Abstract

New architectures for telescopes or powerful lasers require segmented wave front metrology. This paper deals with a new interferometric wave front sensing technique called PISTIL (PISton and TILt), able to recover both piston and tilts of segment beams. The main advantages of the PISTIL technique are the absence of a reference arm and an access to the tilt information. An explanation of the principle, as well as an experimental implementation and the use of a segmented active mirror, are presented. Measurement errors of λ/200 for piston and 40 µrad for tilts have been achieved, well beyond performances requested for the above mentioned applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  21. L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

2017 (1)

2016 (1)

2015 (1)

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

2013 (1)

G. Mourou, B. Brocklesby, T. Toshiki, and J. Limpert, “The future is fiber accelerators,” Nat. Photonics 7(4), 258–261 (2013).

2010 (1)

A. Norton, D. Gavel, and D. Dillon, “High-power visible-laser effect on a 37-segment Iris AO deformable mirror,” Proc. SPIE 7595, 759506 (2010).

2008 (1)

2007 (1)

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).

2006 (2)

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Performance of a high-stroke segmented MEMS deformable-mirror technology,” Proc. SPIE 6113, 61130L (2006).

2005 (1)

2004 (1)

J.-C. Chanteloup and M. Cohen, “Compact high resolution four wave lateral shearing interferometer,” Proc. SPIE 5252, 5252 (2004).

2001 (1)

M. G. Löfdahl and H. Eriksson, “Algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40(6), 984–990 (2001).

1998 (1)

1982 (1)

1980 (1)

1955 (1)

R. Penrose and J. A. Todd, “A generalized inverse for matrices,” Math. Proc. Camb. Philos. Soc. 51(3), 406–413 (1955).

Antier, M.

Antier-Murgey, M.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

Bellanger, C.

Bellanger, S.

J. Le Dortz, A. Heilmann, M. Antier, J. Bourderionnet, C. Larat, I. Fsaifes, L. Daniault, S. Bellanger, C. Simon Boisson, J.-C. Chanteloup, E. Lallier, and A. Brignon, “Highly scalable femtosecond coherent beam combining demonstrated with 19 fibers,” Opt. Lett. 42(10), 1887–1890 (2017).

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

Bourderionnet, J.

J. Le Dortz, A. Heilmann, M. Antier, J. Bourderionnet, C. Larat, I. Fsaifes, L. Daniault, S. Bellanger, C. Simon Boisson, J.-C. Chanteloup, E. Lallier, and A. Brignon, “Highly scalable femtosecond coherent beam combining demonstrated with 19 fibers,” Opt. Lett. 42(10), 1887–1890 (2017).

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

Brignon, A.

J. Le Dortz, A. Heilmann, M. Antier, J. Bourderionnet, C. Larat, I. Fsaifes, L. Daniault, S. Bellanger, C. Simon Boisson, J.-C. Chanteloup, E. Lallier, and A. Brignon, “Highly scalable femtosecond coherent beam combining demonstrated with 19 fibers,” Opt. Lett. 42(10), 1887–1890 (2017).

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

Brocklesby, B.

G. Mourou, B. Brocklesby, T. Toshiki, and J. Limpert, “The future is fiber accelerators,” Nat. Photonics 7(4), 258–261 (2013).

Chanan, G.

Chanteloup, J.-C.

J. Le Dortz, A. Heilmann, M. Antier, J. Bourderionnet, C. Larat, I. Fsaifes, L. Daniault, S. Bellanger, C. Simon Boisson, J.-C. Chanteloup, E. Lallier, and A. Brignon, “Highly scalable femtosecond coherent beam combining demonstrated with 19 fibers,” Opt. Lett. 42(10), 1887–1890 (2017).

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

J.-C. Chanteloup and M. Cohen, “Compact high resolution four wave lateral shearing interferometer,” Proc. SPIE 5252, 5252 (2004).

Cohen, M.

J.-C. Chanteloup and M. Cohen, “Compact high resolution four wave lateral shearing interferometer,” Proc. SPIE 5252, 5252 (2004).

Daniault, L.

J. Le Dortz, A. Heilmann, M. Antier, J. Bourderionnet, C. Larat, I. Fsaifes, L. Daniault, S. Bellanger, C. Simon Boisson, J.-C. Chanteloup, E. Lallier, and A. Brignon, “Highly scalable femtosecond coherent beam combining demonstrated with 19 fibers,” Opt. Lett. 42(10), 1887–1890 (2017).

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

Dekens, F.

Deprez, M.

Dillon, D.

A. Norton, D. Gavel, and D. Dillon, “High-power visible-laser effect on a 37-segment Iris AO deformable mirror,” Proc. SPIE 7595, 759506 (2010).

Doble, N.

A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Performance of a high-stroke segmented MEMS deformable-mirror technology,” Proc. SPIE 6113, 61130L (2006).

Dohlen, K.

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

Eriksson, H.

M. G. Löfdahl and H. Eriksson, “Algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40(6), 984–990 (2001).

Esposito, S.

Fsaifes, I.

Gavel, D.

