Abstract

We have constructed two-wavelength phase-shifting interferometry that is insensitive to the intensity changes in interferograms associated with the current variations in two laser-diode (LD) sources by using a newly developed phase-extraction algorithm. The tested phase at a synthetic wavelength can be measured from six interferograms with different phase shifts. The algorithm becomes a simple form for seven interferograms and reduces to a minimum of five phase-shifted data in the proper conditions. We shifted the phases equally in opposite directions to one another by separately varying the stepwise currents in dual LD’s on an unbalanced interferometer. The measurement accuracy has been improved compared with that of the two-wavelength four-step method. The phase error caused by the power changes in the dual LD’s has been investigated theoretically and experimentally. The experimental results are shown to measure a step object with a 4.6-μm synthetic wavelength.

© 1994 Optical Society of America

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  1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  2. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, London, 1988), Vol. 26, pp. 349–393.
    [CrossRef]
  3. Y. Ishii, “Recent developments in laser-diode interferometry,” Opt. Laser Eng. 14, 293–309 (1991).
    [CrossRef]
  4. Y. Y. Cheng, J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23, 4539–4543 (1984).
    [CrossRef] [PubMed]
  5. K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26, 2810–2816 (1987).
    [CrossRef] [PubMed]
  6. C. Polhemus, “Two-wavelength interferometry,” Appl. Opt. 12, 2071–2074 (1973).
    [CrossRef] [PubMed]
  7. A. F. Fercher, H. Z. Hu, U. Vry, “Rough surface interferometry with a two-wavelength heterodyne speckle interferometer,” Appl. Opt. 24, 2181–2188 (1985).
    [CrossRef] [PubMed]
  8. C. C. Williams, H. K. Wickramasinghe, “Optical ranging by wavelength multiplexed interferometry,” J. Appl. Phys. 60, 1900–1903 (1986).
    [CrossRef]
  9. R. Dändliker, R. Thalmann, D. Prongué, “Two-wavelength laser interferometry using superheterodyne detection,” Opt. Lett. 13, 339–341 (1988).
    [CrossRef] [PubMed]
  10. Z. Sodnik, E. Fischer, T. Ittner, H. J. Tiziani, “Two-wavelength double heterodyne interferometry using a matched grating technique,” Appl. Opt. 30, 3139–3144 (1991).
    [CrossRef] [PubMed]
  11. A. J. den Boef, “Two-wavelength scanning spot interferometer using single-frequency diode lasers,” Appl. Opt. 27, 306–311 (1988).
    [CrossRef]
  12. O. Sasaki, H. Sasazaki, T. Suzuki, “Two-wavelength sinusoidal phase/modulating laser-diode interferometer insensitive to external disturbances,” Appl. Opt. 30, 4040–4045 (1991).
    [CrossRef] [PubMed]
  13. L. Bartolini, G. Fornetti, M. FerriDeCollibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
    [CrossRef]
  14. Y. Ishii, R. Onodera, “Two-wavelength laser-diode interferometry that uses phase-shifting techniques,” Opt. Lett. 16, 1523–1525 (1991).
    [CrossRef] [PubMed]
  15. P. Hariharan, “Phase-stepping interferometry with laser diodes: effect of changes in laser power with output wavelength,” Appl. Opt. 28, 27–29 (1989).
    [CrossRef] [PubMed]
  16. K. Tatsuno, Y. Tsunoda, “Diode laser direct modulation heterodyne interferometer,” Appl. Opt. 26, 37–40 (1987).
    [CrossRef] [PubMed]
  17. Y. Ishii, J. Chen, K. Murata, “Digital phase-measuring interferometry with a tunable laser diode,” Opt. Lett. 12, 233–235 (1987).
    [CrossRef] [PubMed]
  18. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  19. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

1991 (4)

1990 (1)

L. Bartolini, G. Fornetti, M. FerriDeCollibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

1989 (1)

1988 (2)

1987 (3)

1986 (1)

C. C. Williams, H. K. Wickramasinghe, “Optical ranging by wavelength multiplexed interferometry,” J. Appl. Phys. 60, 1900–1903 (1986).
[CrossRef]

1985 (1)

1984 (2)

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

Y. Y. Cheng, J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23, 4539–4543 (1984).
[CrossRef] [PubMed]

1982 (1)

1974 (1)

1973 (1)

Bartolini, L.

