Abstract

Unexpected mechanical vibrations can significantly degrade the otherwise high accuracy of phase-shifting interferometry. Fourier analysis of phase-shift algorithms is shown to provide the analytical means of predicting measurement errors as a function of the frequency, the phase, and the amplitude of vibrations. The results of this analysis are concisely represented by a phase-error transfer function, which may be multiplied by the noise spectrum to predict the response of an interferometer to various forms of vibration. Analytical forms for the phase error are derived for several well-known algorithms, and the results are supported by numerical simulations and experiments with an interference microscope.

© 1995 Optical Society of America

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Corrections

Peter J. de Groot, "Vibration in phase-shifting interferometry: errata," J. Opt. Soc. Am. A 12, 2212-2212 (1995)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-12-10-2212

References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.
  2. P. Hariharan, Optical Interferometry (Academic, Orlando, Fla., 1985).
  3. Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef] [PubMed]
  4. Y. Y. Cheng, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
    [CrossRef]
  5. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
    [CrossRef]
  6. C. Ai, J. C. Wyant, “Effect of spurious reflection on phase shift interferometry,” Appl. Opt. 27, 3039–3045 (1988).
    [CrossRef] [PubMed]
  7. P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26, 2506–2507 (1987).
    [CrossRef] [PubMed]
  8. K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. Soc. Photo-Opt. Instrum. Eng.680, 19–28 (1986).
    [CrossRef]
  9. J. van Wingerden, H. H. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  10. K. Kinnstätter, Q. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef]
  11. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  12. 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]
  13. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  14. G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), p. 682.
  15. J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–107 (1984).
    [CrossRef]
  16. J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus (May1982), pp. 65–71.
  17. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  18. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  19. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992).
  20. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a date sampling window,” Appl. Opt. (to be published).
  21. P. de Groot, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).
    [CrossRef]
  22. K. G. Larkin, B. F. Oreb, “A new seven-sample symmetrical phase-shifting algorithm,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 2–11 (1992).
    [CrossRef]
  23. The orthogonality of the functions P1(ν), P2(ν) in phase space is obvious for symmetric sampling functions because in this case P1(ν) is purely real and P2(ν) purely imaginary. The orthogonality of these functions for asymmetric sampling functions follows from the observation that the effect of asymmetry in common PSI algorithms can be reduced to a simultaneous phase shift of both P1(ν) and P2(ν).

1992 (1)

1991 (1)

1990 (2)

1988 (2)

1987 (3)

1985 (1)

1984 (1)

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–107 (1984).
[CrossRef]

1983 (1)

1982 (1)

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus (May1982), pp. 65–71.

1974 (1)

Ai, C.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), p. 682.

Brangaccio, D. J.

Brophy, C. P.

Bruning, J. H.

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]

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992).

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

Burow, R.

Bushan, B.

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–107 (1984).
[CrossRef]

Cheng, Y. Y.

Creath, K.

K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. Soc. Photo-Opt. Instrum. Eng.680, 19–28 (1986).
[CrossRef]

de Groot, P.

P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a date sampling window,” Appl. Opt. (to be published).

P. de Groot, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).
[CrossRef]

Eiju, T.

Elssner, K.-E.

Frankena, H. H.

Freischlad, K.

Gallagher, J. E.

George, O. E.

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–107 (1984).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992).

Grzanna, J.

Hariharan, P.

Herriott, D. R.

Kinnstätter, K.

Koliopoulos, C. L.

K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
[CrossRef]

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–107 (1984).
[CrossRef]

Larkin, K. G.

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

K. G. Larkin, B. F. Oreb, “A new seven-sample symmetrical phase-shifting algorithm,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 2–11 (1992).
[CrossRef]

Lohmann, Q. W.

Merkel, K.

Oreb, B. F.

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

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

K. G. Larkin, B. F. Oreb, “A new seven-sample symmetrical phase-shifting algorithm,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 2–11 (1992).
[CrossRef]

Rosenfeld, D. P.

