Abstract

A method to extract the visibility of celestial objects in real time using a parallel-shear interferometer is described. Such an instrument produces fringes of constant visibility but with random atmospheric-induced phases. The fringes are modulated internally, and synchronuous detection with many parallel channels recovers their contrast. First, we perform parallel sine and cosine phase-locked accumulation for the short period in which the atmosphere is presumed frozen. The visibility amplitude is then calculated allowing for Poisson noise. We find that a 7.8m star can be resolved by a single detector to the diffraction limit of the telescope; with twenty such detectors, the limit is 10m.

© 1985 Optical Society of America

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  1. E. Ribak, E. Leibowitz, E. K. Hege, “Shearing Stellar Interferometer. 2: Optoelectronic Phase-Locked System,” Appl. Opt. 24, 3094 (1985).
    [Crossref] [PubMed]
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1976).
  3. S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., London, 1983).
  4. A. Labeyrie, “High Resolution Techniques in Optical Astronomy,” Prog. Opt. 13, 49 (1976).
  5. W. J. Tango, R. Q. Twiss, “Michelson Stellar Interferometry,” Prog. Opt. 17, 239 (1980).
    [Crossref]
  6. F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
    [Crossref]
  7. R. H. T. Bates, “Astronomical Speckle Imaging,” Phys. Rep. 90, 203 (1982).
    [Crossref]
  8. N. J. Woolf, “High Resolution Imaging from the Ground,” Ann. Rev. Astron. Astrophys. 20, 367 (1982).
    [Crossref]
  9. J. C. Dainty, Ed., “Stellar Speckle Interferometry,” in Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1983).
  10. C. Roddier, F. Roddier, “Fringe Visibility in a Michelson Interferometer,” J. Opt. Soc. Am. 66, 580 (1976).
    [Crossref]
  11. J. J. Burke, J. B. Breckinridge, “Passive Imaging Through the Turbulent Atmosphere: Fundamental Limits on the Spatial Frequency Resolution of a Rotational Shearing Interferometer,” J. Soc. Opt. Am. 68, 67 (1978).
    [Crossref]
  12. J. L. Elliot, I. S. Glass, “A Quantitative Fringe Detector for Stellar Interferometry,” Astron. J. 75, 1123 (1970).
    [Crossref]
  13. W. S. Finsen, “Twenty Years of Double Star Interferometry and its Lessons,” Astrophys. Space Sci. 11, 13 (1971).
    [Crossref]
  14. W. C. Wickes, R. H. Dicke, “An Automatic Interferometer for Double Star Observations,” Astron. J. 78, 757 (1973).
    [Crossref]
  15. W. C. Wickes, R. H. Dicke, “Achromatic Double Star Interferometry,” Astron. J. 79, 1433 (1974).
    [Crossref]
  16. D. G. Currie, “On the Amplitude Interferometer Program at the University of Maryland,” IAU Colloquium 50, 7-1 (Maryland, 1978).
  17. E. S. Kulagin, “A Superposed-Ray Interferometer,” Sov. Astron. 13, 1023 (1970).
  18. D. Kelsall, “Optical “Seeing” Through the Atmosphere by an Interferometric Technique,” J. Opt. Soc. Am. 63, 1472 (1973).
    [Crossref]
  19. J. C. Dainty, R. J. Scadden, “Measurement of the Atmospheric Transfer Function at Mauna Kea, Hawaii,” Mon. Not. R. Astron. Soc. 170, 519 (1975).
  20. J. B. Breckinridge, “A Two-Dimensional White-Light Interferometer,” IAU Colloquium50, 31 (Maryland, 1978).
  21. F. Roddier, “Rotation-Shearing Interferometry,” IAU Colloquium50, 32-1 (Maryland, 1978).
  22. A. A. Tokovinin, “A Phase-Grating Stellar Interferometer,” Sov. Astron. Lett. 4, 229 (1979).
  23. E. Ribak, S. G. Lipson, “Complex Spatial Coherence Function: Its Measurement by Means of a Phase-Modulated Shearing Interferometer,” Appl. Opt. 20, 1102 (1981).
    [Crossref] [PubMed]
  24. M. Dugan, “Shearing Interferometer for the Measurement of Atmospheric MTF, M. Sc. Thesis, U. Rochester, New York (1982).
  25. C. J. Oliver, E. R. Pike, “Statistical Accuracy in the Photon Counting Structure Function of Fluctuating Light Fields,” Opt. Acta 28, 1345 (1981).
    [Crossref]
  26. K. Schatzel, “Noise in Photon Correlation and Photon Structure Function,” Opt. Acta 30, 155 (1983).
    [Crossref]
  27. A. A. Tokovinin, “The Influence of Turbulence on the Operation of a Stellar Interferometer,” Sov. Astron. Lett. 6, 386 (1980).
  28. C. W. Allen, Astrophysical Quantities (Clowes, London, 1972), p. 197.
  29. S. M. Kozel, “On the Fluctuation Resolution Limit of an Optical Modulation Interferometer,” Sov. Phys. JETP 5, 609 (1957).
  30. D. V. Korolkov, O. I. Krat, “On the Sensitivity of a Stellar Interferometer with Incoherent Accumulation of the Signal,” Sov. Astron. 20, 370 (1976).
  31. A. H. Greenaway, J. C. Dainty, “The Formal Equivalence Between Autocorrelation and Power Spectra Analysis of Photon-Limited Data,” Opt. Commun. 35, 307 (1980).
    [Crossref]
  32. K. Itoh, Y. Ohtsuka, “Photon-Noise Limitations in Wave-Front-Folding Interferometry,” J. Opt. Soc. Am. 73, 479 (1983).
    [Crossref]

