Abstract

In many metrological applications the data being measured is associated to the phase difference codified in two fringe patterns. This phase difference can be recovered directly with what are called Differential Phase Shifting Algorithms (DPSAs) by using a combination of irradiance values from both patterns in the arctangent argument. Use of such algorithms requires characterisation mechanisms to inform of their sensitivity to the various random and systematic error sources, which is the same as for well-studied Phase Shifting Algorithms (PSAs). Thus, we present a new analysis of error propagation for DPSAs taking into account the frequency shifting property of the employed arctangent function. The general analysis is verified for significant specific cases associated to large errors that appear during phase difference evaluation using the Monte Carlo method, which provides a characterisation of a DPSA’s sensitivity; this is an alternative to spatial and temporal techniques but has wholly coinciding results. Monte Carlo simulation opens up the possibilities for the analysis of other error types for any DPSA.

© 2010 OSA

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References

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  1. B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), 33–55 (1999).
    [CrossRef]
  2. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor&Francis Group 2005).
  3. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
  4. D. C. Ghiglia, and M. D. Pritt, Two-dimensional phase unwrapping, Wiley, New York, 1998.
  5. K. A. Stetson, “Theory and applications of electronic holography,” in SEM Conference on Hologram Interferometry and Speckle Metrology: 294–300 (1990).
  6. M. Owner-Petersen, “Digital speckle pattern shearing interferometry: limitations and prospects,” Appl. Opt. 30(19), 2730–2738 (1991).
    [CrossRef] [PubMed]
  7. C. S. Vikram, W. K. Witherow, and J. D. Trolinger, “Algorithm for phase-difference measurement in phase-shifting interferometry,” Appl. Opt. 32(31), 6250–6252 (1993).
    [CrossRef] [PubMed]
  8. K. A. Stetson and W. R. Brohinsky, “Electrooptic holography and its application to hologram interferometry,” Appl. Opt. 24(21), 3631–3637 (1985).
    [CrossRef] [PubMed]
  9. H. Saldner, N. Molin, and K. Stetson, “Fourier-transform evaluation of phase data in spatially phase-biased TV holograms,” Appl. Opt. 35(2), 332–336 (1996).
    [CrossRef] [PubMed]
  10. J. Burke and H. Helmers, “Complex division as a common basis for calculating phase differences in electronic speckle pattern interferometry in one step,” Appl. Opt. 37(13), 2589–2590 (1998).
    [CrossRef]
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    [CrossRef]
  12. M. Miranda, and B. V. Dorrío, “Error behaviour in Differential Phase-Shifting Algorithms,” Proc. SPIE 7102, 71021B–1 - 71021B–9 (2008).
  13. M. Miranda, and B. V. Dorrío, “Design and assessment of Differential Phase-Shifting Algorithms by means of their Fourier representation,” in Fringe 2009 (Elsevier, Stuttgart, 2009) 153–159.
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    [CrossRef]
  22. J. V. Wingerden, H. J. Frankena, and C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30(19), 2718–2729 (1991).
    [CrossRef] [PubMed]
  23. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983).
    [CrossRef] [PubMed]
  24. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
    [CrossRef] [PubMed]
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    [CrossRef]
  26. M. A. Herrador, A. G. Asuero, and G. González, “Estimation of the uncertainty of indirect measurements from the propagation of distributions by using the Monte-Carlo method: An overview,” Chemom. Intell. Lab. Syst. 79(1-2), 115–122 (2005).
    [CrossRef]
  27. M. G. Cox and B. R. L. Siebert, “The use of a Monte Carlo method for evaluating uncertainty and expanded uncertainty,” Metrologia 43(4), S178–S188 (2006).
    [CrossRef]
  28. V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
    [CrossRef]
  29. R. R. Cordero, J. Molimard, A. Martínez, and F. Labbe, “Uncertainty analysis of temporal phase-stepping algorithms for interferometry,” Opt. Commun. 275(1), 144–155 (2007).
    [CrossRef]

2009 (6)

2007 (2)