L. Bartolini, G. Fornetti, M. FerriDeCollibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Chen, J.

Cheng, Y. Y.

Creath, K.

K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26, 2810–2816 (1987).
[CrossRef] [PubMed]

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, London, 1988), Vol. 26, pp. 349–393.
[CrossRef]

Dändliker, R.

den Boef, A. J.

Fercher, A. F.

FerriDeCollibus, M.

L. Bartolini, G. Fornetti, M. FerriDeCollibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Fischer, E.

Fornetti, G.

L. Bartolini, G. Fornetti, M. FerriDeCollibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Gallagher, J. E.

Greivenkamp, J. E.

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

Hariharan, P.

Herriott, D. R.

Hu, H. Z.

Ishii, Y.

Ittner, T.

Morgan, C. J.

Murata, K.

Occhionero, G.

L. Bartolini, G. Fornetti, M. FerriDeCollibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Onodera, R.

Papetti, F.

L. Bartolini, G. Fornetti, M. FerriDeCollibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Polhemus, C.

Prongué, D.

Rosenfeld, D. P.

Sasaki, O.

Sasazaki, H.

Sodnik, Z.

Suzuki, T.

Tatsuno, K.

Thalmann, R.

Tiziani, H. J.

Tsunoda, Y.

Vry, U.

White, A. D.

Wickramasinghe, H. K.

C. C. Williams, H. K. Wickramasinghe, “Optical ranging by wavelength multiplexed interferometry,” J. Appl. Phys. 60, 1900–1903 (1986).
[CrossRef]

Williams, C. C.

C. C. Williams, H. K. Wickramasinghe, “Optical ranging by wavelength multiplexed interferometry,” J. Appl. Phys. 60, 1900–1903 (1986).
[CrossRef]

Wyant, J. C.

Appl. Opt. (10)

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

Y. Y. Cheng, J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23, 4539–4543 (1984).
[CrossRef] [PubMed]

A. F. Fercher, H. Z. Hu, U. Vry, “Rough surface interferometry with a two-wavelength heterodyne speckle interferometer,” Appl. Opt. 24, 2181–2188 (1985).
[CrossRef] [PubMed]

K. Tatsuno, Y. Tsunoda, “Diode laser direct modulation heterodyne interferometer,” Appl. Opt. 26, 37–40 (1987).
[CrossRef] [PubMed]

K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26, 2810–2816 (1987).
[CrossRef] [PubMed]

A. J. den Boef, “Two-wavelength scanning spot interferometer using single-frequency diode lasers,” Appl. Opt. 27, 306–311 (1988).
[CrossRef]

Z. Sodnik, E. Fischer, T. Ittner, H. J. Tiziani, “Two-wavelength double heterodyne interferometry using a matched grating technique,” Appl. Opt. 30, 3139–3144 (1991).
[CrossRef] [PubMed]

O. Sasaki, H. Sasazaki, T. Suzuki, “Two-wavelength sinusoidal phase/modulating laser-diode interferometer insensitive to external disturbances,” Appl. Opt. 30, 4040–4045 (1991).
[CrossRef] [PubMed]

C. Polhemus, “Two-wavelength interferometry,” Appl. Opt. 12, 2071–2074 (1973).
[CrossRef] [PubMed]

P. Hariharan, “Phase-stepping interferometry with laser diodes: effect of changes in laser power with output wavelength,” Appl. Opt. 28, 27–29 (1989).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

C. C. Williams, H. K. Wickramasinghe, “Optical ranging by wavelength multiplexed interferometry,” J. Appl. Phys. 60, 1900–1903 (1986).
[CrossRef]

Opt. Eng. (1)

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

Opt. Laser Eng. (1)

Y. Ishii, “Recent developments in laser-diode interferometry,” Opt. Laser Eng. 14, 293–309 (1991).
[CrossRef]

Opt. Lett. (4)

Rev. Sci. Instrum. (1)

L. Bartolini, G. Fornetti, M. FerriDeCollibus, G. Occhionero, F. Papetti, “Two-wavelength infrared heterodyne transceiver with a continuous phase tracking system,” Rev. Sci. Instrum. 61, 1177–1181 (1990).
[CrossRef]

Other (1)

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, London, 1988), Vol. 26, pp. 349–393.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Experimental setup for a two-wavelength phase-shifting interferometer with dual frequency-modulation LD’s.