1985 (1)

1983 (2)

K. Itoh, Y. Ohtsuka, “Photon-Noise Limitations in Wave-Front-Folding Interferometry,” J. Opt. Soc. Am. 73, 479 (1983).
[Crossref]

K. Schatzel, “Noise in Photon Correlation and Photon Structure Function,” Opt. Acta 30, 155 (1983).
[Crossref]

1982 (2)

R. H. T. Bates, “Astronomical Speckle Imaging,” Phys. Rep. 90, 203 (1982).
[Crossref]

N. J. Woolf, “High Resolution Imaging from the Ground,” Ann. Rev. Astron. Astrophys. 20, 367 (1982).
[Crossref]

1981 (3)

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
[Crossref]

C. J. Oliver, E. R. Pike, “Statistical Accuracy in the Photon Counting Structure Function of Fluctuating Light Fields,” Opt. Acta 28, 1345 (1981).
[Crossref]

E. Ribak, S. G. Lipson, “Complex Spatial Coherence Function: Its Measurement by Means of a Phase-Modulated Shearing Interferometer,” Appl. Opt. 20, 1102 (1981).
[Crossref] [PubMed]

1980 (3)

A. A. Tokovinin, “The Influence of Turbulence on the Operation of a Stellar Interferometer,” Sov. Astron. Lett. 6, 386 (1980).

A. H. Greenaway, J. C. Dainty, “The Formal Equivalence Between Autocorrelation and Power Spectra Analysis of Photon-Limited Data,” Opt. Commun. 35, 307 (1980).
[Crossref]

W. J. Tango, R. Q. Twiss, “Michelson Stellar Interferometry,” Prog. Opt. 17, 239 (1980).
[Crossref]

1979 (1)

A. A. Tokovinin, “A Phase-Grating Stellar Interferometer,” Sov. Astron. Lett. 4, 229 (1979).

1978 (1)

J. J. Burke, J. B. Breckinridge, “Passive Imaging Through the Turbulent Atmosphere: Fundamental Limits on the Spatial Frequency Resolution of a Rotational Shearing Interferometer,” J. Soc. Opt. Am. 68, 67 (1978).
[Crossref]

1976 (3)

A. Labeyrie, “High Resolution Techniques in Optical Astronomy,” Prog. Opt. 13, 49 (1976).

D. V. Korolkov, O. I. Krat, “On the Sensitivity of a Stellar Interferometer with Incoherent Accumulation of the Signal,” Sov. Astron. 20, 370 (1976).