R. R. Cordero, J. Molimard, A. Martínez, and F. Labbe, “Uncertainty analysis of temporal phase-stepping algorithms for interferometry,” Opt. Commun. 275(1), 144–155 (2007).
[CrossRef]

M. Miranda and B. V. Dorrío, “Error-phase compensation properties of differential phase-shifting algorithms for Fizeau fringe patterns ,” in RIAO/OPTILAS 2007 AIP Conf. Proc. 992, 993–998 (2007).
[CrossRef]

2006 (1)

M. G. Cox and B. R. L. Siebert, “The use of a Monte Carlo method for evaluating uncertainty and expanded uncertainty,” Metrologia 43(4), S178–S188 (2006).
[CrossRef]

2005 (2)

M. G. Cox and P. Harris, “An outline of Supplement 1 to the Guide to the Expression of Uncertainty in Measurement on numerical methods for the propagation of distributions,” Meas. Tech. 48(4), 336–345 (2005).
[CrossRef]

M. A. Herrador, A. G. Asuero, and G. González, “Estimation of the uncertainty of indirect measurements from the propagation of distributions by using the Monte-Carlo method: An overview,” Chemom. Intell. Lab. Syst. 79(1-2), 115–122 (2005).
[CrossRef]

1999 (1)

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), 33–55 (1999).
[CrossRef]

1998 (1)

1997 (1)

J. M. Huntley, “Random phase measurements errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26(2-3), 131–150 (1997).
[CrossRef]

1996 (1)

1993 (1)

1992 (1)

K. G. Larkin and B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” Proc. SPIE 1775, 219–227 (1992).

1991 (2)

1987 (1)

1985 (1)

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

1983 (1)

Álvarez-Valado, V.

V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
[CrossRef]

Asuero, A. G.

M. A. Herrador, A. G. Asuero, and G. González, “Estimation of the uncertainty of indirect measurements from the propagation of distributions by using the Monte-Carlo method: An overview,” Chemom. Intell. Lab. Syst. 79(1-2), 115–122 (2005).
[CrossRef]

Brohinsky, W. R.

Burke, J.

Burow, R.

Cordero, R. R.

R. R. Cordero, J. Molimard, A. Martínez, and F. Labbe, “Uncertainty analysis of temporal phase-stepping algorithms for interferometry,” Opt. Commun. 275(1), 144–155 (2007).
[CrossRef]

Cox, M. G.

M. G. Cox and B. R. L. Siebert, “The use of a Monte Carlo method for evaluating uncertainty and expanded uncertainty,” Metrologia 43(4), S178–S188 (2006).
[CrossRef]

M. G. Cox and P. Harris, “An outline of Supplement 1 to the Guide to the Expression of Uncertainty in Measurement on numerical methods for the propagation of distributions,” Meas. Tech. 48(4), 336–345 (2005).
[CrossRef]

Cywiak, M.

Doblado, D. M.

Dorrío, B. V.

V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
[CrossRef]

M. Miranda and B. V. Dorrío, “Error-phase compensation properties of differential phase-shifting algorithms for Fizeau fringe patterns ,” in RIAO/OPTILAS 2007 AIP Conf. Proc. 992, 993–998 (2007).
[CrossRef]

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), 33–55 (1999).
[CrossRef]

Eiju, T.

Elssner, K. E.

Estrada, J. C.

Fernández, J. L.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), 33–55 (1999).
[CrossRef]

Frankena, H. J.

González, G.

M. A. Herrador, A. G. Asuero, and G. González, “Estimation of the uncertainty of indirect measurements from the propagation of distributions by using the Monte-Carlo method: An overview,” Chemom. Intell. Lab. Syst. 79(1-2), 115–122 (2005).
[CrossRef]

González-Jorge, H.

V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Grzanna, J.

Hariharan, P.

Harris, P.

M. G. Cox and P. Harris, “An outline of Supplement 1 to the Guide to the Expression of Uncertainty in Measurement on numerical methods for the propagation of distributions,” Meas. Tech. 48(4), 336–345 (2005).
[CrossRef]

Helmers, H.

