Abstract

High-resolution phase-contrast wave-front sensors based on phase spatial light modulators and micromirror/liquid-crystal arrays are introduced. Wave-front sensor performance is analyzed for atmospheric-turbulence-induced phase distortions described by the Kolmogorov and the Andrews models. A high-resolution phase-contrast wave-front sensor (nonlinear Zernike filter) based on an optically controlled liquid-crystal phase spatial light modulator is experimentally demonstrated. The results demonstrate high-resolution visualization of dynamically changing phase distortions within the sensor time response of ∼10 ms.

© 2001 Optical Society of America

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    [CrossRef]

2001 (1)

2000 (1)

1999 (1)

1998 (1)

1997 (3)

R. Dou, M. A. Vorontsov, V. P. Sivokon, M. K. Giles, “Iterative technique for high-resolution phase distortion compensation in adaptive interferometers,” Opt. Eng. 36, 3327–3335 (1997).
[CrossRef]

M. C. Wu, “Micromachining for optical and opto-electronic systems,” Proc. IEEE 85, 1833–1856 (1997).
[CrossRef]

J. Glückstad, L. Lading, H. Toyoda, T. Hara, “Lossless light projection,” Opt. Lett. 22, 1373–1375 (1997).
[CrossRef]

1996 (1)

A. G. Andreou, K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits Signal Process. 9, 141–166 (1996).
[CrossRef]

1995 (2)

J. Glückstad, “Adaptive array illumination and structured light generated by spatial zero-order self-phase modulation in Kerr medium,” Opt. Commun. 120, 194–203 (1995).
[CrossRef]

G. V. Vdovin, P. M. Sarro, “Flexible mirror micromachined in silicon,” Appl. Opt. 34, 2968–2972 (1995).
[CrossRef] [PubMed]

1994 (2)

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

R. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368, 203–207 (1994).
[CrossRef]

1992 (2)

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

V. Yu. Ivanov, V. P. Sivokon, M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992).
[CrossRef]

1989 (1)

M. A. Vorontsov, A. F. Naumov, V. P. Katulin, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
[CrossRef]

1988 (2)

1983 (1)

A. D. Fisher, C. Warde, “Technique for real-time high-resolution adaptive phase compensation,” Opt. Lett. 87, 353–355 (1983).
[CrossRef]

1982 (1)

1978 (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1977 (1)

1976 (1)

1975 (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxon, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1965 (1)

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef] [PubMed]

Akhmanov, S. A.

S. A. Akhmanov, S. Yu. Nikitin, Physical Optics (Clarendon, Oxford, UK, 1997).

Andreou, A. G.

A. G. Andreou, K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits Signal Process. 9, 141–166 (1996).
[CrossRef]

Andrews, L. C.

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

Angel, R.

R. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368, 203–207 (1994).
[CrossRef]

Arnold, R.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Barrett, T.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Beresnev, L. A.

Boahen, K. A.

A. G. Andreou, K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits Signal Process. 9, 141–166 (1996).
[CrossRef]

Carhart, G. W.

E. W. Justh, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
[CrossRef]

G. W. Carhart, M. A. Vorontsov, E. W. Justh, “Opto-electronic Zernike filter for high-resolution wavefront analysis using a phase-only liquid crystal spatial light modulator,” in High-Resolution Wavefront Control: Methods, Devices, and Applications II, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE4124, 138–147 (2000).
[CrossRef]

Chigrinov, V. G.

V. G. Chigrinov, Liquid Crystal Devices: Physics and Applications (Artech House, Boston, Mass., 1999).

Cuellar, L.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Dou, R.

R. Dou, M. A. Vorontsov, V. P. Sivokon, M. K. Giles, “Iterative technique for high-resolution phase distortion compensation in adaptive interferometers,” Opt. Eng. 36, 3327–3335 (1997).
[CrossRef]

Fienup, J. R.

Fisher, A. D.

A. D. Fisher, C. Warde, “Technique for real-time high-resolution adaptive phase compensation,” Opt. Lett. 87, 353–355 (1983).
[CrossRef]

Fried, D. L.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxon, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Giles, M. K.