C. Roddier, F. Roddier, “Fringe Visibility in a Michelson Interferometer,” J. Opt. Soc. Am. 66, 580 (1976).
[Crossref]

1975 (1)

J. C. Dainty, R. J. Scadden, “Measurement of the Atmospheric Transfer Function at Mauna Kea, Hawaii,” Mon. Not. R. Astron. Soc. 170, 519 (1975).

1974 (1)

W. C. Wickes, R. H. Dicke, “Achromatic Double Star Interferometry,” Astron. J. 79, 1433 (1974).
[Crossref]

1973 (2)

W. C. Wickes, R. H. Dicke, “An Automatic Interferometer for Double Star Observations,” Astron. J. 78, 757 (1973).
[Crossref]

D. Kelsall, “Optical “Seeing” Through the Atmosphere by an Interferometric Technique,” J. Opt. Soc. Am. 63, 1472 (1973).
[Crossref]

1971 (1)

W. S. Finsen, “Twenty Years of Double Star Interferometry and its Lessons,” Astrophys. Space Sci. 11, 13 (1971).
[Crossref]

1970 (2)

J. L. Elliot, I. S. Glass, “A Quantitative Fringe Detector for Stellar Interferometry,” Astron. J. 75, 1123 (1970).
[Crossref]

E. S. Kulagin, “A Superposed-Ray Interferometer,” Sov. Astron. 13, 1023 (1970).

1957 (1)

S. M. Kozel, “On the Fluctuation Resolution Limit of an Optical Modulation Interferometer,” Sov. Phys. JETP 5, 609 (1957).

Allen, C. W.

C. W. Allen, Astrophysical Quantities (Clowes, London, 1972), p. 197.

Bates, R. H. T.

R. H. T. Bates, “Astronomical Speckle Imaging,” Phys. Rep. 90, 203 (1982).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1976).

Breckinridge, J. B.

J. J. Burke, J. B. Breckinridge, “Passive Imaging Through the Turbulent Atmosphere: Fundamental Limits on the Spatial Frequency Resolution of a Rotational Shearing Interferometer,” J. Soc. Opt. Am. 68, 67 (1978).
[Crossref]

J. B. Breckinridge, “A Two-Dimensional White-Light Interferometer,” IAU Colloquium50, 31 (Maryland, 1978).

Burke, J. J.

J. J. Burke, J. B. Breckinridge, “Passive Imaging Through the Turbulent Atmosphere: Fundamental Limits on the Spatial Frequency Resolution of a Rotational Shearing Interferometer,” J. Soc. Opt. Am. 68, 67 (1978).
[Crossref]

Currie, D. G.

D. G. Currie, “On the Amplitude Interferometer Program at the University of Maryland,” IAU Colloquium 50, 7-1 (Maryland, 1978).

Dainty, J. C.

A. H. Greenaway, J. C. Dainty, “The Formal Equivalence Between Autocorrelation and Power Spectra Analysis of Photon-Limited Data,” Opt. Commun. 35, 307 (1980).
[Crossref]

J. C. Dainty, R. J. Scadden, “Measurement of the Atmospheric Transfer Function at Mauna Kea, Hawaii,” Mon. Not. R. Astron. Soc. 170, 519 (1975).

Dicke, R. H.

W. C. Wickes, R. H. Dicke, “Achromatic Double Star Interferometry,” Astron. J. 79, 1433 (1974).
[Crossref]

W. C. Wickes, R. H. Dicke, “An Automatic Interferometer for Double Star Observations,” Astron. J. 78, 757 (1973).
[Crossref]

Dugan, M.

M. Dugan, “Shearing Interferometer for the Measurement of Atmospheric MTF, M. Sc. Thesis, U. Rochester, New York (1982).

Elliot, J. L.

J. L. Elliot, I. S. Glass, “A Quantitative Fringe Detector for Stellar Interferometry,” Astron. J. 75, 1123 (1970).
[Crossref]

Finsen, W. S.

W. S. Finsen, “Twenty Years of Double Star Interferometry and its Lessons,” Astrophys. Space Sci. 11, 13 (1971).
[Crossref]

Glass, I. S.