Fig. 2
Fig. 2

Variations in the bias intensities of two interferograms as a function of phase shift for seven steps. The phase shifts are marked by circles.

Fig. 3
Fig. 3

Schematic explanation of the phase-extraction algorithm that is insensitive to output-power changes in dual LD’s. The intensity of the interferogram is changed to the intensity caused by the power alteration of the LD’s from the intensity in the constant power of the LD’s. The difference between the changes in interferometric intensities is designated by k j (j = 1, …, 7).

Fig. 4
Fig. 4

Timing chart for the calibration of phase shifts: (a) the feedback loop is opened, (b) the error signal becomes maximum when the bias current is adjusted, (c) the phase shift is matched to the phase ranging from −π to π by the feedback system.

Fig. 5
Fig. 5

Variations in the calibrated currents of two LD’s (the upper two traces) and the corresponding interference signals (the lower two traces). Two LD’s are driven by mutually inverted triangular currents to satisfy the phase-shift condition of Eq. (19).

Fig. 6
Fig. 6

Three-dimensional phase maps of a step-height object when a four-step algorithm (top) and a seven-step algorithm (bottom) are used. The rms phase error caused by the changes in LD power is reduced from Λ/35 to Λ/81.

Fig. 7
Fig. 7

Phase error Φ′ − Φ as a function of the phase Φ for a four-step method: (a) the experiment, (b) the numerical result from the intensity-change rate of ρ1 = 8%. Both results show fairly good agreement.

Fig. 8
Fig. 8

Phase error as a function of phase Φ for a seven-step method. The rms phase error decreases to Λ/81, which has been verified by applying the intensity-insensitive phase-calculation method to the two-wavelength interferometer.

Equations (42)