Schwider, J.

Smorenburg, C.

Spolaczyk, R.

Streibl, N.

van Wingerden, J.

White, A. D.

Wyant, J. C.

Appl. Opt. (9)

Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

Y. Y. Cheng, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
[CrossRef]

C. Ai, J. C. Wyant, “Effect of spurious reflection on phase shift interferometry,” Appl. Opt. 27, 3039–3045 (1988).
[CrossRef] [PubMed]

P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26, 2506–2507 (1987).
[CrossRef] [PubMed]

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

K. Kinnstätter, Q. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

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]

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

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

ASLE Trans. (1)

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–107 (1984).
[CrossRef]

J. Opt. Soc. Am. A (3)

Laser Focus (1)

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus (May1982), pp. 65–71.

Other (9)

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), p. 682.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992).

P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a date sampling window,” Appl. Opt. (to be published).

P. de Groot, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).
[CrossRef]

K. G. Larkin, B. F. Oreb, “A new seven-sample symmetrical phase-shifting algorithm,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1755, 2–11 (1992).
[CrossRef]

The orthogonality of the functions P1(ν), P2(ν) in phase space is obvious for symmetric sampling functions because in this case P1(ν) is purely real and P2(ν) purely imaginary. The orthogonality of these functions for asymmetric sampling functions follows from the observation that the effect of asymmetry in common PSI algorithms can be reduced to a simultaneous phase shift of both P1(ν) and P2(ν).

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

P. Hariharan, Optical Interferometry (Academic, Orlando, Fla., 1985).

K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. Soc. Photo-Opt. Instrum. Eng.680, 19–28 (1986).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Experimental phase-error map for an interferometer in the presence of a 1-Hz vibration having an amplitude of approximately one-quarter fringe. There were approximately three fringes in the field of view.

Fig. 2
Fig. 2

Vibrational noise signal used for a numerical simulation of phase error.

Fig. 3
Fig. 3

Comparison of a numerical simulation of phase error with the theoretical predictions based on the phase-error transfer function P(ν, θ). The close correspondence is typical of phase errors of less than 1 rad.

Fig. 4
Fig. 4

Predicted rms phase-measurement errors for the three-bucket algorithm with instantaneous phase sampling (β = 0). The vertical scale is normalized to the vibrational amplitude, and the horizontal scale is normalized to the phase-shift rate, in units of 2π rad/unit time. The upper graph refers to the errors Eν(r) periodic in the phase angle, whereas the lower graph refers to the constant offset error Eν(c).

Fig. 5
Fig. 5

Predicted rms phase-measurement errors for the three-bucket algorithm calculated with an integrating bucket of length β = π/2. The gradual decline in sensitivity with higher frequency is due to the smoothing effect of the integrating bucket.

Fig. 6
Fig. 6

Predicted rms phase-measurement errors for the four-bucket algorithm calculated with an integrating bucket of length β = π/2.

Fig. 7
Fig. 7

Predicted rms phase-measurement errors for the Schwider–Hariharan five-bucket algorithm calculated with an integrating bucket of length β = π/2.

Fig. 8
Fig. 8

Predicted rms phase-measurement errors for the seven-bucket algorithm calculated with an integrating bucket of length β = π/2. This algorithm has the lowest overall sensitivity to vibration, particularly in the low-frequency region (ν < 1).

Fig. 9
Fig. 9

Predicted rms phase-measurement errors for the Larkin–Oreb algorithm, calculated with an integrating bucket of length β = π/3. The peak sensitivities above ν = 3 are not the same as for algorithms based on π/2 phase shifts.

Fig. 10
Fig. 10

Experimental system for introducing controlled amounts of vibrational noise. The system is a conventional interferometric microscope with an additional PZT driven by a sinusoidal waveform generator.

