Abstract

We present a noncontact optical metrology measuring the pistons and tip/tilt angles of the 61 hexagonal segments of a compact-sized segmented mirror. The instrument has been developed within the scope of a design study for the European Extremely Large Telescope (E-ELT). It is used as reference sensor for cophasing of the mirror segments in closed-loop control. The mirror shape is also measured by different types of stellar light-based phasing cameras whose performances will be evaluated with regard to a future E-ELT. Following a description of the system architecture, the second part of the paper presents experimental results demonstrating the achieved precision: 0.48nmrms in piston and 74nradrms in tip/tilt.

© 2008 Optical Society of America

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References

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  1. P.Y.Bely, ed., The Design and Construction of Large Optical Telescopes (Springer Verlag, 2003).
  2. F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
    [CrossRef]
  3. C. Dupuy, C. Frank, and F. Y. J. Gonté, “ASM: a scaled-down active segmented mirror for the active phasing experiment,” Proc. SPIE 7012, 70123B (2008).
    [CrossRef]
  4. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).
  5. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005).
    [CrossRef]
  6. B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
    [CrossRef]
  7. R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-364 (1984).
  8. C. Dunsby, Y. Gu, and P. M. W. French, “Single-shot phase-stepped wide-field coherence-gated imaging,” Opt. Express 105, 105-115 (2003).
    [CrossRef]
  9. C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 472-480 (1992).
  10. J. Oberfell, MICOS GmbH, Eschbach, Germany (personal communication, 2005).
  11. K. G. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9, 236-253 (2001).
    [CrossRef] [PubMed]
  12. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping. Theory, Algorithms, and Software (Wiley, 1998).
  13. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832-843 (1996).
    [CrossRef]

2008

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

C. Dupuy, C. Frank, and F. Y. J. Gonté, “ASM: a scaled-down active segmented mirror for the active phasing experiment,” Proc. SPIE 7012, 70123B (2008).
[CrossRef]

2005

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005).
[CrossRef]

J. Oberfell, MICOS GmbH, Eschbach, Germany (personal communication, 2005).

2003

C. Dunsby, Y. Gu, and P. M. W. French, “Single-shot phase-stepped wide-field coherence-gated imaging,” Opt. Express 105, 105-115 (2003).
[CrossRef]

P.Y.Bely, ed., The Design and Construction of Large Optical Telescopes (Springer Verlag, 2003).

2001

1999

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

1998

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

1996

1992

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 472-480 (1992).

1984

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-364 (1984).

Araujo, C.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Bourtembourg, R.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Brast, R.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Chueca, S.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Derie, F.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Dorrío, B. V.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Duhoux, P.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Dunsby, C.

C. Dunsby, Y. Gu, and P. M. W. French, “Single-shot phase-stepped wide-field coherence-gated imaging,” Opt. Express 105, 105-115 (2003).
[CrossRef]

Dupuy, C.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

C. Dupuy, C. Frank, and F. Y. J. Gonté, “ASM: a scaled-down active segmented mirror for the active phasing experiment,” Proc. SPIE 7012, 70123B (2008).
[CrossRef]

Fernández, J. L.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Frank, C.

C. Dupuy, C. Frank, and F. Y. J. Gonté, “ASM: a scaled-down active segmented mirror for the active phasing experiment,” Proc. SPIE 7012, 70123B (2008).
[CrossRef]

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

French, P. M. W.

C. Dunsby, Y. Gu, and P. M. W. French, “Single-shot phase-stepped wide-field coherence-gated imaging,” Opt. Express 105, 105-115 (2003).
[CrossRef]

Ghiglia, D. C.

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

Gonté, F. Y. J.

C. Dupuy, C. Frank, and F. Y. J. Gonté, “ASM: a scaled-down active segmented mirror for the active phasing experiment,” Proc. SPIE 7012, 70123B (2008).
[CrossRef]

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Gu, Y.

C. Dunsby, Y. Gu, and P. M. W. French, “Single-shot phase-stepped wide-field coherence-gated imaging,” Opt. Express 105, 105-115 (2003).
[CrossRef]

Karban, R.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Koliopoulos, C. L.

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 472-480 (1992).

Larkin, K. G.

Luong, B.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Malacara, D.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005).
[CrossRef]

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005).
[CrossRef]

Mazzoleni, R.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Moore, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-364 (1984).

Noethe, L.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Oberfell, J.

J. Oberfell, MICOS GmbH, Eschbach, Germany (personal communication, 2005).

Pinna, E.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Pritt, M. D.

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

Servín, M.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005).
[CrossRef]

Smythe, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-364 (1984).

Surdej, I.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Vigan, A.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Wilhelm, R.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

Yaitskova, N.

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Opt. Eng.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-364 (1984).

