional phase conjugation.

Fig. 7
Fig. 7

Simulation of the Fourier-based correction technique. Left, resulting PSF. Right, horizontal slice through the center of the ideal, aberrated, conventional, phase-conjugation-corrected and Fourier-decomposition-based-correction PSFs.

Fig. 8
Fig. 8

Simulation of the PSF of the system with both phase and amplitude aberrations. Left, aberrated system. Right, simulation of conventional phase conjugation.

Fig. 9
Fig. 9

Simulation of the Fourier-based correction technique applied to the case of a system with both amplitude and phase aberrations. Left, resulting PSF. Right, horizontal slice through the center of the ideal, aberrated, conventional, phase-conjugation-corrected and Fourier-decomposition-based-correction PSFs.

Equations (49)

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E ( ξ , η ) = S e 2 π i ( x ξ + y η ) A ( x , y ) d x d y ,
S = { ( x , y ) : 1 2 x 1 2 , 1 2 y 1 2 } .
E ( ξ , η ) = S e 2 π i ( x ξ + y η ) A ( x , y ) e γ ( x , y ) d x d y ,
E ( ξ , η ) = S e 2 π i ( x ξ + y η ) A ( x , y ) [ 1 + α ( x , y ) + i ϕ ( x , y ) ] d x d y = E 0 ( ξ , η ) + E ( ξ , η ) ,
E ( ξ , η ) = e 2 π i ( x ξ + y η ) A ( x , y ) [ α ( x , y ) + i ϕ ( x , y ) ] d x d y .
ϕ ( x , y ) = m = n = a m , n e i 2 π ( m x + n y ) .
E ( ξ , η ) = i m = n = a m , n e 2 π i ( x ξ + y η ) A ( x , y ) e i 2 π ( m x + n y ) d x d y .
ϕ ( x , y ) = ( m = a m e i 2 π m x ) ( n = a n e i 2 π n y ) .
E ( ξ , η ) = i [ m = a m e 2 π i x ξ A x ( x ) e i 2 π m x d x ] [ n = a n e 2 π i y ζ A y ( y ) e i 2 π n y d y ] .
E ξ ( ξ ) = e i 2 π x ξ A x ( x ) e i ϕ ( x ) d x ,
e i ϕ ( x ) = n = A n e i 2 π n x ,
A m = 1 / 2 1 / 2 e i ϕ ( x ) e i 2 π m x d x = 1 / 2 1 / 2 { 1 + i ϕ ( x ) + [ i ϕ ( x ) ] 2 2 ! + [ i ϕ ( x ) ] 3 3 ! + O ( α 4 ) } e i 2 π m x d x = 1 / 2 1 / 2 e i 2 π m x d x + i 1 / 2 1 / 2 ϕ x ( x ) e i 2 π m x d x 1 2 ! 1 / 2 1 / 2 [ ( n = a n e i 2 π n x ) ( k = a k e i 2 π k x ) ] e i 2 π m x d x i 3 ! 1 / 2 1 / 2 [ ( n = a n e i 2 π n x ) ( k = a k e i 2 π k x ) ( l = a l e i 2 π l x ) ] e i 2 π m x d x + O ( α 4 ) = 0 + i a m 1 2 ! n , k = a n a k 1 / 2 1 / 2 e i 2 π ( n + k m ) x d x i 3 ! n , k , l = a n a k a l 1 / 2 1 / 2 e i 2 π ( n + k + l m ) x d x + O ( α 4 ) ,
δ ( k ) = { 1 for k = 0 0 otherwise } ,
A m = i a m 1 2 ! n , k = a n a k δ ( n + k m ) i 3 ! n , k , l = a n a k a l δ ( n + k + l m ) + O ( α 4 ) .
E ξ ( ξ ) = e 2 π i x ξ A x ( x ) e i [ ϕ ( x ) ϕ r ( x ) ] d x .
I ( ξ ) = 1 / 2 1 / 2 A ( x ) e i ϕ ( x ) e 2 π i x ξ d x 2 .
A ( x ) = I 0 ( β 1 4 x 2 ) I 0 ( β ) ,
ϕ 1 ( x ) = n = 0 N [ A n cos ( 2 π n x ) + B n sin ( 2 π n x ) ] ,
ϕ 2 ( x ) = n = M N [ A n cos ( 2 π n x ) + B n sin ( 2 π n x ) ] ,
E ( ρ , φ ) = 0 1 2 0 2 π e 2 π i r ρ cos ( 0 φ ) E i ( r , θ ) r d θ d r ,