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I k ( x , y ) = I M k ( x , y ) { 1 + γ k ( x , y ) cos [ ϕ k ( x , y ) ] } ,
ϕ k + Δ ϕ k = 2 π λ k ( 2 w + l ) - ( 2 π l Δ λ k λ k 2 + 2 π 2 w Δ λ k λ k 2 ) .
2 δ k = 2 π l Δ λ k λ k 2 = 2 π l α k Δ i k λ k 2 ,
I k j ( x , y ) = I M k ( x , y ) { 1 + γ k ( x , y ) cos [ ϕ k ( x , y ) - 2 δ k j ] } .
i j ( x , y ) = I 1 j ( x , y ) + I 2 j ( x , y ) = I M 1 ( x , y ) + I M 2 ( x , y ) + γ 1 ( x , y ) I M 1 ( x , y ) cos [ ϕ 1 ( x , y ) - 2 δ 1 j ] + γ 2 ( x , y ) I M 2 ( x , y ) cos [ ϕ 2 ( x , y ) - 2 δ 2 j ] .
i j ( x , y ) = I M 1 ( x , y ) + I M 2 ( x , y ) + 2 γ 1 ( x , y ) I M 1 ( x , y ) cos { [ ϕ 1 ( x , y ) + ϕ 2 ( x , y ) ] / 2 - δ 1 j - δ 2 j } cos { [ ϕ 1 ( x , y ) - ϕ 2 ( x , y ) ] / 2 - δ 1 j + δ 2 j } .
δ 1 j = - δ 2 j ,
i j ( x , y ) = I M 1 ( x , y ) + I M 2 ( x , y ) + 2 γ 1 ( x , y ) I M 1 ( x , y ) × cos [ Ψ ( x , y ) ] cos [ Φ ( x , y ) - 2 δ 1 j ] ,
Φ ( x , y ) = tan - 1 [ i 1 ( x , y ) - i 3 ( x , y ) i 4 ( x , y ) - i 2 ( x , y ) ] .
I k j ( x , y ) = [ I M k 0 ( x , y ) + 4 δ k j Δ I k ( x , y ) / π ] × { 1 + γ k ( x , y ) cos [ ϕ k ( x , y ) - 2 δ k j ] }
i j ( x , y ) = i 1 j ( x , y ) + I 2 j ( x , y ) = [ I M 1 0 ( x , y ) + 4 δ 1 j Δ I 1 ( x , y ) / π ] × { 1 + γ 1 ( x , y ) cos [ ϕ 1 ( x , y ) - 2 δ 1 j ] } + [ I M 2 0 ( x , y ) - 4 δ 1 j Δ I 2 ( x , y ) / π ] × { 1 + γ 2 ( x , y ) cos [ ϕ 2 ( x , y ) + 2 δ 1 j ] } ,
i j ( x , y ) = a 1 ( x , y ) f 1 ( δ 1 j ) + a 2 ( x , y ) f 2 ( δ 1 j ) + a 3 ( x , y ) f 3 ( δ 1 j ) + a 4 ( x , y ) f 4 ( δ 1 j ) + a 5 ( x , y ) f 5 ( δ 1 j ) + a 6 ( x , y ) f 6 ( δ 1 j ) ,
a 1 ( x , y ) = I M 1 0 ( x , y ) + I M 2 0 ( x , y ) , f 1 ( δ 1 j ) = 1 , a 2 ( x , y ) = γ 1 ( x , y ) I M 1 0 ( x , y ) cos [ ϕ 1 ( x , y ) ] + γ 2 ( x , y ) I M 2 0 ( x , y ) cos [ ϕ 2 ( x , y ) ] , f 2 ( δ 1 j ) = cos ( 2 δ 1 j ) , a 3 ( x , y ) = γ 1 ( x , y ) I M 1 0 ( x , y ) sin [ ϕ 1 ( x , y ) ] - γ 2 ( x , y ) I M 2 0 ( x , y ) sin [ ϕ 2 ( x , y ) ] , f 3 ( δ 1 j ) = sin ( 2 δ 1 j ) , a 4 ( x , y ) = Δ I 1 ( x , y ) - Δ I 2 ( x , y ) , f 4 ( δ 1 j ) = 4 δ 1 j / π , a 5 ( x , y ) = γ 1 ( x , y ) Δ I 1 ( x , y ) cos [ ϕ 1 ( x , y ) ] - γ 2 ( x , y ) Δ I 2 ( x , y ) cos [ ϕ 2 ( x , y ) ] , f 5 ( δ 1 j ) = ( 4 δ 1 j / π ) cos ( 2 δ 1 j ) , a 6 ( x , y ) = γ 1 ( x , y ) Δ I 1 ( x , y ) sin [ ϕ 1 ( x , y ) ] + γ 2 ( x , y ) Δ I 2 ( x , y ) sin [ ϕ 2 ( x , y ) ] , f 6 ( δ 1 j ) = ( 4 δ 1 j / π ) sin ( 2 δ 1 j ) .