Fig. 11
Fig. 11

Comparison of theoretical and experimental sensitivity of the five-bucket algorithm with vibration as a function of frequency, for a vibrational amplitude of one-quarter fringe. These data were obtained with the apparatus shown in Fig. 10.

Fig. 12
Fig. 12

Experimental demonstration of the improvement in performance when the seven-bucket algorithm is used. This is the phase-error map for an interferometer in the presence of the same 1-Hz vibration used when generating Fig. 1. The periodic error is suppressed, as predicted by the theory.

Equations (78)

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g ( θ ) = Q [ 1 + V cos ( θ ) ] ,
θ = 4 π L / λ .
ϕ ( t ) = 4 π δ L ( t ) λ .
t = ϕ / 2 π .
g ( θ , ϕ ) = Q [ 1 + V cos ( θ + ϕ ) ] .
g ( θ , ϕ ) = Q { 1 + V cos [ ϕ + θ - n ( ϕ ) ] } ,
Δ φ ( θ , ) = φ ( θ , ) - θ .
φ ( θ ) = tan - 1 [ s ( θ ) / c ( θ ) ] ,
N ( ν ) = FT { n ( ϕ ) } ,
n ( ϕ ) = FT - 1 { N ( ν ) } .
FT { n ( ϕ ) } = 1 2 π - n ( ϕ ) exp ( - i ν ϕ ) d ϕ ,
FT - 1 { N ( ν ) } = - N ( ν ) exp ( i ν ϕ ) d ν ,
g ( θ , ϕ ) = Q + Q V { cos [ n ( ϕ ) ] cos ( ϕ + θ ) + sin [ n ( ϕ ) ] × sin ( ϕ + θ ) } .
g ( θ , ϕ ) g ( θ , ϕ ) + Δ g ( θ , ϕ ) ,
Δ g ( θ , ϕ ) = Q V n ( ϕ ) sin ( ϕ + θ ) .
Δ G ( θ , ν ^ ) = FT { n ( ϕ ) sin ( θ + ϕ ) } Q V .
Δ G ( θ , ν ^ ) = Q V N ( ν ^ ) ( i / 2 ) [ δ ( ν ^ - 1 ) exp ( i θ ) - δ ( ν ^ + 1 ) × exp ( - i θ ) ] ,
Δ G ( θ , ν ^ ) = Q V ( i / 2 ) [ N ( ν ^ - 1 ) exp ( i θ ) - N ( ν ^ + 1 ) × exp ( - i θ ) ] .
Δ φ ( θ ) = - N ( ν ) P ( ν , θ ) d ν ,
φ ( θ ) = tan - 1 [ s ( θ ) / c ( θ ) ] .
s ( θ ) = - f S ( ϕ ) g ( θ , ϕ ) d ϕ , c ( θ ) = - f C ( ϕ ) g ( θ , ϕ ) d ϕ .
s ( θ ) Q V q sin ( θ ) , c ( θ ) Q V q cos ( θ ) ,
q = - f S ( ϕ ) sin ( - ϕ ) d ϕ = - f C ( ϕ ) cos ( ϕ ) d ϕ .
Δ φ ( θ ) [ Δ s ( θ ) d φ d s + Δ c ( θ ) d φ d c ] .
Δ φ ( θ ) 1 Q V q [ Δ s ( θ ) cos ( θ ) - Δ c ( θ ) sin ( θ ) ] .