R. Dou, M. A. Vorontsov, V. P. Sivokon, M. K. Giles, “Iterative technique for high-resolution phase distortion compensation in adaptive interferometers,” Opt. Eng. 36, 3327–3335 (1997).
[CrossRef]

A. Seward, F. Lacombe, M. K. Giles, “Focal plane masks in adaptive optics systems,” in Adaptive Optics Systems and Technology, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 283–293 (1999).
[CrossRef]

Glückstad, J.

P. C. Mogensen, J. Glückstad, “Phase-only optical encryption,” Opt. Lett. 25, 566–568 (2000).
[CrossRef]

J. Glückstad, L. Lading, H. Toyoda, T. Hara, “Lossless light projection,” Opt. Lett. 22, 1373–1375 (1997).
[CrossRef]

J. Glückstad, “Adaptive array illumination and structured light generated by spatial zero-order self-phase modulation in Kerr medium,” Opt. Commun. 120, 194–203 (1995).
[CrossRef]

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Hara, T.

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
[CrossRef]

Ivanov, V. Yu.

Johnson, P.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Justh, E. W.

E. W. Justh, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
[CrossRef]

G. W. Carhart, M. A. Vorontsov, E. W. Justh, “Opto-electronic Zernike filter for high-resolution wavefront analysis using a phase-only liquid crystal spatial light modulator,” in High-Resolution Wavefront Control: Methods, Devices, and Applications II, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE4124, 138–147 (2000).
[CrossRef]

Katulin, V. P.

M. A. Vorontsov, A. F. Naumov, V. P. Katulin, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
[CrossRef]

Koliopoulos, C. L.

J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
[CrossRef]

K. Underwood, J. C. Wyant, C. L. Koliopoulos, “Self-referencing wavefront sensor,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds., Proc. SPIE351, 108–114 (1982).
[CrossRef]

Krishnaprasad, P. S.

Lacombe, F.

A. Seward, F. Lacombe, M. K. Giles, “Focal plane masks in adaptive optics systems,” in Adaptive Optics Systems and Technology, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 283–293 (1999).
[CrossRef]

Lading, L.

Lefebvre, J. E.

Lefebvre, M.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

McKnight, D.

S. Serati, G. Sharp, R. Serati, D. McKnight, J. Stockley, “128×128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 378–387 (1995).
[CrossRef]

Mogensen, P. C.

Naumov, A. F.

M. A. Vorontsov, A. F. Naumov, V. P. Katulin, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
[CrossRef]

Nikitin, S. Yu.

S. A. Akhmanov, S. Yu. Nikitin, Physical Optics (Clarendon, Oxford, UK, 1997).

Paxman, R. G.

Rego, A.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Roddier, F.

Roggemann, M. C.

M. C. Roggemann, B. WelshImaging through Turbulence (CRC Press, Boca Raton, Fla., 1995).

Rousset, G.

G. Rousset, “Wavefront sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, New York, 1999), pp. 91–130.

Sandler, D. G.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Sarro, P. M.

Saxon, W. O.

R. W. Gerchberg, W. O. Saxon, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Serati, R.

S. Serati, G. Sharp, R. Serati, D. McKnight, J. Stockley, “128×128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 378–387 (1995).
[CrossRef]

Serati, S.

S. Serati, G. Sharp, R. Serati, D. McKnight, J. Stockley, “128×128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 378–387 (1995).
[CrossRef]

Seward, A.

A. Seward, F. Lacombe, M. K. Giles, “Focal plane masks in adaptive optics systems,” in Adaptive Optics Systems and Technology, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 283–293 (1999).
[CrossRef]

Sharp, G.

S. Serati, G. Sharp, R. Serati, D. McKnight, J. Stockley, “128×128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 378–387 (1995).
[CrossRef]

Sivokon, V. P.

Smartt, R. N.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
[CrossRef]

Smith, G.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Spiv, G.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Steel, W. H.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
[CrossRef]

Stockley, J.

S. Serati, G. Sharp, R. Serati, D. McKnight, J. Stockley, “128×128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 378–387 (1995).
[CrossRef]

Taylor, B.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, B. Taylor, G. Spiv, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Toyoda, H.

