Abstract

The past decade has seen a significant growth in research targeted at space-based observatories for imaging exosolar planets. The challenge is in designing an imaging system for high contrast. Even with a perfect coronagraph that modifies the point spread function to achieve high contrast, wavefront sensing and control is needed to correct the errors in the optics and generate a “dark hole.” The high-contrast imaging laboratory at Princeton University is equipped with two Boston Micromachines Kilo-DMs. We review here an algorithm designed to achieve high contrast on both sides of the image plane while minimizing the stroke necessary from each deformable mirror (DM). This algorithm uses the first DM to correct for amplitude aberrations and uses the second DM to create a flat wavefront in the pupil plane. We then show the first results obtained at Princeton with this correction algorithm, and we demonstrate a symmetric dark hole in monochromatic light.

© 2009 Optical Society of America

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  1. F. Malbet, J. W. Yu, and M. Shao, “High dynamic range imaging using a deformable mirror for space coronagraphy,” Publ. Astron. Soc. Pac. 107, 386-396 (1995).
    [CrossRef]
  2. R. A. Brown and C. J. Burrows, “On the feasibility of detecting extrasolar planets by reflected starlight using the Hubble Space Telescope,” Icarus 87, 484-491 (1990).
    [CrossRef]
  3. J. Trauger, “The Eclipse Mission of imaging of nearby planetary systems: concept and laboratory validation,” American Astronomical Society Meeting 205 (American Astronomical Society, 2004), Vol. 36, p. 1344.
  4. P. J. Bordé and W. A. Traub, “High-contrast imaging from space: speckle nulling in a low-aberration regime,” Astrophys. J. 638, 488-501 (2006).
    [CrossRef]
  5. A. Give'on, B. Kern, S. Shaklan, D. C. Moody, and L. Pueyo, “Electric field conjugation--a broadband wavefront correction algorithm for high-contrast imaging systems,” American Astronomical Society Meeting Abstracts 211 (American Astronomical Society, 2007), Vol. 39, p. 975.
  6. A. Give'on, N. Kasdin, and R. Vanderbei, “Closed-loop wavefront correction for high contrast imaging: the peek-a-boo algorithm,” in Proceedings of the International Astronomical Union (Cambridge U. Press, 2006), Vol. 200, pp. 541-546.
  7. L. Pueyo and N. J. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609-625 (2007).
    [CrossRef]
  8. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. O. Flannery, Numerical Recipes in C (Cambridge U. Press, 1995).
  9. S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 11-19 (2006).
  10. S. B. Shaklan and J. J. Green, “Reflectivity and optical surface height requirements in a broadband coronagraph. 1. Contrast floor due to controllable spatial frequencies,” Appl. Opt. 45, 5143-5156 (2006).
    [CrossRef] [PubMed]

2007

L. Pueyo and N. J. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609-625 (2007).
[CrossRef]

2006

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 11-19 (2006).

S. B. Shaklan and J. J. Green, “Reflectivity and optical surface height requirements in a broadband coronagraph. 1. Contrast floor due to controllable spatial frequencies,” Appl. Opt. 45, 5143-5156 (2006).
[CrossRef] [PubMed]

P. J. Bordé and W. A. Traub, “High-contrast imaging from space: speckle nulling in a low-aberration regime,” Astrophys. J. 638, 488-501 (2006).
[CrossRef]

1995

F. Malbet, J. W. Yu, and M. Shao, “High dynamic range imaging using a deformable mirror for space coronagraphy,” Publ. Astron. Soc. Pac. 107, 386-396 (1995).
[CrossRef]

1990

R. A. Brown and C. J. Burrows, “On the feasibility of detecting extrasolar planets by reflected starlight using the Hubble Space Telescope,” Icarus 87, 484-491 (1990).
[CrossRef]

Bordé, P. J.

P. J. Bordé and W. A. Traub, “High-contrast imaging from space: speckle nulling in a low-aberration regime,” Astrophys. J. 638, 488-501 (2006).
[CrossRef]

Brown, R. A.

R. A. Brown and C. J. Burrows, “On the feasibility of detecting extrasolar planets by reflected starlight using the Hubble Space Telescope,” Icarus 87, 484-491 (1990).
[CrossRef]

Burrows, C. J.