Δ s ( θ ) = - f S ( ϕ ) Δ g ( θ , ϕ ) d ϕ , Δ c ( θ ) = - f C ( ϕ ) Δ g ( θ , ϕ ) d ϕ .
Δ s ( θ ) = q - F S * ( ν ^ ) Δ G ( θ , ν ^ ) d ν ^ , Δ c ( θ ) = q - F C * ( ν ^ ) Δ G ( θ , ν ^ ) d ν ^ ,
Δ φ ( θ ) = 1 Q V - [ F S * ( ν ^ ) cos ( θ ) - F C * ( ν ^ ) sin ( θ ) ] Δ G ( θ , ν ^ ) d ν ^ .
Δ φ ( θ ) = i 2 - N ( ν ^ - 1 ) [ F S * ( ν ^ ) cos ( θ ) - F C * ( ν ^ ) sin ( θ ) ] × exp ( + i θ ) d ν ^ - i 2 - N ( ν ^ + 1 ) [ F S * ( ν ^ ) cos ( θ ) - F C * ( ν ^ ) sin ( θ ) ] exp ( - i θ ) d ν ^ .
Δ φ ( θ ) = i 2 - N ( ν ) { [ F S * ( ν + 1 ) cos ( θ ) - F C * ( ν + 1 ) sin ( θ ) ] × exp ( + i θ ) - [ F S * ( ν - 1 ) cos ( θ ) - F C * ( ν - 1 ) × sin ( θ ) ] exp ( - i θ ) } d ν .
exp ( ± i θ ) = cos ( θ ) ± i sin ( θ ) , 2 sin ( θ ) cos ( θ ) = sin ( 2 θ ) , 2 cos 2 ( θ ) = 1 + cos ( 2 θ ) , 2 sin 2 ( θ ) = 1 - cos ( 2 θ ) .
Δ φ ( θ ) = - N ( ν ) P ( ν , θ ) d ν ,
P ( ν , θ ) = P 0 ( ν ) + P 1 ( ν ) cos ( 2 θ ) + P 2 ( ν ) sin ( 2 θ ) , P 0 ( ν ) = ¼ { F C * ( ν + 1 ) + F C * ( ν - 1 ) + i [ F S * ( ν + 1 ) - F S * ( ν - 1 ) ] } , P 1 ( ν ) = ¼ { - F C * ( ν + 1 ) - F C * ( ν - 1 ) + i [ F S * ( ν + 1 ) - F S * ( ν - 1 ) ] } , P 2 ( ν ) = ¼ { - F S * ( ν + 1 ) - F S * ( ν - 1 ) - i [ F C * ( ν + 1 ) - F C * ( ν - 1 ) ] } .
f S ( ϕ ) = h S ( ϕ ) b ( ϕ ) , f C ( ϕ ) = h C ( ϕ ) b ( ϕ ) .
F S ( ν ^ ) = H S ( ν ^ ) B ( ν ^ ) ,
b ( ϕ ) = { 1 ϕ β / 2 0 ϕ > β / 2 ,
B ( ν ^ ) = - b ( ϕ ) exp ( - i ϕ ν ^ ) d ϕ - b ( ϕ ) exp ( - i ϕ ) d ϕ ,
B ( ν ^ ) = sin ( ν ^ β / 2 ) ν ^ sin ( ν ^ β / 2 ) .
B ν ± 1 = B ( ν ± 1 ) .
h S ( ϕ ) = j h S , j δ ( ϕ - ϕ j ) , h C ( ϕ ) = j h C , j δ ( ϕ - ϕ j ) .
θ = tan - 1 [ j h S , j g ( θ , ϕ j ) j h C , j g ( θ , ϕ j ) ] .
H S ( ν ^ ) = 1 q j h S , j exp ( - i ϕ j ν ^ ) , H C ( ν ^ ) = 1 q j h C , j exp ( - i ϕ j ν ^ ) ,
q = j h S , j sin ( - ϕ j ) = j h C , j cos ( ϕ j ) .
θ = tan - 1 ( g 3 - g 2 g 1 - g 2 ) ,
g j = g [ θ , ( j π / 2 - π / 4 ) ]