Underwood, K.

K. Underwood, J. C. Wyant, C. L. Koliopoulos, “Self-referencing wavefront sensor,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds., Proc. SPIE351, 108–114 (1982).
[CrossRef]

Vdovin, G. V.

Vorontsov, M. A.

E. W. Justh, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
[CrossRef]

M. A. Vorontsov, “High-resolution adaptive phase distortion compensation using a diffractive-feedback system: experimental results,” J. Opt. Soc. Am. A 16, 2567–2573 (1999).
[CrossRef]

V. P. Sivokon, M. A. Vorontsov, “High-resolution adaptive phase distortion suppression based solely on intensity information,” J. Opt. Soc. Am. A 15, 234–247 (1998).
[CrossRef]

R. Dou, M. A. Vorontsov, V. P. Sivokon, M. K. Giles, “Iterative technique for high-resolution phase distortion compensation in adaptive interferometers,” Opt. Eng. 36, 3327–3335 (1997).
[CrossRef]

V. Yu. Ivanov, V. P. Sivokon, M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992).
[CrossRef]

M. A. Vorontsov, A. F. Naumov, V. P. Katulin, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
[CrossRef]

G. W. Carhart, M. A. Vorontsov, E. W. Justh, “Opto-electronic Zernike filter for high-resolution wavefront analysis using a phase-only liquid crystal spatial light modulator,” in High-Resolution Wavefront Control: Methods, Devices, and Applications II, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE4124, 138–147 (2000).
[CrossRef]

Warde, C.

A. D. Fisher, C. Warde, “Technique for real-time high-resolution adaptive phase compensation,” Opt. Lett. 87, 353–355 (1983).
[CrossRef]

Welsh, B.

M. C. Roggemann, B. WelshImaging through Turbulence (CRC Press, Boca Raton, Fla., 1995).

Wu, M. C.

M. C. Wu, “Micromachining for optical and opto-electronic systems,” Proc. IEEE 85, 1833–1856 (1997).
[CrossRef]

Wyant, J. C.

K. Underwood, J. C. Wyant, C. L. Koliopoulos, “Self-referencing wavefront sensor,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds., Proc. SPIE351, 108–114 (1982).
[CrossRef]

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef] [PubMed]

Analog Integr. Circuits Signal Process. (1)

A. G. Andreou, K. A. Boahen, “Translinear circuits in subthreshold MOS,” Analog Integr. Circuits Signal Process. 9, 141–166 (1996).
[CrossRef]

Appl. Opt. (3)

J. Mod. Opt. (1)

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (5)

Jpn. J. Appl. Phys. (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
[CrossRef]

Nature (1)

R. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368, 203–207 (1994).
[CrossRef]

Opt. Commun. (2)

M. A. Vorontsov, A. F. Naumov, V. P. Katulin, “Wavefront control by an optical-feedback interferometer,” Opt. Commun. 71, 35–38 (1989).
[CrossRef]

J. Glückstad, “Adaptive array illumination and structured light generated by spatial zero-order self-phase modulation in Kerr medium,” Opt. Commun. 120, 194–203 (1995).
[CrossRef]

Opt. Eng. (1)

R. Dou, M. A. Vorontsov, V. P. Sivokon, M. K. Giles, “Iterative technique for high-resolution phase distortion compensation in adaptive interferometers,” Opt. Eng. 36, 3327–3335 (1997).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Basic wave-front sensor schematic: (a) conventional, (b) differential, (c) nonlinear, (d) optoelectronic Zernike filters.

Fig. 2
Fig. 2

Output intensity patterns for the optoelectronic Zernike filter corresponding to the input phase realization (a) with Andrews spectrum for different σφ values, (b) σφ=0.23 rad, (c) σφ=0.41, (d) σφ=2.45.

Fig. 3
Fig. 3

Contrast enhancement in the optoelectronic Zernike filter with intensity thresholding. Output intensity patterns for the optoelectronic Zernike filter (a) without and (b) with intensity thresholding (=0.75) corresponding to the input phase realization with Andrews spectrum shown in Fig. 2a. In both cases the standard deviation for the aperture-averaged phase deviation is σφ=1.48 rad (St=0.1).

