Abstract

The dispersed fringe sensor (DFS) has been demonstrated as an effective means of measuring mirror segment piston error for telescopes with primary mirror apertures below 10 meters. With larger proposed telescopes such as The Thirty Meter Telescope (TMT) and The European Large Telescope (ELT) including ever more segments, there is a need for improvement in the co-phasing capability for segmented primary mirrors. In this paper a novel DFS that employs polarization phase shifting technology is introduced. This novel technology provides system designers and engineers with a new tool to extend the dynamic range of a DFS.

© 2012 OSA

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  1. F. Shi, G. Chanan, C. Ohara, M. Troy, and D. C. Redding, “Experimental verification of dispersed fringe sensing as a segment phasing technique using the Keck telescope,” Appl. Opt. 43(23), 4474–4481 (2004).
    [CrossRef] [PubMed]
  2. D. C. Zimmerman, “Feasibility studies for the alignment of the Thirty Meter Telescope,” Appl. Opt. 49(18), 3485–3498 (2010).
    [CrossRef] [PubMed]
  3. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (John Wiley & Sons, 2007).
  4. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
    [CrossRef]
  5. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
    [CrossRef] [PubMed]
  6. G. Olczak and C. Merle, “Polarization modulated image conjugate piston sensing and phase retrieval system,” ITT Manufacturing Enterprises, Inc., US Patent 7,864,333 (2011)
  7. F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
    [CrossRef]
  8. W. Zhao and G. Cao, “Active cophasing and aligning testbed with segmented mirrors,” Opt. Express 19(9), 8670–8683 (2011).
    [CrossRef] [PubMed]
  9. S. P. Walborn, M. O. Terra Cunha, S. Pa’dua, and C. H. Monken, “Double-slit quantum eraser,” Phys. Rev. A 65(3), 033818 (2002).
    [CrossRef]
  10. M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24(24), 4439–4442 (1985).
    [CrossRef] [PubMed]
  11. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase-shifting by a rotating polarizer,” Opt. Commun. 154(5-6), 249–254 (1998).
    [CrossRef]
  12. M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37(6), 631–641 (2002).
    [CrossRef]
  13. M. Roy and P. Hariharan, “White-light geometric phase interferometer for surface profiling,” Proc. SPIE 2544, 64–73 (1995).
    [CrossRef]
  14. M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 126–133 (2005).
  15. C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992).
    [CrossRef]
  16. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory Algorithms, and Software (John Wiley and Sons, Inc. 1998)
  17. L. Koechlin, P. R. Lawson, D. Mourard, A. Blazit, D. Bonneau, F. Morand, Ph. Stee, I. Tallon-Bosc, and F. Vakili, “Dispersed fringe tracking with the multi-ro apertures of the Grand Interféromètre à 2 Télescopes,” Appl. Opt. 35(16), 3002–3009 (1996).
    [CrossRef] [PubMed]
  18. J. A. Spechler, D. J. Hoppe, N. Sigrist, F. Shi, B.-J. Seo, and S. Bikkannavar, “Advanced DFS: a dispersed fringe sensing algorithm insensitive to small calibration errors,” Proc. SPIE 7731, 773155 (2010).
  19. G. Olczak, “Recent advances in wavefront sensing at ITT,” ITT Internal Report (2009).

2011 (1)

2010 (2)

D. C. Zimmerman, “Feasibility studies for the alignment of the Thirty Meter Telescope,” Appl. Opt. 49(18), 3485–3498 (2010).
[CrossRef] [PubMed]

J. A. Spechler, D. J. Hoppe, N. Sigrist, F. Shi, B.-J. Seo, and S. Bikkannavar, “Advanced DFS: a dispersed fringe sensing algorithm insensitive to small calibration errors,” Proc. SPIE 7731, 773155 (2010).

2005 (2)

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 126–133 (2005).

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

2004 (2)

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

F. Shi, G. Chanan, C. Ohara, M. Troy, and D. C. Redding, “Experimental verification of dispersed fringe sensing as a segment phasing technique using the Keck telescope,” Appl. Opt. 43(23), 4474–4481 (2004).
[CrossRef] [PubMed]

2003 (1)

F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
[CrossRef]

2002 (2)

S. P. Walborn, M. O. Terra Cunha, S. Pa’dua, and C. H. Monken, “Double-slit quantum eraser,” Phys. Rev. A 65(3), 033818 (2002).
[CrossRef]

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37(6), 631–641 (2002).
[CrossRef]

1998 (1)

S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase-shifting by a rotating polarizer,” Opt. Commun. 154(5-6), 249–254 (1998).
[CrossRef]

1996 (1)

1995 (1)

M. Roy and P. Hariharan, “White-light geometric phase interferometer for surface profiling,” Proc. SPIE 2544, 64–73 (1995).
[CrossRef]

1992 (1)

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992).
[CrossRef]

1985 (1)

Basinger, S. A.

F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
[CrossRef]

Bikkannavar, S.