j = 1 N ( i j * - i j ) 2
[ x 11 ( δ 1 j ) x 12 ( δ 1 j ) x 13 ( δ 1 j ) x 14 ( δ 1 j ) x 15 ( δ 1 j ) x 16 ( δ 1 j ) x 21 ( δ 1 j ) x 22 ( δ 1 j ) x 23 ( δ 1 j ) x 24 ( δ 1 j ) x 25 ( δ 1 j ) x 26 ( δ 1 j ) x 31 ( δ 1 j ) x 32 ( δ 1 j ) x 33 ( δ 1 j ) x 34 ( δ 1 j ) x 35 ( δ 1 j ) x 36 ( δ 1 j ) x 41 ( δ 1 j ) x 42 ( δ 1 j ) x 43 ( δ 1 j ) x 44 ( δ 1 j ) x 45 ( δ 1 j ) x 46 ( δ 1 j ) x 51 ( δ 1 j ) x 52 ( δ 1 j ) x 53 ( δ 1 j ) x 54 ( δ 1 j ) x 55 ( δ 1 j ) x 56 ( δ 1 j ) x 61 ( δ 1 j ) x 62 ( δ 1 j ) x 63 ( δ 1 j ) x 64 ( δ 1 j ) x 65 ( δ 1 j ) x 66 ( δ 1 j ) ] × [ a 1 ( x , y ) a 2 ( x , y ) a 3 ( x , y ) a 4 ( x , y ) a 5 ( x , y ) a 6 ( x , y ) ] = [ i j * ( x , y ) f 1 ( δ 1 j ) i j * ( x , y ) f 2 ( δ 1 j ) i j * ( x , y ) f 3 ( δ 1 j ) i j * ( x , y ) f 4 ( δ 1 j ) i j * ( x , y ) f 5 ( δ 1 j ) i j * ( x , y ) f 6 ( δ 1 j ) ] ,
X ( δ 1 j ) [ a 1 ( x , y ) a 2 ( x , y ) a 3 ( x , y ) a 4 ( x , y ) a 5 ( x , y ) a 6 ( x , y ) ] = C ( x , y , δ 1 j ) ,
[ a 1 ( x , y ) a 2 ( x , y ) a 3 ( x , y ) a 4 ( x , y ) a 5 ( x , y ) a 6 ( x , y ) ] = X - 1 ( δ 1 j ) C ( x , y , δ 1 j ) .
a 3 ( x , y ) a 2 ( x , y ) = γ 1 ( x , y ) I M 1 0 ( x , y ) sin [ ϕ 1 ( x , y ) ] - γ 2 ( x , y ) I M 2 0 ( x , y ) sin [ ϕ 2 ( x , y ) ] γ 1 ( x , y ) I M 1 0 ( x , y ) cos [ ϕ 1 ( x , y ) ] + γ 2 ( x , y ) I M 2 0 ( x , y ) cos [ ϕ 2 ( x , y ) ] .
γ 1 ( x , y ) I M 1 0 ( x , y ) = γ 2 ( x , y ) I M 2 0 ( x , y ) ,
a 3 ( x , y ) a 3 ( x , y ) = sin [ ϕ 1 ( x , y ) ] - sin [ ϕ 2 ( x , y ) ] cos [ ϕ 1 ( x , y ) ] + cos [ ϕ 2 ( x , y ) ] = 2 cos [ Ψ ( x , y ) sin [ Φ ( x , y ) ] 2 cos [ Ψ ( x , y ) ] cos [ Φ ( x , y ) ] = tan [ Φ ( x , y ) ]
Φ ( x , y ) = tan - 1 [ a 3 ( x , y ) a 2 ( x , y ) ] ,
X = [ 6 0 0 14 2 - 2 3 0 3 0 2 8 - 3 / 3 0 0 3 - 2 3 - 3 / 3 6 14 2 - 2 3 364 / 9 40 / 3 - 8 3 2 8 - 3 / 3 40 / 3 226 / 9 - 4 3 / 3 - 2 3 - 3 / 3 6 - 8 3 - 4 3 / 3 46 / 3 ] .
2 δ 1 j = - 2 δ 2 j = ( j - 4 ) π / 2             ( j = 1 , , 7 ) .
X = [ 7 - 1 0 0 0 - 4 - 1 3 0 0 0 0 0 0 4 - 4 0 0 0 0 - 4 28 - 8 0 0 0 0 - 8 8 0 - 4 0 0 0 0 20 ] .