A. Norton, D. Gavel, and D. Dillon, “High-power visible-laser effect on a 37-segment Iris AO deformable mirror,” Proc. SPIE 7595, 759506 (2010).

Guérineau, N.

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).

Haïdar, R.

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).

Hart, M.

A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Performance of a high-stroke segmented MEMS deformable-mirror technology,” Proc. SPIE 6113, 61130L (2006).

Heilmann, A.

Helmbrecht, A.

A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Performance of a high-stroke segmented MEMS deformable-mirror technology,” Proc. SPIE 6113, 61130L (2006).

Hernandez, E.

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

Ina, H.

Juneau, T.

A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Performance of a high-stroke segmented MEMS deformable-mirror technology,” Proc. SPIE 6113, 61130L (2006).

Kirkman, D.

Kobayashi, S.

Lallier, E.

J. Le Dortz, A. Heilmann, M. Antier, J. Bourderionnet, C. Larat, I. Fsaifes, L. Daniault, S. Bellanger, C. Simon Boisson, J.-C. Chanteloup, E. Lallier, and A. Brignon, “Highly scalable femtosecond coherent beam combining demonstrated with 19 fibers,” Opt. Lett. 42(10), 1887–1890 (2017).

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

Langlois, M.

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

Lanzoni, P.

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

Larat, C.

J. Le Dortz, A. Heilmann, M. Antier, J. Bourderionnet, C. Larat, I. Fsaifes, L. Daniault, S. Bellanger, C. Simon Boisson, J.-C. Chanteloup, E. Lallier, and A. Brignon, “Highly scalable femtosecond coherent beam combining demonstrated with 19 fibers,” Opt. Lett. 42(10), 1887–1890 (2017).

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

Le Dortz, J.

J. Le Dortz, A. Heilmann, M. Antier, J. Bourderionnet, C. Larat, I. Fsaifes, L. Daniault, S. Bellanger, C. Simon Boisson, J.-C. Chanteloup, E. Lallier, and A. Brignon, “Highly scalable femtosecond coherent beam combining demonstrated with 19 fibers,” Opt. Lett. 42(10), 1887–1890 (2017).

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

Limpert, J.

G. Mourou, B. Brocklesby, T. Toshiki, and J. Limpert, “The future is fiber accelerators,” Nat. Photonics 7(4), 258–261 (2013).

Löfdahl, M. G.

M. G. Löfdahl and H. Eriksson, “Algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40(6), 984–990 (2001).

Lombard, L.

Mast, T.

Mazzanti, S.

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

Mercier, R.

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).

Michaels, S.

Montoya, L.

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

Mourou, G.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

G. Mourou, B. Brocklesby, T. Toshiki, and J. Limpert, “The future is fiber accelerators,” Nat. Photonics 7(4), 258–261 (2013).

Nelson, J.

Norton, A.

A. Norton, D. Gavel, and D. Dillon, “High-power visible-laser effect on a 37-segment Iris AO deformable mirror,” Proc. SPIE 7595, 759506 (2010).

Penrose, R.

R. Penrose and J. A. Todd, “A generalized inverse for matrices,” Math. Proc. Camb. Philos. Soc. 51(3), 406–413 (1955).

Pinna, E.

Primot, J.

M. Deprez, C. Bellanger, L. Lombard, B. Wattellier, and J. Primot, “Piston and tilt interferometry for segmented wavefront sensing,” Opt. Lett. 41(6), 1078–1081 (2016).

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).

Puglisi, A.

Reyes, M.

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

Scipioni, M.

Simon Boisson, C.

Simon-Boisson, C.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

Southwell, W. H.

Stefanini, P.

Surdej, I.

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

Takeda, M.

Todd, J. A.

R. Penrose and J. A. Todd, “A generalized inverse for matrices,” Math. Proc. Camb. Philos. Soc. 51(3), 406–413 (1955).

Toshiki, T.

G. Mourou, B. Brocklesby, T. Toshiki, and J. Limpert, “The future is fiber accelerators,” Nat. Photonics 7(4), 258–261 (2013).

Toulon, B.

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).

Tozzi, A.

Troy, M.

Tyson, R. K.

Velghe, S.

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).

Viegas, J.

Vigan, A.

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

Wattellier, B.

Yaitskova, N.

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

Appl. Opt. (2)

Eur. Phys. J. Spec. Top. (1)

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, E. Lallier, C. Larat, M. Antier-Murgey, J.-C. Chanteloup, A. Brignon, C. Simon-Boisson, and G. Mourou, “XCAN – A coherent amplification network of femtosecond fiber-chirped-pulse amplifiers,” Eur. Phys. J. Spec. Top. 224(13), 2609–2613 (2015).

J. Opt. Soc. Am. (2)

Math. Proc. Camb. Philos. Soc. (1)

R. Penrose and J. A. Todd, “A generalized inverse for matrices,” Math. Proc. Camb. Philos. Soc. 51(3), 406–413 (1955).

Nat. Photonics (1)

G. Mourou, B. Brocklesby, T. Toshiki, and J. Limpert, “The future is fiber accelerators,” Nat. Photonics 7(4), 258–261 (2013).