h S ( ϕ ) = δ ( ϕ - 5 π / 4 ) - δ ( ϕ - 3 π / 4 ) , h C ( ϕ ) = δ ( ϕ - π / 4 ) - δ ( ϕ - 3 π / 4 ) .
H S ( ν ^ ) = - i 2 sin ( π ν ^ / 4 ) exp ( - i π ν ^ ) , H C ( ν ^ ) = i 2 sin ( π ν ^ / 4 ) exp ( - i π ν ^ / 2 ) .
P 0 ( ν ) = i exp ( i 3 ν π / 4 ) cos ( ν π 4 ) 2 ( B ν + 1 + B ν - 1 ) × cos ( ν π / 4 ) + ( B ν + 1 - B ν - 1 ) sin ( ν π / 4 ) , P 1 ( ν ) = i exp ( i 3 ν π / 4 ) cos ( ν π 4 ) 2 ( B ν + 1 + B ν - 1 ) × cos ( ν π / 4 ) + ( B ν + 1 - B ν - 1 ) sin ( ν π / 4 ) , P 2 ( ν ) = - exp ( i 3 ν π / 4 ) sin ( ν π 4 ) 2 ( B ν + 1 - B ν - 1 ) × cos ( ν π / 4 ) + ( B ν + 1 + B ν - 1 ) sin ( ν π / 4 ) .
θ = tan - 1 ( g 2 - g 4 g 3 - g 1 ) ,
g j = g [ θ , ( j π / 2 - π ) ]
H S ( ν ^ ) = i sin ( π ν ^ / 2 ) , H C ( ν ^ ) = sin 2 ( π ν ^ / 2 ) - ( i / 2 ) sin ( π ν ^ ) .
P 0 ( ν ) = ( B ν + 1 + B ν - 1 ) 2 [ cos ( ν π / 2 ) cos 2 ( ν π / 4 ) - i 4 sin ( ν π ) ] , P 1 ( ν ) = ( B ν + 1 + B ν - 1 ) 2 [ cos ( ν π / 2 ) sin 2 ( ν π / 4 ) + i 4 sin ( ν π ) ] , P 2 ( ν ) = ( B ν + 1 - B ν - 1 ) 2 [ cos ( ν π / 2 ) sin 2 ( ν π / 4 ) + i 4 sin ( ν π ) ] .
θ = tan - 1 [ 2 ( g 2 - g 4 ) 2 g 3 - g 1 - g 5 ] ,
g j = g [ θ , ( j π / 2 - 3 π / 2 ) ]
H S ( ν ^ ) = i sin ( π ν ^ / 2 ) , H C ( ν ^ ) = sin 2 ( π ν ^ / 2 ) .
P 0 ( ν ) = ( B ν + 1 + B ν - 1 ) 2 cos ( ν π / 2 ) cos 2 ( ν π / 4 ) , P 1 ( ν ) = ( B ν + 1 + B ν - 1 ) 2 cos ( ν π / 2 ) sin 2 ( ν π / 4 ) , P 2 ( ν ) = i ( B ν + 1 - B ν - 1 ) 2 cos ( ν π / 2 ) sin 2 ( ν π / 4 ) .
θ = tan - 1 [ 7 ( g 2 - g 4 ) - ( g 0 - g 6 ) 8 g 3 - 4 ( g 1 + g 5 ) ] ,
g j = g [ θ , ( j π / 2 - 3 π / 2 ) ]
H S ( ν ^ ) = i sin ( π ν ^ / 2 ) - ( i / 8 ) [ sin ( 3 π ν ^ / 2 ) + sin ( π ν ^ / 2 ) ] , H C ( ν ^ ) = sin 2 ( π ν ^ / 2 ) .
P 0 ( ν ) = ( B ν + 1 + B ν - 1 ) [ cos ( ν π / 2 ) cos 2 ( ν π / 4 ) 2 + cos ( 3 ν π / 2 ) - cos ( ν π / 2 ) 32 ] , P 1 ( ν ) = ( B ν + 1 + B ν - 1 ) [ cos ( ν π / 2 ) sin 2 ( ν π / 4 ) 2 + cos ( 3 ν π / 2 ) - cos ( ν π / 2 ) 32 ] , P 2 ( ν ) = i ( B ν + 1 - B ν - 1 ) [ cos ( ν π / 2 ) sin 2 ( ν π / 4 ) 2 + cos ( 3 ν π / 2 ) - cos ( ν π / 2 ) 32 ] .