J. L. Elliot, I. S. Glass, “A Quantitative Fringe Detector for Stellar Interferometry,” Astron. J. 75, 1123 (1970).
[Crossref]

Greenaway, A. H.

A. H. Greenaway, J. C. Dainty, “The Formal Equivalence Between Autocorrelation and Power Spectra Analysis of Photon-Limited Data,” Opt. Commun. 35, 307 (1980).
[Crossref]

Hege, E. K.

Itoh, K.

Kelsall, D.

Korolkov, D. V.

D. V. Korolkov, O. I. Krat, “On the Sensitivity of a Stellar Interferometer with Incoherent Accumulation of the Signal,” Sov. Astron. 20, 370 (1976).

Kozel, S. M.

S. M. Kozel, “On the Fluctuation Resolution Limit of an Optical Modulation Interferometer,” Sov. Phys. JETP 5, 609 (1957).

Krat, O. I.

D. V. Korolkov, O. I. Krat, “On the Sensitivity of a Stellar Interferometer with Incoherent Accumulation of the Signal,” Sov. Astron. 20, 370 (1976).

Kulagin, E. S.

E. S. Kulagin, “A Superposed-Ray Interferometer,” Sov. Astron. 13, 1023 (1970).

Labeyrie, A.

A. Labeyrie, “High Resolution Techniques in Optical Astronomy,” Prog. Opt. 13, 49 (1976).

Leibowitz, E.

Lipson, H.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., London, 1983).

Lipson, S. G.

Ohtsuka, Y.

Oliver, C. J.

C. J. Oliver, E. R. Pike, “Statistical Accuracy in the Photon Counting Structure Function of Fluctuating Light Fields,” Opt. Acta 28, 1345 (1981).
[Crossref]

Pike, E. R.

C. J. Oliver, E. R. Pike, “Statistical Accuracy in the Photon Counting Structure Function of Fluctuating Light Fields,” Opt. Acta 28, 1345 (1981).
[Crossref]

Ribak, E.

Roddier, C.

Roddier, F.

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
[Crossref]

C. Roddier, F. Roddier, “Fringe Visibility in a Michelson Interferometer,” J. Opt. Soc. Am. 66, 580 (1976).
[Crossref]

F. Roddier, “Rotation-Shearing Interferometry,” IAU Colloquium50, 32-1 (Maryland, 1978).

Scadden, R. J.

J. C. Dainty, R. J. Scadden, “Measurement of the Atmospheric Transfer Function at Mauna Kea, Hawaii,” Mon. Not. R. Astron. Soc. 170, 519 (1975).

Schatzel, K.

K. Schatzel, “Noise in Photon Correlation and Photon Structure Function,” Opt. Acta 30, 155 (1983).
[Crossref]

Tango, W. J.

W. J. Tango, R. Q. Twiss, “Michelson Stellar Interferometry,” Prog. Opt. 17, 239 (1980).
[Crossref]

Tokovinin, A. A.

A. A. Tokovinin, “The Influence of Turbulence on the Operation of a Stellar Interferometer,” Sov. Astron. Lett. 6, 386 (1980).

A. A. Tokovinin, “A Phase-Grating Stellar Interferometer,” Sov. Astron. Lett. 4, 229 (1979).

Twiss, R. Q.

W. J. Tango, R. Q. Twiss, “Michelson Stellar Interferometry,” Prog. Opt. 17, 239 (1980).
[Crossref]

Wickes, W. C.

W. C. Wickes, R. H. Dicke, “Achromatic Double Star Interferometry,” Astron. J. 79, 1433 (1974).
[Crossref]

W. C. Wickes, R. H. Dicke, “An Automatic Interferometer for Double Star Observations,” Astron. J. 78, 757 (1973).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1976).

Woolf, N. J.