x 15 = j = 1 7 f 1 ( δ 1 j ) f 5 ( δ 1 j ) = j = 1 7 ( 4 δ 1 j / π ) cos ( 2 δ 1 j ) = 4 / π [ - δ 17 cos ( - 2 δ 17 ) - δ 16 cos ( - 2 δ 16 ) - δ 15 cos ( - 2 δ 15 ) + δ 15 cos ( 2 δ 15 ) + δ 16 cos ( 2 δ 16 ) + δ 17 cos ( 2 δ 17 ) ] = 0.
x 13 = x 31 = sin ( 2 δ 1 j ) = 0 , x 46 = x 64 = ( 4 δ 1 j / π ) 2 sin ( 2 δ 1 j ) = 0.
x 23 = j = 1 7 f 2 ( δ 1 j ) f 3 ( δ 1 j ) = j = 1 7 cos ( 2 δ 1 j ) sin ( 2 δ 1 j ) = j = 1 7 1 2 sin ( 4 δ 1 j ) = 0.
C = [ i 1 * + i 2 * + i 3 * + i 4 * + i 5 * + i 6 * + i 7 * - i 2 * + i 4 * - i 6 * i 1 * - i 3 * + i 5 * - i 7 * - 3 i 1 * - 2 i 2 * - i 3 * + i 5 * + 2 i 6 * + 3 i 7 * 2 i 2 * - 2 i 6 * - 3 i 1 * + i 3 * + i 5 * - 3 i 7 * ] .
Φ ( x , y ) = tan - 1 { [ 3 i 5 * ( x , y ) + i 1 * ( x , y ) ] - [ 3 i 3 * ( x , y ) + i 7 * ( x , y ) ] 4 i 4 * ( x , y ) - 2 [ i 6 * ( x , y ) + i 2 * ( x , y ) ] } .
k 1 = - 3 D 1 , k 2 = - 2 D 3 , k 3 = D 2 , k 4 = 0 , k 5 = D 1 , k 6 = 2 D 3 , k 7 = - 3 D 2 ,
( 3 i 5 + i 1 ) - ( 3 i 3 + i 7 ) = 8 ( γ 1 I M 1 0 sin ϕ 1 - γ 2 I M 2 0 sin ϕ 2 ) , 4 i 4 - 2 ( i 6 + i 2 ) = 8 ( γ 1 I M 1 0 cos ϕ 1 + γ 2 I M 2 0 cos ϕ 2 ) ,
Δ I 1 = Δ I 2 .
[ a 1 ( x , y ) a 2 ( x , y ) a 3 ( x , y ) a 5 ( x , y ) a 6 ( x , y ) ] = Y - 1 ( δ 1 j ) D ( x , y , δ 1 j ) ,
2 δ 1 j = - 2 δ 2 j = ( j - 3 ) π / 2             ( j = 1 , , 5 ) .
Y - 1 = [ 3 / 8 1 / 8 0 0 - 3 / 8 1 / 8 3 / 8 0 0 - 1 / 8 0 0 1 / 2 0 0 0 0 0 1 / 8 0 - 3 / 8 - 1 / 8 0 0 7 / 8 ] , D = [ i 1 * + i 2 * + i 3 * + i 4 * + i 5 * - i 1 * + i 3 * - i 5 * - i 2 * + i 4 * 2 i 1 * - 2 i 5 * i 2 * + i 4 * ] .
Φ ( x , y ) = tan - 1 [ - 2 i 2 * ( x , y ) + 2 i 4 * ( x , y ) - i 1 * ( x , y ) + 2 i 3 * ( x , y ) - i 5 * ( x , y ) ] .
i j = I M 1 0 ( 1 + 4 δ 1 j ρ 1 / π ) [ 1 + γ 1 cos ( ϕ 1 - 2 δ 1 j ] + I M 2 0 ( 1 - 4 δ 1 j ρ 2 / π ) [ 1 + γ 2 cos ( ϕ 2 + 2 δ 1 j ) ] ,
i j = 2 I M 1 0 + I M 1 0 γ 1 [ cos ( ϕ 1 - j π / 2 ) + cos ( ϕ 2 + j π / 2 ) ] + I M 1 0 γ 1 ( j - 4 ) ρ 1 [ cos ( ϕ 1 - j π / 2 ) - cos ( ϕ 2 + j π / 2 ) ] ,
tan Φ = i 1 - i 3 i 4 - i 2 = tan Φ - 2 ρ 1 tan Ψ 1 + ρ 1 tan Ψ tan Φ .
tan Φ - tan Φ = - ρ 1 1 + ρ 1 tan Ψ tan Φ 1 cos 2 Φ × 3 + cos 2 Φ 2 tan Ψ ,
sin ( Φ - Φ ) = - ρ 1 1 + ρ 1 tan Ψ tan Φ cos Φ cos Φ × 3 + cos 2 Φ 2 tan Ψ .
Φ - Φ - ρ 1 3 + cos 2 Φ 2 tan Ψ .

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