J. A. Spechler, D. J. Hoppe, N. Sigrist, F. Shi, B.-J. Seo, and S. Bikkannavar, “Advanced DFS: a dispersed fringe sensing algorithm insensitive to small calibration errors,” Proc. SPIE 7731, 773155 (2010).

Blazit, A.

Bolcar, M. R.

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 126–133 (2005).

Bonneau, D.

Bowers, C. W.

F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
[CrossRef]

Brock, N.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Burns, L. A.

F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
[CrossRef]

Cao, G.

Chanan, G.

Cherel, L.

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37(6), 631–641 (2002).
[CrossRef]

Delisle, C.

Fienup, J. R.

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 126–133 (2005).

Hariharan, P.

M. Roy and P. Hariharan, “White-light geometric phase interferometer for surface profiling,” Proc. SPIE 2544, 64–73 (1995).
[CrossRef]

Hayes, J.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Helen, S.

S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase-shifting by a rotating polarizer,” Opt. Commun. 154(5-6), 249–254 (1998).
[CrossRef]

Hoppe, D. J.

J. A. Spechler, D. J. Hoppe, N. Sigrist, F. Shi, B.-J. Seo, and S. Bikkannavar, “Advanced DFS: a dispersed fringe sensing algorithm insensitive to small calibration errors,” Proc. SPIE 7731, 773155 (2010).

Koechlin, L.

Koliopoulos, C. L.

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992).
[CrossRef]

Kothiyal, M. P.

S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase-shifting by a rotating polarizer,” Opt. Commun. 154(5-6), 249–254 (1998).
[CrossRef]

M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24(24), 4439–4442 (1985).
[CrossRef] [PubMed]

Lawson, P. R.

Lowman, A. E.

F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
[CrossRef]

Millerd, J.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Monken, C. H.

S. P. Walborn, M. O. Terra Cunha, S. Pa’dua, and C. H. Monken, “Double-slit quantum eraser,” Phys. Rev. A 65(3), 033818 (2002).
[CrossRef]

Morand, F.

Mourard, D.

North-Morris, M.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Novak, M.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Ohara, C.

Ohara, C. M.

F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
[CrossRef]

Pa’dua, S.

S. P. Walborn, M. O. Terra Cunha, S. Pa’dua, and C. H. Monken, “Double-slit quantum eraser,” Phys. Rev. A 65(3), 033818 (2002).
[CrossRef]

Petrone, P.

F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
[CrossRef]

Redding, D. C.

F. Shi, G. Chanan, C. Ohara, M. Troy, and D. C. Redding, “Experimental verification of dispersed fringe sensing as a segment phasing technique using the Keck telescope,” Appl. Opt. 43(23), 4474–4481 (2004).
[CrossRef] [PubMed]

F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
[CrossRef]

Roy, M.

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37(6), 631–641 (2002).
[CrossRef]

M. Roy and P. Hariharan, “White-light geometric phase interferometer for surface profiling,” Proc. SPIE 2544, 64–73 (1995).
[CrossRef]

Seo, B.-J.

J. A. Spechler, D. J. Hoppe, N. Sigrist, F. Shi, B.-J. Seo, and S. Bikkannavar, “Advanced DFS: a dispersed fringe sensing algorithm insensitive to small calibration errors,” Proc. SPIE 7731, 773155 (2010).

Sheppard, C. J. R.

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37(6), 631–641 (2002).
[CrossRef]

Shi, F.

J. A. Spechler, D. J. Hoppe, N. Sigrist, F. Shi, B.-J. Seo, and S. Bikkannavar, “Advanced DFS: a dispersed fringe sensing algorithm insensitive to small calibration errors,” Proc. SPIE 7731, 773155 (2010).

F. Shi, G. Chanan, C. Ohara, M. Troy, and D. C. Redding, “Experimental verification of dispersed fringe sensing as a segment phasing technique using the Keck telescope,” Appl. Opt. 43(23), 4474–4481 (2004).
[CrossRef] [PubMed]

F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
[CrossRef]

Sigrist, N.

J. A. Spechler, D. J. Hoppe, N. Sigrist, F. Shi, B.-J. Seo, and S. Bikkannavar, “Advanced DFS: a dispersed fringe sensing algorithm insensitive to small calibration errors,” Proc. SPIE 7731, 773155 (2010).

Sirohi, R. S.

S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase-shifting by a rotating polarizer,” Opt. Commun. 154(5-6), 249–254 (1998).
[CrossRef]

Spechler, J. A.

J. A. Spechler, D. J. Hoppe, N. Sigrist, F. Shi, B.-J. Seo, and S. Bikkannavar, “Advanced DFS: a dispersed fringe sensing algorithm insensitive to small calibration errors,” Proc. SPIE 7731, 773155 (2010).

Stee, Ph.

Svahn, P.

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37(6), 631–641 (2002).
[CrossRef]

Tallon-Bosc, I.

Terra Cunha, M. O.