N. J. Woolf, “High Resolution Imaging from the Ground,” Ann. Rev. Astron. Astrophys. 20, 367 (1982).
[Crossref]

Ann. Rev. Astron. Astrophys. (1)

N. J. Woolf, “High Resolution Imaging from the Ground,” Ann. Rev. Astron. Astrophys. 20, 367 (1982).
[Crossref]

Appl. Opt. (2)

Astron. J. (3)

J. L. Elliot, I. S. Glass, “A Quantitative Fringe Detector for Stellar Interferometry,” Astron. J. 75, 1123 (1970).
[Crossref]

W. C. Wickes, R. H. Dicke, “An Automatic Interferometer for Double Star Observations,” Astron. J. 78, 757 (1973).
[Crossref]

W. C. Wickes, R. H. Dicke, “Achromatic Double Star Interferometry,” Astron. J. 79, 1433 (1974).
[Crossref]

Astrophys. Space Sci. (1)

W. S. Finsen, “Twenty Years of Double Star Interferometry and its Lessons,” Astrophys. Space Sci. 11, 13 (1971).
[Crossref]

J. Opt. Soc. Am. (3)

J. Soc. Opt. Am. (1)

J. J. Burke, J. B. Breckinridge, “Passive Imaging Through the Turbulent Atmosphere: Fundamental Limits on the Spatial Frequency Resolution of a Rotational Shearing Interferometer,” J. Soc. Opt. Am. 68, 67 (1978).
[Crossref]

Mon. Not. R. Astron. Soc. (1)

J. C. Dainty, R. J. Scadden, “Measurement of the Atmospheric Transfer Function at Mauna Kea, Hawaii,” Mon. Not. R. Astron. Soc. 170, 519 (1975).

Opt. Acta (2)

C. J. Oliver, E. R. Pike, “Statistical Accuracy in the Photon Counting Structure Function of Fluctuating Light Fields,” Opt. Acta 28, 1345 (1981).
[Crossref]

K. Schatzel, “Noise in Photon Correlation and Photon Structure Function,” Opt. Acta 30, 155 (1983).
[Crossref]

Opt. Commun. (1)

A. H. Greenaway, J. C. Dainty, “The Formal Equivalence Between Autocorrelation and Power Spectra Analysis of Photon-Limited Data,” Opt. Commun. 35, 307 (1980).
[Crossref]

Phys. Rep. (1)

R. H. T. Bates, “Astronomical Speckle Imaging,” Phys. Rep. 90, 203 (1982).
[Crossref]

Prog. Opt. (3)

A. Labeyrie, “High Resolution Techniques in Optical Astronomy,” Prog. Opt. 13, 49 (1976).

W. J. Tango, R. Q. Twiss, “Michelson Stellar Interferometry,” Prog. Opt. 17, 239 (1980).
[Crossref]

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
[Crossref]

Sov. Astron. (2)

E. S. Kulagin, “A Superposed-Ray Interferometer,” Sov. Astron. 13, 1023 (1970).

D. V. Korolkov, O. I. Krat, “On the Sensitivity of a Stellar Interferometer with Incoherent Accumulation of the Signal,” Sov. Astron. 20, 370 (1976).

Sov. Astron. Lett. (2)

A. A. Tokovinin, “A Phase-Grating Stellar Interferometer,” Sov. Astron. Lett. 4, 229 (1979).

A. A. Tokovinin, “The Influence of Turbulence on the Operation of a Stellar Interferometer,” Sov. Astron. Lett. 6, 386 (1980).

Sov. Phys. JETP (1)

S. M. Kozel, “On the Fluctuation Resolution Limit of an Optical Modulation Interferometer,” Sov. Phys. JETP 5, 609 (1957).

Other (8)

M. Dugan, “Shearing Interferometer for the Measurement of Atmospheric MTF, M. Sc. Thesis, U. Rochester, New York (1982).

J. B. Breckinridge, “A Two-Dimensional White-Light Interferometer,” IAU Colloquium50, 31 (Maryland, 1978).

F. Roddier, “Rotation-Shearing Interferometry,” IAU Colloquium50, 32-1 (Maryland, 1978).

C. W. Allen, Astrophysical Quantities (Clowes, London, 1972), p. 197.

D. G. Currie, “On the Amplitude Interferometer Program at the University of Maryland,” IAU Colloquium 50, 7-1 (Maryland, 1978).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1976).

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., London, 1983).