Opt. Express

C. Dunsby, Y. Gu, and P. M. W. French, “Single-shot phase-stepped wide-field coherence-gated imaging,” Opt. Express 105, 105-115 (2003).
[CrossRef]

K. G. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9, 236-253 (2001).
[CrossRef] [PubMed]

Proc. SPIE

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 472-480 (1992).

F. Y. J. Gonté, C. Araujo, R. Bourtembourg, R. Brast, F. Derie, P. Duhoux, C. Dupuy, C. Frank, R. Karban, R. Mazzoleni, L. Noethe, I. Surdej, N. Yaitskova, R. Wilhelm, B. Luong, E. Pinna, S. Chueca, and A. Vigan, “Active Phasing Experiment: preliminary results,” Proc. SPIE 7012, 70120Z (2008).
[CrossRef]

C. Dupuy, C. Frank, and F. Y. J. Gonté, “ASM: a scaled-down active segmented mirror for the active phasing experiment,” Proc. SPIE 7012, 70123B (2008).
[CrossRef]

Other

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis, 2005).
[CrossRef]

J. Oberfell, MICOS GmbH, Eschbach, Germany (personal communication, 2005).

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

P.Y.Bely, ed., The Design and Construction of Large Optical Telescopes (Springer Verlag, 2003).

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Figures (12)

Fig. 1
Fig. 1

ASM with 61 segments, each equipped with three piezo actuators. The flat-to-flat distance of a single segment is d = 17 mm . The total diameter of the mirror (corner to corner) is D = 154 mm .

Fig. 2
Fig. 2

Functional block diagram of the metrology system. Optical signals (in fibers or free space) are represented by dashed lines. Electrical signals are represented by solid lines. S PA and S RA are the probe and reference signals (electric field) whose phase difference φ carries information about the OPD between the two arms. S 1 S 4 are the four CCD images (interferograms) that are acquired by the image acquisition board in the computer. h, α, and β are the piston, tip, and tilt measured for each mirror segment, respectively.

Fig. 3
Fig. 3

Optical layout of the BCM (compare with Figs. 2, 4).

Fig. 4
Fig. 4

Optical layout of the interferometer. The dimensions of the rectangular breadboard are 1200 mm × 600 mm . The internal magnification between the exit pupil (top right) and each of the CCDs is M int = 1 / 3 . The segmented mirror is imaged onto the exit pupil by a beam expander (not shown here) with magnification of M ext = 1 / 10 . The gray squares represent the local state of polarization. The polarization of the probe beam is represented by a solid line, while the polarization of the reference beam is represented by a dashed line. A closer look to the BCM is provided in Fig. 3.

Fig. 5
Fig. 5

Systematic error for the calibration of phase differences between CCDs due to the nonuniformity of the translation stage movement for the three phase differences as a function of the number of CCD frames used.

Fig. 6
Fig. 6

Phase shift ( θ 2 ) between CCD2 and CCD1. On the left-hand side is the raw phase-shift, and on the right-hand side is a constant piecewise phase shift per segment that is effectively used to compute the instantaneous phases during the measurement step.

Fig. 7
Fig. 7

Matching of the images using the contour line of the fringe contrast (white polygonal curves). The vertices of this polygon are extracted and matched to ideal positions of the vertices (cross markers). The original image (left) is matched to an ideal image (without transformation) by using a bicubic polynomial scheme (right).

Fig. 8
Fig. 8

Patch of 4 × 4 neighboring pixels used for tensorial bicubic polynomial interpolation at point ( x , y ) . Interpolation is performed to match the fringe images acquired by four CCDs.

Fig. 9
Fig. 9

Block diagram of the control loop of the APE experiment. The part of the diagram highlighted by the dashed line corresponds to the block diagram in Fig. 2.

Fig. 10
Fig. 10

Left: Variation of the OPD at λ 1 during approximately 7 h of measurement. Right: Differential OPD ( δ OPD 1 21 δ OPD 0 21 ) of segment 1 relative to the central segment as a function of the OPD at λ 1 .

Fig. 11
Fig. 11

Errors of estimation of phase (top row), amplitudes (middle row), and background (bottom row). The left-hand column shows the error obtained when using Hariharan’s method. For phase and amplitude the right-hand column refers to Larkin’s method, whereas for the background it shows the error related to the background estimation formula [Eq. (A2)].

Fig. 12
Fig. 12

Standard deviation of piston and tip/tilt in closed loop during 7 h . While the piston is measured for segments 0 , , 60 relative to the central segment, the tip and tilt are measured for all 61 segments. The standard deviation of pistons of all segments all together is 0.48 nm rms . The standard deviation of tip/tilt of all segments all together is 74 nrad rms .