R. A. Brown and C. J. Burrows, “On the feasibility of detecting extrasolar planets by reflected starlight using the Hubble Space Telescope,” Icarus 87, 484-491 (1990).
[CrossRef]

Flannery, B. O.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. O. Flannery, Numerical Recipes in C (Cambridge U. Press, 1995).

Give'on, A.

A. Give'on, B. Kern, S. Shaklan, D. C. Moody, and L. Pueyo, “Electric field conjugation--a broadband wavefront correction algorithm for high-contrast imaging systems,” American Astronomical Society Meeting Abstracts 211 (American Astronomical Society, 2007), Vol. 39, p. 975.

A. Give'on, N. Kasdin, and R. Vanderbei, “Closed-loop wavefront correction for high contrast imaging: the peek-a-boo algorithm,” in Proceedings of the International Astronomical Union (Cambridge U. Press, 2006), Vol. 200, pp. 541-546.

Green, J. J.

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 11-19 (2006).

S. B. Shaklan and J. J. Green, “Reflectivity and optical surface height requirements in a broadband coronagraph. 1. Contrast floor due to controllable spatial frequencies,” Appl. Opt. 45, 5143-5156 (2006).
[CrossRef] [PubMed]

Kasdin, N.

A. Give'on, N. Kasdin, and R. Vanderbei, “Closed-loop wavefront correction for high contrast imaging: the peek-a-boo algorithm,” in Proceedings of the International Astronomical Union (Cambridge U. Press, 2006), Vol. 200, pp. 541-546.

Kasdin, N. J.

L. Pueyo and N. J. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609-625 (2007).
[CrossRef]

Kern, B.

A. Give'on, B. Kern, S. Shaklan, D. C. Moody, and L. Pueyo, “Electric field conjugation--a broadband wavefront correction algorithm for high-contrast imaging systems,” American Astronomical Society Meeting Abstracts 211 (American Astronomical Society, 2007), Vol. 39, p. 975.

Malbet, F.

F. Malbet, J. W. Yu, and M. Shao, “High dynamic range imaging using a deformable mirror for space coronagraphy,” Publ. Astron. Soc. Pac. 107, 386-396 (1995).
[CrossRef]

Moody, D. C.

A. Give'on, B. Kern, S. Shaklan, D. C. Moody, and L. Pueyo, “Electric field conjugation--a broadband wavefront correction algorithm for high-contrast imaging systems,” American Astronomical Society Meeting Abstracts 211 (American Astronomical Society, 2007), Vol. 39, p. 975.

Palacios, D. M.

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 11-19 (2006).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. O. Flannery, Numerical Recipes in C (Cambridge U. Press, 1995).

Pueyo, L.

L. Pueyo and N. J. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609-625 (2007).
[CrossRef]

A. Give'on, B. Kern, S. Shaklan, D. C. Moody, and L. Pueyo, “Electric field conjugation--a broadband wavefront correction algorithm for high-contrast imaging systems,” American Astronomical Society Meeting Abstracts 211 (American Astronomical Society, 2007), Vol. 39, p. 975.

Shaklan, S.

A. Give'on, B. Kern, S. Shaklan, D. C. Moody, and L. Pueyo, “Electric field conjugation--a broadband wavefront correction algorithm for high-contrast imaging systems,” American Astronomical Society Meeting Abstracts 211 (American Astronomical Society, 2007), Vol. 39, p. 975.

Shaklan, S. B.

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 11-19 (2006).

S. B. Shaklan and J. J. Green, “Reflectivity and optical surface height requirements in a broadband coronagraph. 1. Contrast floor due to controllable spatial frequencies,” Appl. Opt. 45, 5143-5156 (2006).
[CrossRef] [PubMed]

Shao, M.

F. Malbet, J. W. Yu, and M. Shao, “High dynamic range imaging using a deformable mirror for space coronagraphy,” Publ. Astron. Soc. Pac. 107, 386-396 (1995).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. O. Flannery, Numerical Recipes in C (Cambridge U. Press, 1995).

Traub, W. A.