J. C. Dainty, Ed., “Stellar Speckle Interferometry,” in Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1983).

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Figures (3)

Fig. 1
Fig. 1

Modulated signal entering the detector (top) and after the amplifier (center). Each counter collects pulses for one quarter of a cycle (bottom).

Fig. 2
Fig. 2

Cyclic channeling of photon pulses into four counters.

Fig. 3
Fig. 3

Schemes to achieve sine and cosine approximations by adding and subtracting every two of the four (quarter cycle) counters for each (top) or by adding and subtracting every second one (center). The traight counter adds them all up (bottom).

Equations (30)

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

I = I 1 + I 2 + 2 ( I 1 I 2 ) 1 / 2 | g 12 | cos ( ϕ + θ ) ,
V = I max I min I max + I min = 2 | g 12 | ( I 1 / I 2 ) 1 / 2 + ( I 2 / I 1 ) 1 / 2 .
I ( t ) = I 1 + I 2 + 2 ( I 1 I 2 ) 1 / 2 | g 12 | cos ( ϕ + k cos t ) ,
f ( t ) = 1 A A d 2 r [ U ( r , t ) + V ( r , t ) cos ( 2 π t T + ϕ ) ] ,
f C = 0 T 1 d t f ( t ) cos 2 π t T = V cos ϕ .
p j = T 4 i = 1 K 1 U i j + V i j 2 ϕ T 1 d t 1 A A d 2 r cos [ ϕ ( r , t ) + π j / 2 ] .
s j = p j = T 1 4 [ U + 2 V cos ( ϕ + π j / 2 ) ] .
S 0 = k = 1 K 2 ( p 1 k p 3 k ) 2 + ( p 2 k p 4 k ) 2 ,
N 0 = k = 1 K 2 p 1 k + p 2 k + p 3 k + p 4 k .
N = E ( N 0 ) = var ( N 0 ) = T 2 U ,
S = E ( S 0 ) = N ( 1 + W m 2 / 2 ) ,
var ( S 0 ) = N ( 1 + W + 2 W m 2 + W 2 m 2 ) ,
m = ( V / U ) ( sin ϕ 2 + cos ϕ 2 ) 1 / 2 .
P 0 = 2 K 2 S 0 N 0 2 N 0 2 .
P = E ( P 0 ) = m 2 ( 1 + 3 N ) 12 K 2 N 3 = m 2 ( 1 + 3 N ) 12 N W 2 ,
var ( P 0 ) = 4 K 2 [ 2 K 2 + N ( 1 + 4 m 2 ) ] N 3 + 12 N = 4 [ 2 + W ( 1 + 4 m 2 ) ] N W 2 + 12 N ,
m υ = log 2 . 51 ( 10 8 A T 1 Δ λ η / U ) = 8 . 14 2 . 5 log 10 ( U ) ,
m υ = 7 . 81 + 5 / 3 log 10 m ( bias < 3 % ) ,
m υ = 7 . 81 + 10 / 3 log 10 m ( SNR > 5 ) .
N = N 0 = K 2 p 1 + p 2 + p 3 + p 4 = K 2 ( s 1 + s 2 + s 3 + s 4 ) = K 2 T 1 U = k 2 W = T 2 U ,
var ( N ) = n 0 = N .
m 1 ( u ) = exp [ ( b + s ) ( e u 1 ) ]
m 3 ( u ) = exp [ ( b s ) ( e u 1 ) ] .
m 13 ( u ) = m 1 ( u ) m 3 ( u ) = exp [ ( b + s ) e u + ( b s ) e u 2 b ] .
m 13 ( 0 ) = 2 b + 4 s 2 ,
m ( 4 ) ( 0 ) [ m 13 ( 0 ) ] 2 = 8 b 2 + 2 b + 32 s 2 b + 16 s 2 .
S = E ( S 0 ) = 4 K 2 ( b + s 2 + c 2 ) = T 2 U ( 1 + W m 2 / 2 ) = N ( 1 + W m 2 / 2 ) ,
m = ( V / U ) ( sin ϕ 2 + cos ϕ 2 ) 1 / 2 .
f 2 = f 2 + f .
var ( S 0 ) = N ( 1 + W + 2 W m 2 + W 2 m 2 ) .

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