Hernández, D. M.

Herrador, M. A.

M. A. Herrador, A. G. Asuero, and G. González, “Estimation of the uncertainty of indirect measurements from the propagation of distributions by using the Monte-Carlo method: An overview,” Chemom. Intell. Lab. Syst. 79(1-2), 115–122 (2005).
[CrossRef]

Huntley, J. M.

J. M. Huntley, “Random phase measurements errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26(2-3), 131–150 (1997).
[CrossRef]

Labbe, F.

R. R. Cordero, J. Molimard, A. Martínez, and F. Labbe, “Uncertainty analysis of temporal phase-stepping algorithms for interferometry,” Opt. Commun. 275(1), 144–155 (2007).
[CrossRef]

Larkin, K. G.

K. G. Larkin and B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” Proc. SPIE 1775, 219–227 (1992).

Martínez, A.

R. R. Cordero, J. Molimard, A. Martínez, and F. Labbe, “Uncertainty analysis of temporal phase-stepping algorithms for interferometry,” Opt. Commun. 275(1), 144–155 (2007).
[CrossRef]

Merkel, K.

Miranda, M.

V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
[CrossRef]

M. Miranda and B. V. Dorrío, “Error-phase compensation properties of differential phase-shifting algorithms for Fizeau fringe patterns ,” in RIAO/OPTILAS 2007 AIP Conf. Proc. 992, 993–998 (2007).
[CrossRef]

Molimard, J.

R. R. Cordero, J. Molimard, A. Martínez, and F. Labbe, “Uncertainty analysis of temporal phase-stepping algorithms for interferometry,” Opt. Commun. 275(1), 144–155 (2007).
[CrossRef]

Molin, N.

Mosiño, J. F.

Oreb, B. F.

K. G. Larkin and B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” Proc. SPIE 1775, 219–227 (1992).

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
[CrossRef] [PubMed]

Owner-Petersen, M.

Quiroga, J. A.

Rodriguez, J.

V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
[CrossRef]

Rodríguez, F.

V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
[CrossRef]

Rodríguez J, J.

V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
[CrossRef]

Saldner, H.

Schwider, J.

Servin, M.

Siebert, B. R. L.

M. G. Cox and B. R. L. Siebert, “The use of a Monte Carlo method for evaluating uncertainty and expanded uncertainty,” Metrologia 43(4), S178–S188 (2006).
[CrossRef]

Smorenburg, C.

Spolaczyk, R.

Stetson, K.

Stetson, K. A.

Trolinger, J. D.

Valencia, J. L.

V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
[CrossRef]

Vikram, C. S.

Wingerden, J. V.

Witherow, W. K.

Yebra, F. J.

V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
[CrossRef]

Appl. Opt. (8)

M. Owner-Petersen, “Digital speckle pattern shearing interferometry: limitations and prospects,” Appl. Opt. 30(19), 2730–2738 (1991).
[CrossRef] [PubMed]

C. S. Vikram, W. K. Witherow, and J. D. Trolinger, “Algorithm for phase-difference measurement in phase-shifting interferometry,” Appl. Opt. 32(31), 6250–6252 (1993).
[CrossRef] [PubMed]

K. A. Stetson and W. R. Brohinsky, “Electrooptic holography and its application to hologram interferometry,” Appl. Opt. 24(21), 3631–3637 (1985).
[CrossRef] [PubMed]

H. Saldner, N. Molin, and K. Stetson, “Fourier-transform evaluation of phase data in spatially phase-biased TV holograms,” Appl. Opt. 35(2), 332–336 (1996).
[CrossRef] [PubMed]

J. Burke and H. Helmers, “Complex division as a common basis for calculating phase differences in electronic speckle pattern interferometry in one step,” Appl. Opt. 37(13), 2589–2590 (1998).
[CrossRef]

J. V. Wingerden, H. J. Frankena, and C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30(19), 2718–2729 (1991).
[CrossRef] [PubMed]

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
[CrossRef] [PubMed]

Chemom. Intell. Lab. Syst. (1)