P. J. Bordé and W. A. Traub, “High-contrast imaging from space: speckle nulling in a low-aberration regime,” Astrophys. J. 638, 488-501 (2006).
[CrossRef]

Trauger, J.

J. Trauger, “The Eclipse Mission of imaging of nearby planetary systems: concept and laboratory validation,” American Astronomical Society Meeting 205 (American Astronomical Society, 2004), Vol. 36, p. 1344.

Vanderbei, R.

A. Give'on, N. Kasdin, and R. Vanderbei, “Closed-loop wavefront correction for high contrast imaging: the peek-a-boo algorithm,” in Proceedings of the International Astronomical Union (Cambridge U. Press, 2006), Vol. 200, pp. 541-546.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. O. Flannery, Numerical Recipes in C (Cambridge U. Press, 1995).

Yu, J. W.

F. Malbet, J. W. Yu, and M. Shao, “High dynamic range imaging using a deformable mirror for space coronagraphy,” Publ. Astron. Soc. Pac. 107, 386-396 (1995).
[CrossRef]

Appl. Opt.

Astrophys. J.

P. J. Bordé and W. A. Traub, “High-contrast imaging from space: speckle nulling in a low-aberration regime,” Astrophys. J. 638, 488-501 (2006).
[CrossRef]

L. Pueyo and N. J. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609-625 (2007).
[CrossRef]

Icarus

R. A. Brown and C. J. Burrows, “On the feasibility of detecting extrasolar planets by reflected starlight using the Hubble Space Telescope,” Icarus 87, 484-491 (1990).
[CrossRef]

Proc. SPIE

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 11-19 (2006).

Publ. Astron. Soc. Pac.

F. Malbet, J. W. Yu, and M. Shao, “High dynamic range imaging using a deformable mirror for space coronagraphy,” Publ. Astron. Soc. Pac. 107, 386-396 (1995).
[CrossRef]

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. O. Flannery, Numerical Recipes in C (Cambridge U. Press, 1995).

J. Trauger, “The Eclipse Mission of imaging of nearby planetary systems: concept and laboratory validation,” American Astronomical Society Meeting 205 (American Astronomical Society, 2004), Vol. 36, p. 1344.

A. Give'on, B. Kern, S. Shaklan, D. C. Moody, and L. Pueyo, “Electric field conjugation--a broadband wavefront correction algorithm for high-contrast imaging systems,” American Astronomical Society Meeting Abstracts 211 (American Astronomical Society, 2007), Vol. 39, p. 975.

A. Give'on, N. Kasdin, and R. Vanderbei, “Closed-loop wavefront correction for high contrast imaging: the peek-a-boo algorithm,” in Proceedings of the International Astronomical Union (Cambridge U. Press, 2006), Vol. 200, pp. 541-546.

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Figures (10)

Fig. 1
Fig. 1

Numerical results of the stroke minimization algorith using a 10 - 10 shaped pupil. (a) DM deformation in radians. (b) Log(corrected PSF).

Fig. 2
Fig. 2

Comparison between the peak-to-valley actuator strokes necessary to correct a given wavefront error using energy minimization and stroke minimization.

Fig. 3
Fig. 3

Optical layout of the Princeton high-contrast imaging test bed. For the experiment presented in Subsection 2D, only one DM is used for wavefront control and only the image plane camera is used for wavefront sensing.

Fig. 4
Fig. 4

(a) Aberrated and (b) corrected PSF on the Princeton test bed in log(contrast). The wavefront is flattened so that half of the image plane exhibits a dark hole over a specified region, in this case X = 7 10 λ / D and Y = 2.5 2.5 λ / D . This monochromatic experiment used an illumination wavelength of 635 nm .

Fig. 5
Fig. 5

Experimental results for six different target contrasts. The top two curves correspond respectively to the maximum of the intensity in the dark hole and the maximum of the estimated intensity in the dark hole. The bottom two correspond to the average intensity in the dark hole and the average estimated intensity in the dark hole. Note that, when the algorithm converges, the average estimated intensity is equal to the target contrast. Also, note that these results were obtained on the Princeton testbed prior to the installation of the second DM and, therefore, the contrast limit is slightly better than other results shown in this paper.