Tables (1)

Tables Icon

Table 1 Achieved Specifications of the Metrology

Equations (35)

Equations on this page are rendered with MathJax. Learn more.

Λ s = λ 1 λ 2 | λ 1 λ 2 | ,
I i k ( x , y ) = B i k ( x , y ) + A i k ( x , y ) sin [ φ k ( x , y ) + θ i k ( x , y ) ] ,
B i k ( x , y ) = I i , probe k + I i , ref k ,
A i k ( x , y ) = 2 ( I i , probe k I i , ref k ) 1 / 2 .
( θ 1 , θ 2 , θ 3 , θ 4 ) ( 0 , π , π 2 , π 2 )
δ θ = 4 π λ Δ p M t o t a l δ x · G TT ,
I ( x , y ) = p , q a p q x p y q , with ( p , q ) = 0 , , 3 .
I ˜ i k ( x , y ) = I i k ( x , y ) B i , cal k ( x , y ) A i , c a l k ( x , y ) I i k ( x , y ) B i k ( x , y ) A i k ( x , y ) = sin ( φ k ( x , y ) + θ i k ( x , y ) ) .
I ˜ i k ( x , y ) = A ˜ k ( x , y ) sin ( φ k ( x , y ) + θ i k ( x , y ) ) + B ˜ k ( x , y ) ,
( φ k ( x , y ) , A ˜ k , B ˜ k ) = argmin i = 1 , , 4 ( I i k , m o d e l ( φ k , A ˜ , B ˜ ) I ˜ i k , d a t a ) 2 ,
z m ( x , y ) = h m + α m x + β m y + δ z m ( x , y ) ,
φ ˜ k m ( x , y ) = φ k m ( x , y ) 4 π δ z m ( x , y ) λ k .
z ˜ m ( x , y ) = z m ( x , y ) δ z m ( x , y ) = h m + α m x + β m y .
φ ˜ m k x = M t o t a l Δ p N p i x e l i , j ( φ ˜ m k ( x i + 1 , y j ) φ ˜ m k ( x i , y j ) + f i , j 2 π ) ,
| φ ˜ m k ( x i + 1 , y j ) φ ˜ m k ( x i , y j ) + f i , j 2 π | π .
φ ˜ m k y = M t o t a l Δ p N p i x e l i , j ( φ ˜ m k ( x i , y j + 1 ) φ ˜ m k ( x i , y j ) + g i , j 2 π ) ,
4 π h m λ k = φ m k ( φ ˜ m k ( x i , y j ) φ ˜ m k x x i φ ˜ m k y y j ) × mod 2 π ,     pixel i , j .
φ m k tan 1 [ i , j sin ( φ ˜ m k ( x i , y j ) φ ˜ m k x x i φ ˜ m k y y j ) i , j cos ( φ ˜ m k ( x i , y j ) φ ˜ m k x x i φ ˜ m k y y j ) ] .
α m = 1 8 π ( λ 1 φ ˜ m 1 x + λ 2 φ ˜ m 2 x ) ,
β m = 1 8 π ( λ 1 φ ˜ m 1 y + λ 2 φ ˜ m 2 y ) .
4 π h m λ k = φ m k + c m k 2 π , ( k = 1 , 2 ) .
OPD m k = λ k 2 π unwrap φ m k , for     m = 0 , , 60 and k = 1 , 2 .
δ OPD m 21 = OPD m 2 OPD m 1 , for     m = 0 , , 60 .
σ m , h = ( 1 N j = 1 , , N h m , j 2 ) 1 2 .
σ m , γ = ( 1 N j = 1 , , N γ m , j 2 ) 1 2 ,
γ m , j = α m , j 2 + β m , j 2 .
z j = ( j 3 ) λ 8 ( 1 + Δ v ) , for     j = 1 , , 5 .
B e s t i m a t e = ( I 1 + 2 I 3 + I 5 ) / 4 .
B e s t i m a t e = B t r u e + sin ( θ ) A C Δ v 2 ,
( φ 1 2 π + c 1 ) λ 1 = ( φ 2 2 π + c 2 ) λ 2 .
( φ 1 + δ φ 1 2 π + c 1 + Δ c ) λ 1 = ( φ 2 + δ φ 2 2 π + c 2 + Δ c ) λ 2 .
δ OPD 2 δ OPD 1 = Δ c ( λ 2 λ 1 ) ,
δ OPD 21 λ 2 λ 1 = Δ c .
3 2 ( λ 2 λ 1 ) δ OPD 21 1 2 ( λ 2 λ 1 ) .
5 2 ( λ 2 λ 1 ) < 3 2 ( λ 2 λ 1 ) < 1 2 ( λ 2 λ 1 ) < + 1 2 ( λ 2 λ 1 ) < + 3 2 ( λ 2 λ 1 ) <

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