M. A. Herrador, A. G. Asuero, and G. González, “Estimation of the uncertainty of indirect measurements from the propagation of distributions by using the Monte-Carlo method: An overview,” Chemom. Intell. Lab. Syst. 79(1-2), 115–122 (2005).
[CrossRef]

in RIAO/OPTILAS 2007 AIP Conf. Proc. (1)

M. Miranda and B. V. Dorrío, “Error-phase compensation properties of differential phase-shifting algorithms for Fizeau fringe patterns ,” in RIAO/OPTILAS 2007 AIP Conf. Proc. 992, 993–998 (2007).
[CrossRef]

Meas. Sci. Technol. (1)

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), 33–55 (1999).
[CrossRef]

Meas. Tech. (1)

M. G. Cox and P. Harris, “An outline of Supplement 1 to the Guide to the Expression of Uncertainty in Measurement on numerical methods for the propagation of distributions,” Meas. Tech. 48(4), 336–345 (2005).
[CrossRef]

Metrologia (2)

M. G. Cox and B. R. L. Siebert, “The use of a Monte Carlo method for evaluating uncertainty and expanded uncertainty,” Metrologia 43(4), S178–S188 (2006).
[CrossRef]

V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009).
[CrossRef]

Opt. Commun. (1)

R. R. Cordero, J. Molimard, A. Martínez, and F. Labbe, “Uncertainty analysis of temporal phase-stepping algorithms for interferometry,” Opt. Commun. 275(1), 144–155 (2007).
[CrossRef]

Opt. Eng. (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Opt. Express (5)

Opt. Lasers Eng. (1)

J. M. Huntley, “Random phase measurements errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26(2-3), 131–150 (1997).
[CrossRef]

Proc. SPIE (1)

K. G. Larkin and B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” Proc. SPIE 1775, 219–227 (1992).

Other (6)

M. Miranda and B.V. Dorrío, “Fourier analysis of two-stage phase-shifting algorithms,” J. Opt. Soc. Am. A, doc. ID 115412 (posted 1 December 2009, in press).

D. C. Ghiglia, and M. D. Pritt, Two-dimensional phase unwrapping, Wiley, New York, 1998.

K. A. Stetson, “Theory and applications of electronic holography,” in SEM Conference on Hologram Interferometry and Speckle Metrology: 294–300 (1990).

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor&Francis Group 2005).

M. Miranda, and B. V. Dorrío, “Error behaviour in Differential Phase-Shifting Algorithms,” Proc. SPIE 7102, 71021B–1 - 71021B–9 (2008).

M. Miranda, and B. V. Dorrío, “Design and assessment of Differential Phase-Shifting Algorithms by means of their Fourier representation,” in Fringe 2009 (Elsevier, Stuttgart, 2009) 153–159.

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Figures (8)

Fig. 1
Fig. 1

Uniform distribution of values of εq and χr .

Fig. 2
Fig. 2

Example of an original and modified pattern (m=1 and p=1) at point 2π/5 affected by detuning ε 1=β 1=0.1.

Fig. 3
Fig. 3

Example of an original and modified pattern (m=1 and p=1) at point 2π/5 affected by second order phase shift error ε 2=β 2=0.1.

Fig. 4
Fig. 4

Second order additional shift error for the AGDPSA (a), and the Schwider-Hariharan SGDPSA1 (b) and SGDPSA2 (c) comparing the analytical value from Table 1 (dashed) with that obtained with the MCM (solid).

Fig. 6
Fig. 6

Example of an original and modified pattern (m=1 and p=1) at point 2π/5 with the presence of undesired second order harmonics a 2=b 2=0.1.

Fig. 5
Fig. 5

Uniform distribution for the values of a 1 and b 1 or a 2 and b 2.

Fig. 7
Fig. 7

Example of an original and modified pattern (m=1 and p=1) at point 2π/5 affected by second order phase shift error a 3=b 3=0.1.

Fig. 8
Fig. 8

Presence of undesired third order harmonics for the AGDPSA (a), and Schwider-Hariharan SGDPSA1 (b) and SGDPSA (c) comparing the analytical value from Table 2 (dashed) with that obtained with the MCM (solid).