Fig. 6
Fig. 6

DM surfaces in radians obtained using the two-DM stroke minimization algorithm. The algorithm used here is designed to operate monochromatically and does not take advantage of the broadband capabilities of the wavefront controller. Pupil size D = 3 cm and DM separation z = 1 m .

Fig. 7
Fig. 7

Monochromatic PSF resulting from two-DM wavefront correction using a monchromatic stroke minimization algorithm. D = 3 cm and z = 1 m .

Fig. 8
Fig. 8

(a) Aberrated image and (b) corrected image of the two-DM stroke minimization symmetric dark hole experiment. (c) Plot of contrast versus iteration in each of the two dark holes and in the combination of the two.

Fig. 9
Fig. 9

Half dark hole correction using one DM. Left column: a phase aberration can theoretically be compensated at all wavelengths by a matching DM setting to cancel out the total electric field on both sides of the optical axis. Right column: in the case of amplitude aberrations, a DM setting can be found that will exactly compensate the amplitude error on one side of the optical axis, and at a single wavelength.

Fig. 10
Fig. 10

Broadband amplitude correction using two DMs: complex phasor illustration. Left column: one can find a DM1setting that will cancel the amplitude error achromatically on both sides of the optical axis after propagation through the system (equivalent to a phasor rotation in the angular spectrum approximation). Right column: the phase error induced by DM1, together with the phase error accumulated through the system, is then taken out at all wavelengths by the second DM, resulting in broadband two-sided light cancellation.

Equations (65)