θ = tan - 1 { 3 [ g 2 + g 3 - g 5 - g 6 + ( g 7 - g 1 ) / 3 - g 1 - g 2 + g 3 + 2 g 4 + g 5 - g 6 - g 7 ] } ,
g j = g [ θ , ( j π / 3 - 4 π / 3 ) ]
H S ( ν ^ ) = 3 3 i [ 1 3 sin ( π ν ^ ) - sin ( π ν ^ / 3 ) - sin ( π 2 ν ^ / 3 ) ] , H C ( ν ^ ) = 1 3 [ - cos ( π ν ^ ) - cos ( π 2 ν ^ / 3 ) + 1 + cos ( π ν ^ / 3 ) ] .
P 0 ( ν ) = ( B ν + 1 + B ν - 1 ) 6 [ cos ( ν π / 3 ) + cos ( ν 2 π / 3 ) + cos 2 ( ν π / 2 ) ] + 3 ( B ν + 1 - B ν - 1 ) 36 sin ( ν π ) , P 1 ( ν ) = ( B ν + 1 + B ν - 1 ) 12 [ cos ( ν π / 3 ) + cos ( ν 2 π / 3 ) - 2 cos 2 ( ν π / 2 ) ] + 3 ( B ν + 1 - B ν - 1 ) 36 × [ sin ( ν π ) - 3 sin ( ν 2 π / 3 ) + 3 sin ( ν π / 3 ) ] , P 2 ( ν ) = i ( B ν + 1 - B ν - 1 ) 12 [ cos ( ν π / 3 ) + cos ( ν 2 π / 3 ) - 2 cos 2 ( ν π / 2 ) ] + i 3 ( B ν + 1 + B ν - 1 ) 36 × [ sin ( ν π ) - 3 sin ( ν 2 π / 3 ) + 3 sin ( ν π / 3 ) ] .
N ( ν ) = ½ A ν [ exp ( i α ) δ ( ν - ν ) + exp ( - i α ) δ ( ν + ν ) ] .
E ν ( c ) = A ν C ( ν ) ,
E ν ( r ) = A ν R ( ν ) .
[ E ν ( c ) ] 2 = 1 4 π 2 - π π - π π Re [ A ν P 0 ( ν , θ ) exp ( i α ) ] 2 d θ d α .
[ E ν ( r ) ] 2 = 1 4 π 2 - π π - π π Re { A ν [ P ( ν , θ ) - P 0 ( ν , θ ) ] × exp ( i α ) } 2 d θ d α .
C ( ν ) 2 = 1 4 π 2 - π π - π π c ( ν , α , θ ) 2 d θ d α ,
R ( ν ) 2 = 1 4 π 2 - π π - π π r ( ν , α , θ ) 2 d θ d α ,
c ( ν , θ , α ) = Re [ P 0 ( ν ) exp ( i α ) ] ,
r ( ν , θ , α ) = Re [ P 1 ( ν ) exp ( i α ) ] cos ( 2 θ ) + Re [ P 2 ( ν ) exp ( i α ) ] sin ( 2 θ ) .
C ( ν ) = / 2 1 P 0 ( ν ) ,
R ( ν ) = ½ P 1 ( ν ) + P 2 ( ν ) ,
E NET ( c ) = [ ν A ν 2 C ( ν ) 2 ] 1 / 2 ,
E NET ( r ) = [ ν A ν 2 R ( ν ) 2 ] 1 / 2 .
R ( ν ) = [ B ν + 1 2 + B ν - 1 2 2 ] 1 / 2 | cos ( ν π / 2 ) sin 2 ( ν π / 4 ) 2 | .

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