Tables (2)

Tables Icon

Table 1 Analytical expression for error in the additional phases of the Schwider-Hariharan GDPSAs.

Tables Icon

Table 2 Analytical simulation of the presence of undesired harmonics for the Schwider-Hariharan GDPSAs.

Equations (25)

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

s m ( ϕ , α m ) = k = 0 a c k o s [ k ( ϕ + α m ) ]
ϕ = arc tan N 1 { s m ( ϕ , α m ) } D 1 { s m ( ϕ , α m ) } = arc tan N 1 D 1 = arc tan C 1 sin ϕ C 1 cos ϕ + ϕ 0
t p ( ϕ + Δ ϕ , β p ) = g = 0 b g cos [ g ( ϕ + Δ ϕ + β p ) ]
ϕ + Δ ϕ = arctan N 2 { t p ( ϕ + Δ ϕ , β p ) } D 2 { t p ( ϕ + Δ ϕ , β p ) } = arc tan N 2 D 2 = arc tan C 2 sin ( ϕ + Δ ϕ ) C 2 cos ( ϕ + Δ ϕ ) + ϕ 0 + Δ ϕ 0
Δ ϕ = arc tan D 1 { s m ( ϕ , α m ) } N 2 { t p ( ϕ + Δ ϕ , β p ) } N 1 { s m ( ϕ , α m ) } D 2 { t p ( ϕ + Δ ϕ , β p ) } N 1 { s m ( ϕ , α m ) } N 2 { t p ( ϕ + Δ ϕ , β p ) } + D 1 { s m ( ϕ , α m ) } D 2 { t p ( ϕ + Δ ϕ , β p ) } = = arc tan D 1 N 2 N 1 D 2 N 1 N 2 + D 1 D 2
Δ ϕ = 2 arc tan N 2 { t p ( ϕ + Δ ϕ , β p ) } N 1 { s m ( ϕ , α m ) } D 1 { s m ( ϕ , α m ) } + D 2 { t p ( ϕ + Δ ϕ , β p ) } = 2 arc tan N 2 N 1 D 1 + D 2
Δ ϕ = 2 arc tan D 1 { s m ( ϕ , α m ) } D 2 { t p ( ϕ + Δ ϕ , β p ) } N 1 { s m ( ϕ , α m ) } + N 2 { t p ( ϕ + Δ ϕ , β p ) } = 2 arc tan D 1 D 2 N 1 + N 2
tan E Δ ϕ = tan Δ ϕ E tan Δ ϕ 1 + tan Δ ϕ E tan Δ ϕ = N 2 E D 1 E N 1 E D 2 E N 1 E N 2 E + D 1 E D 2 E N 2 D 1 N 1 D 2 N 1 N 2 + D 1 D 2 1 + N 2 E D 1 E N 1 E D 2 E N 1 E N 2 E + D 1 E D 2 E N 2 D 1 N 1 D 2 N 1 N 2 + D 1 D 2
E Δ ϕ = E D 1 C 1 sin ϕ cos ( 2 ϕ + 2 Δ ϕ ) E N 1 C 1 [ cos ϕ + sin ϕ sin ( 2 ϕ + 2 Δ ϕ ) ] + + E N 2 C 2 cos ( ϕ + Δ ϕ ) cos ( 2 ϕ + 2 Δ ϕ ) E D 2 C 2 [ sin ( ϕ + Δ ϕ ) + cos ( ϕ + Δ ϕ ) sin 2 ϕ ]
e N 1 = E N 1 C 1 = μ = 1 e N , μ cos ( μ ϕ + κ N , μ )
e D 1 = E D 1 C 1 = μ = 1 e D , μ cos ( μ ϕ + κ D , μ )
e N 2 = E N 2 C 2 = ν = 1 e N , ν cos [ ν ( ϕ + Δ ϕ ) + κ N , ν ]