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E f = C { E 0 } .
E 0 = A ( x , y ) e α λ ( x , y ) + i 2 π λ β λ ( x , y ) e i 2 π λ ψ ( x , y ) ,
ψ ( x , y ) = λ 0 k = 1 N a k f k ( x , y ) ,
ψ ( x , y ) = λ 0 F ( x , y ) X .
E 0 = A ( x , y ) h ( γ λ ( x , y ) , X ) ,
γ λ = α + i 2 π λ β
E S = E f , E f S = S E f E f * d ξ d η ,
X n + 1 = X n - ν X g ( X n )
X n + 1 = X n - H n - 1 X g ( X n ) ,
X n + 1 = X n - ( X g ( X n ) T X g ( X n ) ) - 1 X g ( X n ) T .
X g ( X n ) = 2 * [ C { E 0 ( X n ) } , i 2 π λ 0 λ C { E 0 ( X n ) F } S ] ,
h ( γ λ ( x , y ) , δ X n ) 1 + γ λ ( x , y ) + i 2 π λ 0 λ F ( x , y ) δ X n ,
i 2 π λ 0 λ F ( x , y ) δ X n γ λ ( x , y ) .
X g ( X n ) 2 * [ E f , i 2 π λ 0 λ C { A F } S ] ,
X g ( X n ) T X g ( X n ) - 2 ( 2 2 π λ 0 λ ) 2 * [ C { A F } , C { A F } S ] .
X n + 1 = X n - ( X g ( X n ) T X g ( X n ) + μ I ) - 1 X g ( X n ) T ,
X n + 1 = X n - ( X g ( X n ) T X g ( X n ) + μ diag ( ( X g ( X n ) T X g ( X n ) ) T ( X g ( X n ) T X g ( X n ) ) ) ) - 1 X g ( X n ) T .
E f - E D = 0.
e i 2 π λ ψ e i 2 π λ ψ n ( 1 + i 2 π λ δ ψ n + 1 ) ,
E ˜ n + i 2 π λ 0 λ C { A h ( γ λ , X n ) F } δ X n + 1 = 0 ,
E ˜ n = E D - C { A e α λ + i 2 π λ β λ + i 2 π λ ψ n } = E D - ( E f ) n
E ˜ n + i 2 π λ 0 λ C { A F } δ X n + 1 = 0.
C { A e α + i 2 π λ β + i 2 π λ ψ n } .
minimize     1 2 k = 1 N a k 2 subject to     E S 10 - C ,
E 0 A ( x , y ) [ h n ( γ λ ( x , y ) , X n ) + h X | X n δ X n + 1 ] ,
J n = h X | X n = i 2 π λ 0 λ e α λ ( x , y ) + i 2 π λ β λ ( x , y ) e i 2 π λ ψ n ( x , y ) F ( x , y ) .
E S S ( ξ , η ) [ C { A h n } + C { A J n } δ X n + 1 ] * [ C { A h n } + C { A J n } δ X n + 1 ] d ξ d η .
E S S C { A h n } * C { A h n } d ξ d η ( E S ) n + 2 { S C { A h n } * C { A J n } δ X n + 1 d ξ d η } + S δ X n + 1 T C { A J n } * C { A J n } T δ X n + 1 d ξ d η ,
minimize     1 2 ( X n + δ X n + 1 ) T W - 1 ( X n + δ X n + 1 ) subject to       δ X n + 1 T M n δ X n + 1 + B n δ X n + 1 + d n 10 - C ,
d n = S C { A h n } * C { A h n } d ξ d η , B n = 2 { S C { A h n } * C { A J n } d ξ d η } , M n = S C { A J n } * C { A J n } T d ξ d η .
J 0 = h X | X n = 0 = i 2 π λ 0 λ F ( x , y ) .
E M = 1 2 ( X n + δ X n + 1 ) T W - 1 ( X n + δ X n + 1 ) + μ ( δ X n + 1 T M 0 δ X n + 1 + B n δ X n + 1 + d n - 10 - C ) .
δ X n + 1 ( μ ) = - ( μ I + 2 W M 0 ) - 1 ( 1 μ X n + W B n T ) ,
B n = 2 { S C { A h n } * C { A J 0 } d ξ d η } = E f , i 2 π λ 0 λ C { A F } S ,
M 0 = S C { A J 0 } * C { A J 0 } T d ξ d η = - ( 2 π λ 0 λ ) 2 C { A F } , C { A F } S .
δ E f + E f ( X n ) + C { A F } · δ X k 2 < I Target ,
δ E f 2 + E f ( X n ) + C { A F } δ X n 2 - 2 [ δ E f , E f ( X n ) + C { A F } · δ X n S ] < I Target .
E f ( X n ) + C { A F } δ X n 2 < I Target - I bias + 2 I Bias E S ,
E DM 2 ( x , y ) = A ( x , y ) h ( 2 ) ( γ λ , X ) ,
h ( 2 ) ( γ λ , X ) = e α λ ( x , y ) + i 2 π λ β λ ( x , y ) z [ e i 2 π λ ψ ( 1 ) ( x , y ) ] e i 2 π λ ψ ( 2 ) ( x , y ) .