S. P. Walborn, M. O. Terra Cunha, S. Pa’dua, and C. H. Monken, “Double-slit quantum eraser,” Phys. Rev. A 65(3), 033818 (2002).
[CrossRef]

Troy, M.

Vakili, F.

Walborn, S. P.

S. P. Walborn, M. O. Terra Cunha, S. Pa’dua, and C. H. Monken, “Double-slit quantum eraser,” Phys. Rev. A 65(3), 033818 (2002).
[CrossRef]

Wyant, J.

Wyant, J. C.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Zhao, W.

Zimmerman, D. C.

Appl. Opt. (5)

Opt. Commun. (1)

S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase-shifting by a rotating polarizer,” Opt. Commun. 154(5-6), 249–254 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Lasers Eng. 37(6), 631–641 (2002).
[CrossRef]

Phys. Rev. A (1)

S. P. Walborn, M. O. Terra Cunha, S. Pa’dua, and C. H. Monken, “Double-slit quantum eraser,” Phys. Rev. A 65(3), 033818 (2002).
[CrossRef]

Proc. SPIE (6)

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

F. Shi, D. C. Redding, A. E. Lowman, C. W. Bowers, L. A. Burns, P. Petrone, C. M. Ohara, and S. A. Basinger, “Segmented mirror coarse phasing with a dispersed fringe sensor: experiment on NGST’s wavefront control testbed,” Proc. SPIE 4850, 318–328 (2003).
[CrossRef]

M. Roy and P. Hariharan, “White-light geometric phase interferometer for surface profiling,” Proc. SPIE 2544, 64–73 (1995).
[CrossRef]

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE 5896, 126–133 (2005).

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992).
[CrossRef]

J. A. Spechler, D. J. Hoppe, N. Sigrist, F. Shi, B.-J. Seo, and S. Bikkannavar, “Advanced DFS: a dispersed fringe sensing algorithm insensitive to small calibration errors,” Proc. SPIE 7731, 773155 (2010).

Other (4)

G. Olczak, “Recent advances in wavefront sensing at ITT,” ITT Internal Report (2009).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory Algorithms, and Software (John Wiley and Sons, Inc. 1998)

G. Olczak and C. Merle, “Polarization modulated image conjugate piston sensing and phase retrieval system,” ITT Manufacturing Enterprises, Inc., US Patent 7,864,333 (2011)

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (John Wiley & Sons, 2007).

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Figures (6)

Fig. 1
Fig. 1

Depiction of a polarization phase shifting dispersed fringe sensor.

Fig. 2
Fig. 2

(a) Phase shifted dispersed fringe images for δ = 0 microns, θ n =  {0, π 4 , π 2 , 3π 4 } from left to right. (b) Phase shifted dispersed fringe images for δ = −250 microns, θ n =  {0, π 4 , π 2 , 3π 4 } from left to right.

Fig. 3
Fig. 3

(a) Phase maps in radians for δ = 0 microns (left) and δ = −250 microns (right). (b) Modulation maps forδ = 0 microns (left) and δ = −250 microns (right). (c) Incoherent images for δ = 0 microns (left) and δ = −250 microns (right).

Fig. 4
Fig. 4

Difference map of Iincoh for δ = 0 microns minus Iincoh for δ = −250 microns.

Fig. 5
Fig. 5

Unwrapped phase maps in radians for δ = 0 microns (left) and δ = −250 microns (right).

Fig. 6
Fig. 6

Piston estimation error for SNR {50, 100, 100} (left to right).

Tables (1)

Tables Icon

Table 1 Simulation Parameters

Equations (8)

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λ ( x )=  λ 0 +  λ x  x=  λ 0 +  C 0  x,
x λ ( λ )=  λ  λ 0 C 0 .
I 0 ( x,y )= 2  [ l w f ] 2 λ min λ max S (λ)  1 λ 2   sinc [ l (x x λ ( λ )) λ f ] 2  sinc [ w y λ f ] 2  { 1+ cos( 2π λ [ η 0 y f + δ ] )  } dλ,
I n ( x,y )=  1 2   [ l w f ] 2 λ min λ max S (λ)  1 λ 2   sinc [ l (x x λ ( λ )) λ f ] 2  sinc [ w y λ f ] 2  { 1+ cos( 2π λ [ η 0 y f + δ ]+ 2 θ n )  } dλ
ϕ( x,y )=  tan 1 [ I 1 (x,y) I 3 (x,y) I 2 (x,y) I 4 (x,y) ].
δ ^ = M 1 Z ^ .
γ(x,y)= 2 ( [ I 1 (x,y) I 3 (x,y)] 2 + [ I 2 (x,y) I 4 (x,y)] 2   I 1 ( x,y )+ I 2 ( x,y ) +  I 3 ( x,y )+ I 4 (x,y) ) 1/2 .
I incoh =  I 1 ( x,y )+ I 2 ( x,y ) +  I 3 ( x,y )+ I 4 (x,y).

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