e D 2 = E D 2 C 2 = ν = 1 e D , ν cos [ ν ( ϕ + Δ ϕ ) + κ D , ν ]
E Δ ϕ = μ = 1 e D , μ 2 { sin [ ( μ 1 ) ϕ + κ D , μ ] sin [ ( μ + 1 ) ϕ + κ D , μ ] } cos ( 2 ϕ + 2 Δ ϕ ) μ = 1 e N , μ 2 { cos [ ( μ 1 ) ϕ + κ N , μ ] + cos [ ( μ + 1 ) ϕ + κ N , μ ] } μ = 1 e N , μ 2 { sin [ ( μ 1 ) ϕ + κ N , μ ] sin [ ( μ + 1 ) ϕ + κ N , μ ] } sin ( 2 ϕ + 2 Δ ϕ ) + + ν = 1 e N , ν 2 { cos [ ( ν 1 ) ( ϕ + Δ ϕ ) + κ N , ν ] + cos [ ( ν + 1 ) ( ϕ + Δ ϕ ) + κ N , ν ] } cos 2 ϕ ν = 1 e D , ν 2 { sin [ ( ν 1 ) ( ϕ + Δ ϕ ) + κ D , ν ] sin [ ( ν + 1 ) ( ϕ + Δ ϕ ) + κ D , ν ] } ν = 1 e D , ν 2 { cos [ ( ν 1 ) ( ϕ + Δ ϕ ) + κ D , ν ] + cos [ ( ν + 1 ) ( ϕ + Δ ϕ ) + κ D , ν ] } sin 2 ϕ
E Δ ϕ = 1 2 cos ( ϕ + Δ ϕ 2 ) { cos Δ ϕ 2 [ μ = 1 e N , μ cos ( μ ϕ + γ N , μ ) ν = 1 e N , ν cos [ ν ( ϕ + Δ ϕ ) + γ ν , p ] ] + + sin Δ ϕ 2 [ μ = 1 e D , μ cos ( μ ϕ + γ D , μ ) + ν = 1 e D , ν cos [ ν ( ϕ + Δ ϕ ) + γ D , ν ] ] }
E Δ ϕ = 1 2 sin ( ϕ + Δ ϕ 2 ) { cos Δ ϕ 2 [ μ = 1 e D , μ cos ( μ ϕ + γ D , μ ) ν = 1 e D , ν cos [ ν ( ϕ + Δ ϕ ) + γ D , ν ] ] sin Δ ϕ 2 [ μ = 1 e N , μ cos ( μ ϕ + γ N , μ ) + ν = 1 e N , ν cos [ ν ( ϕ + Δ ϕ ) + γ N , ν ] ] }
E Δ ϕ = m = 1 M ( Δ ϕ s m ) ( s m α m ) E α m + p = 1 P ( Δ ϕ t p ) ( t p β p ) E β p
α E m = α m + E α m = α m + q = 1 ε q α m q q π q 1
β E p = β p + E β p = β p + r = 1 χ r β p r r π r 1
E Δ ϕ = m = 1 M ( Δ ϕ s m ) E s m + p = 1 P ( Δ ϕ t p ) E t p
E s m = k = 2 a k cos [ k ( ϕ + α m ) ]
E t p = g = 2 b g cos [ g ( ϕ + Δ ϕ + β p ) ]
Δ ϕ = arc tan ( 2 s 3 s 1 s 5 ) ( 2 t 2 2 t 4 ) ( 2 s 2 2 s 4 ) ( 2 t 3 t 1 t 5 ) ( 2 s 3 s 1 s 5 ) ( 2 t 3 t 1 t 5 ) + ( 2 s 2 2 s 4 ) ( 2 t 2 2 t 4 )
Δ ϕ = 2 arc tan ( 2 t 2 2 t 4 ) ( 2 s 2 2 s 4 ) ( 2 s 3 s 1 s 5 ) + ( 2 t 3 t 1 t 5 )
Δ ϕ = 2 arc tan ( 2 s 3 s 1 s 5 ) ( 2 t 3 t 1 t 5 ) ( 2 s 2 2 s 4 ) + ( 2 t 2 2 t 4 )

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