Opt. Commun. (1)

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).

Opt. Eng. (1)

M. G. Löfdahl and H. Eriksson, “Algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40(6), 984–990 (2001).

Opt. Lett. (3)

Proc. SPIE (4)

A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Performance of a high-stroke segmented MEMS deformable-mirror technology,” Proc. SPIE 6113, 61130L (2006).

K. Dohlen, M. Langlois, P. Lanzoni, S. Mazzanti, A. Vigan, L. Montoya, E. Hernandez, M. Reyes, I. Surdej, and N. Yaitskova, “ZEUS: a cophasing sensor based on the Zernike phase contrast method,” Proc. SPIE 6267, 626734 (2006).

J.-C. Chanteloup and M. Cohen, “Compact high resolution four wave lateral shearing interferometer,” Proc. SPIE 5252, 5252 (2004).

A. Norton, D. Gavel, and D. Dillon, “High-power visible-laser effect on a 37-segment Iris AO deformable mirror,” Proc. SPIE 7595, 759506 (2010).

Other (5)

http://phasicscorp.com/

http://www.irisao.com/

http://www.jai.com

https://www.eso.org/sci/facilities/eelt

T. K. Korhonen, “Interferometric method for optical testing and wavefront error sensing,” Proc. SPIE 0444, Advanced Technology Optical Telescopes I (1983).

Supplementary Material (2)

NameDescription
» Visualization 1       Explanation when moving in piston or tilt on a segment on the interferogram
» Visualization 2       Animation which shows the residual after soustraction between command and measurement.

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Figures (11)

Fig. 1
Fig. 1 Schematic view of PISTIL technic with 3 elements in hexagonal mesh.
Fig. 2
Fig. 2 Description of two-wave interference due to lateral shearing with two plan segmented wave fronts. Top left: two plan wave fronts Wi and Wj at z = 0 (just before propagation through the grating). Top right: interference fringe pattern after a propagation.
Fig. 3
Fig. 3 Piston and tip/tilt impact on fringes (see Visualization 1).
Fig. 4
Fig. 4 Left, values of the phase shifts within segments; right, example of reconstituted relative phase difference image from typical pistilogram obtained in Fig. 3.
Fig. 5
Fig. 5 Experimental setup with segmented mirror and high speed camera.
Fig. 6
Fig. 6 Photo of the experimental setup.
Fig. 7
Fig. 7 Experimental pistilogram generate with previous setup.
Fig. 8
Fig. 8 Top left: command sent to the mirror. Top right: Measure with PISTIL interferometer. Bottom left: Residual (difference between top right and top left). Bottom right: Residual zoomed by 100.
Fig. 9
Fig. 9 In the left hand side, the command sent to the deformable mirror, at right the residual between the measure and the command, and at the bottom the residual zoomed by 10 (see Visualization 2).
Fig. 10
Fig. 10 Measurements done with small variations in piston for one segment
Fig. 11
Fig. 11 Representation of measurements of pistons for all segments according to commands sent to the mirror. Regression fit is shown on the plot.

Tables (1)

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Table 1 Noise, repeatability and measurement error in case of piston and tilt for PISTIL interferometer

Equations (13)

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k i =( αsinθ αcosθ 1 ), k j =( αsinθ αcosθ 1 )
sin( α )= mλ p g
α λ p g
cos( θ )= k i . Oy k i . Oy
s i =d k i =( d λ p g sinθ d λ p g cosθ d ), s j =d k j =( d λ p g sinθ d λ p g cosθ d )
U i λ ( x,y,z )= U i 0 (x,y)×exp( i 2π λ k i . r +i 2π λ W i ( r s i ) )
I ij λ ( x,y,z )=2 I 0 +2 I 0 cos( 2π λ ( k i k j ). r + 2π λ ( W i ( r s i ) W j ( r s j ) ) )
W i ( x,y,z )= t X i ( x X i )+ t Y i ( y Y i )+ A i +o( x n , y n ) W j ( x,y,z )= t Xj ( x X j )+ t Y j ( y Y j )+ A j +o( x n , y n )
I ij λ ( x,y,z )=2 I 0 +2 I 0 cos( 2π( ( 2sinθ p g + d t X ij λ )x +( 2cosθ p g + d t Y ij λ )y + 1 λ ( t X i M i X t X j M j X + t Y i M i Y t Y j M j Y +d P ij ) ) )
M i = C i + s i =( X i Y i 0 )+( d λ p g sinθ d λ p g cosθ d ) M j = C j + s j =( X j Y j 0 )+( d λ p g sinθ d λ p g cosθ d )
I ij λ ( x,y,z )=2 I 0 +2 I 0 cos( ax+by+Δφ )
a=2π( 2sinθ p g + d t X ij λ ) b=2π( 2cosθ p g + d t Y ij λ ) Δφ=2π d P ij λ
TF[ I ij λ ( x,y,z ) ]=2 I ˜ 0 +2 I ˜ 0 ×[ δ( υ x a 2π )δ( υ y b 2π )exp( 2iπΔφ )+c.c. ]

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