E f = C [ e α λ ( x , y ) + i 2 π λ β λ ( x , y ) F z [ e i 2 π λ ψ ( 1 ) ( x , y ) ] e i 2 π λ ψ ( 2 ) ( x , y ) ] .
E S S C { A h n ( 2 ) } * C { A h n ( 2 ) } d ξ d η ( E S ) n + 2 { S C { A h n ( 2 ) } * C { A J n } δ X d ξ d η } + S Δ X T C { A J n } * C { A J n } T δ X d ξ d η l ,
J n = h ( 2 ) X | X n .
minimize       1 2 k = 1 N a k 2 subject to     E S 10 - C .
minimize     1 2 ( X n + δ X n + 1 ) T W - 1 ( X n + δ X n + 1 ) subject to       δ X n + 1 T M n δ X n + 1 + B n δ X n + 1 + d n 10 - C .
J n = i 2 π λ 0 λ e α λ ( x , y ) + i 2 π λ β λ ( x , y ) [ e i 2 π λ ψ n ( 2 ) ( x , y ) F z [ e i 2 π λ ψ n ( 1 ) ( x , y ) F ( 1 ) ( x , y ) ] ; F z [ e i 2 π λ ψ ( 1 ) ( x , y ) ] e i 2 π λ ψ n ( 2 ) ( x , y ) F ( 2 ) ( x , y ) ] ,
C { A ( x , y ) ) i 2 π λ 0 λ e α λ ( x , y ) + i 2 π λ β λ ( x , y ) F z [ e i 2 π λ ψ ( 1 ) ( x , y ) ] e i 2 π λ ψ n ( 2 ) ( x , y ) F ( 2 ) ( x , y ) } i 2 π λ 0 λ C { A F ( 2 ) } .
C { i A ( x , y ) 2 π λ 0 λ e α λ ( x , y ) + i 2 π λ β λ ( x , y ) e i 2 π λ ψ n ( 2 ) ( x , y ) F z [ e i 2 π λ ψ n ( 1 ) ( x , y ) ] F ( 1 ) ( x , y ) } e - i π λ z D 2 ( ξ 2 + η 2 ) C { A F ( 1 ) } ,
C { A J n } C { A J 0 } = i 2 π λ 0 λ [ e - i π λ z D 2 ( ξ 2 + η 2 ) C { A F ( 1 ) } C { A F ( 2 ) } ] = [ C { A J 0 ( 1 ) } C { A J 0 ( 2 ) } ] .
minimize     1 2 ( X n + δ X n + 1 ) T W - 1 ( X n + δ X n + 1 ) subject to     δ X n + 1 T M δ X n + 1 + B n δ X n + 1 + d n 10 - C ,
d n = C { A h n ( 2 ) } , C { A h n ( 2 ) } S , B n = 2 [ C { A h n ( 2 ) } , C { A J 0 ( 1 ) } S C { A h n ( 2 ) } , C { A J 0 ( 2 ) ) } S ] , M = [ C { A J 0 ( 1 ) } , C { A J 0 ( 1 ) } S C { A J 0 ( 1 ) } , C { A J 0 ( 2 ) } S C { A J 0 ( 2 ) } , C { A J 0 ( 1 ) } S C { A J 0 ( 2 ) } , C { A J 0 ( 2 ) } S ] .
π λ z D 2 ( ξ 2 + η 2 ) 1 ,
C { A J 0 } [ - 2 π 2 λ 0 z D 2 ( ξ 2 + η 2 ) C { A F ( 1 ) } + i 2 π λ 0 λ C { A F ( 1 } i 2 π λ 0 λ C { A F ( 2 ) } ] .
C { A J ˜ 0 } [ - 2 π 2 λ 0 z D 2 ( ξ 2 + η 2 ) C { A F ( 1 ) } i 2 π λ 0 λ ( C { A F ( 1 } + C { A F ( 2 ) } ) ] .
E abb Pup ( x , y ) = e i λ 0 λ cos ( 2 π D ( m x + n y ) + ϕ ) .
E DM Pup ( x , y ) = e - i λ 0 λ cos ( 2 π D ( m x + n y ) + ϕ ) .
E abb Pup ( x , y ) = cos ( 2 π D ( m x + n y ) + ϕ ) ,
E DM Pup ( x , y ) = - i λ 0 λ sin ( 2 π D ( m x + n y ) + ϕ ) .
E Res Pup ( x , y ) = 1 2 ( 1 - λ λ 0 ) e i ( 2 π D ( m x + n y ) + ϕ ) + 1 2 ( 1 + λ λ 0 ) e - i ( 2 π D ( m x + n y ) + ϕ ) .
E pup , abb ( x , y ) = cos ( 2 π D ( m x + n y ) + ϕ ) .
E DM 1 , pup ( x , y ) = i λ 0 λ D 2 π z ( n 2 + m 2 ) cos ( 2 π D ( m x + n y ) + ϕ ) .
E DM 1 , pup ( x , y ) = - i λ 0 λ D 2 π z λ 0 ( n 2 + m 2 ) e - i π λ z ( n 2 + m 2 ) D 2 cos ( 2 π D ( m x + n y ) + ϕ ) .
π λ z ( n 2 + m 2 ) D 2 1.
E DM 1 , pup ( x , y ) = - i λ 0 λ D 2 π z λ 0 ( n 2 + m 2 ) cos ( 2 π D ( m x + n y ) + ϕ ) - cos ( 2 π D ( m x + n y ) + ϕ ) .
E DM 2 , pup ( x , y ) = i λ 0 λ D 2 π z λ 0 ( n 2 + m 2 ) cos ( 2 π D ( m x + n y ) + ϕ )

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