E ( ρ , φ ) = 0 1 2 0 2 π E i ( r , θ ) J 0 ( 2 π r ρ ) r d r d θ + 2 n = 1 0 1 2 0 2 π E i ( r , θ ) i n J n ( 2 π r ρ ) cos n ( θ φ ) r d r d θ .
E ( ρ , φ ) = i 0 1 2 0 2 π A ( r ) ϕ ( r , θ ) J 0 ( 2 π r ρ ) r d r d θ + 2 i n = 1 0 1 2 0 2 π A ( r ) ϕ ( r , θ ) i n J n ( 2 π r ρ ) cos n ( θ φ ) r d r d θ .
ϕ ( r , θ ) = k = 1 c k ( θ ) J 0 ( α k 2 r ) 0 r 1 2 ,
c k ( θ ) = 8 J 1 2 ( α k ) 0 1 2 ϕ ( r , θ ) J 0 ( 2 α k r ) r d r ,
c k ( θ ) = ( a 0 ) k + m = 1 ( a m ) k cos ( m θ ) + ( b m ) k sin ( m θ ) .
E ( ρ , φ ) = 2 π i k = 1 ( a 0 ) k 0 1 2 A ( r ) J 0 ( 2 α k r ) J 0 ( 2 π r ρ ) r d r + π i n = 1 k = 1 i n [ 0 1 2 A ( r ) J 0 ( 2 α k r ) J n ( 2 π r ρ ) r d r ] [ ( a n ) k cos n φ + ( b n ) k sin n φ ] .
J 0 ( 2 α k r ) = l = 1 A l , k ( n ) J n [ 2 α l ( n ) r ] 0 r 1 2 ,
A l , k = 2 { J n + 1 [ α l ( n ) ] } 2 0 1 J 0 ( α k s ) J n [ s α l ( n ) ] d s .
E ( ρ , φ ) = 2 π i k = 1 ( a 0 ) k A k ( ρ ) + π i n = 1 k = 1 l = 1 i n A l , k ( n ) B l , n ( ρ ) [ ( a n ) k cos n φ + ( b n ) k sin n φ ] ,
A k ( ρ ) = 0 1 2 A ( r ) J 0 ( 2 α k r ) J 0 ( 2 π r ρ ) r d r ,
B l , n ( ρ ) = 0 1 2 A ( r ) J n [ 2 α l ( n ) r ] J n ( 2 π r ρ ) r d r ,
A k ( ρ ) = α k J 1 ( α k ) J 0 ( π ρ ) 4 ( α k 2 π 2 ρ 2 ) ,
B l , n ( ρ ) = α l ( n ) J n 1 [ α l ( n ) ] J n ( π ρ ) 4 ( π 2 ρ 2 α l 2 ( n ) ) .
c k ( θ ) = 2 cos m θ J 1 2 ( α k ) 2 μ 1 α k μ + 1 [ 2 μ Γ ( 1 2 + 1 2 μ ) Γ ( 1 2 1 2 μ ) α k J 1 ( α k ) S μ , 0 ( α k ) ] ,
S μ , 0 ( α k ) = s μ , 0 ( α k ) 2 μ 1 Γ 2 ( 1 2 + 1 2 μ ) cos ( 1 2 μ π ) N 0 ( α k )
s μ , 0 ( α k ) = 2 μ ( α k 2 ) ( μ + 1 ) 2 Γ ( 1 2 + 1 2 μ ) 0 π 2 J 1 2 ( 1 + μ ) ( α k sin θ ) ( sin θ ) 1 2 ( 1 μ ) ( cos θ ) μ d θ ,
N 0 ( α k ) = 2 π 1 cos α k x x 2 1 d x .
s μ , 0 ( α k ) = α k μ + 1 ( μ + 1 ) 2 F 2 1 ( 1 ; μ + 3 2 , μ + 3 2 ; α k 2 4 ) .
S e k 2 σ ϕ 2 ,
J p u p = min ϕ DM ( x , y ) [ p u p ϕ D M ( x , y ) ϕ e s t ( x , y ) 2 d x d y ] .
e γ ( x , y ) = 1 + m , n = a m , n e 2 π i ( m x + n y ) ,
a m , n = S [ e γ ( x , y ) 1 ] e 2 π i ( m x + n y ) d x d y .
E ( ξ , η ) = E 0 ( ξ , η ) + m , n = a m , n S A ( x , y ) e 2 π i ( m x + n y ) e 2 π i ( x ξ + y η ) d x d y = E 0 ( ξ , η ) + m , n = a m , n E 0 ( ξ m , η n ) .
E m , n ( ξ , η ) = E 0 ( ξ m , η n ) ,
I ( ξ , η ) = E * ( ξ , η ) E ( ξ , η ) = I 0 ( ξ , η ) + 2 E 0 ( ξ , η ) m , n = R { a m , n } E m , n ( ξ , η ) + m , n , k , l = a m , n * a k , l E m , n ( ξ , η ) E k , l ( ξ , η ) .
J i m g = min ϕ DM ( x , y ) { m , n = K K a m , n 2 } ,
I ( ξ , η ) = S A ( x , y ) e α ( x , y ) + i [ ϕ ( x , y ) + ϕ DM ( x , y ) ] e 2 π i ( x ξ + y η ) d x d y 2 ,
a m , n = S { e i [ ϕ ( x , y ) + ϕ D M ( x , y ) ] 1 } e 2 π i ( m x + n y ) d x d y .
a m , n = S [ e α ( x , y ) + i [ ϕ ( x , y ) + ϕ DM ( x , y ) ] 1 ] e 2 π i ( m x + n y ) d x d y .

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