Fig. 4
Fig. 4

Output intensity patterns for the nonlinear Zernike filter corresponding to the input phase realization (a) with Andrews spectrum for α=0.5π/IF0 and different σφ values, (b) σφ=0.23 rad, (c) σφ=0.72 rad, and (d) σφ=1.48 rad. The value IF0 is the zero spectral component intensity in the absence of phase aberrations.

Fig. 5
Fig. 5

Standard deviation of output intensity fluctuations σout for different wave-front sensor types versus input phase standard deviation σin (Andrews spectrum). Dashed curves correspond to the following optoelectronic wave-front sensor configurations: point-diffraction interferometer (PDI), Zernike filter (ZF), and optoelectronic Zernike filter with intensity thresholding (Th) for =0.5. Solid curves correspond to nonlinear Zernike filters with α=0.5π/IF0 (NZF1) and α=π/IF0 (NZF2). Numbers in parentheses correspond to Strehl ratio values 〈St〉 for σin.

Fig. 6
Fig. 6

Phase-intensity correlation coefficients C versus input phase standard deviation σin (Andrews spectrum). Notation is the same as in Fig. 5.

Fig. 7
Fig. 7

Wave-front sensor performance metric Q versus input phase standard deviation σin (Andrews spectrum). Notation is the same as in Fig. 5.

Fig. 8
Fig. 8

Wave-front sensor performance metric Q versus input phase standard deviation σin (Kolmogorov spectrum). Notation is the same as in Fig. 5.

Fig. 9
Fig. 9

Modulation characteristics of the LCLV: Curves 1, 2, and 4 are for λ=0.514 μm; curve 3 is for λ=0.63 μm. Applied voltage amplitudes are the following: curve 1, V=8.5 V; curve 2, V=13 V; curve 3, V=8.5 V; and curve 4, V=16 V. The schematic for the LCLV is shown at top left.

Fig. 10
Fig. 10

Experimental setup of the LCLV-based nonlinear Zernike filter with a photo of the LCLV.

Fig. 11
Fig. 11

Output patterns for the nonlinear Zernike filter with LCLV: left column, V=0 V; right column, V=8.5 V. Visualizations of phase distortions are generated by (a), (b) HEX127 LC phase SLM; (c), (d) Xinξtics mirror; (e), (f) heater and fan. Pictures (c) and (d) correspond to λ=0.63 μm, and the others to λ=0.514 μm.

Equations (20)

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T(q)=γ exp(iθ)|q|q0,T(q)=1otherwise.
Aout(q)=A(q)[1-δ(q)]+γ exp(iθ)A(q)δ(q),
Aout(r)=Ain(r)-[1-γ exp(iθ)]A¯,
A¯= Ain(r)d2r,
Aout(r)=Ain(r)-[1-γ exp(iθ)] A¯0 exp(iφ¯).
Iout(r)=I0(r)+(2πF)2IF(0)(1+γ2-2γ cos θ)-4πFI01/2(r)IF1/2(0){cos[φ˜(r)-Δ]-γ cos[φ˜(r)-Δ-θ]}.
Iout(r)=I0(r)+(2πF)2IF(0)-4πFI01/2(r)IF1/2(0)cos[φ˜(r)-Δ].
Iout(r)Izer(+)(r)=I0(r)+2(2πF)2IF(0)-4πFI01/2(r)IF1/2(0)×{cos[φ˜(r)-Δ]-sin[φ˜(r)-Δ]}.
ΓPDI(r)=4πFI01/2(r)IF1/2(0)/[I0(r)+(2πF)2IF(0)],
ΓZF(r)=42πFI01/2(r)IF1/2(0)/[I0(r)+2(2πF)2IF(0)].
σI=S-1  I˜out2(r)d2r1/2/I¯out,
cφ,I=1SσφσII¯out φ˜(r)I˜out(r)d2r,
σφ=[S-1 φ˜2(r)d2r]1/2.
Qφ=cφ,IσI=1σφI¯outS φ˜(r)I˜out(r)d2r.
Iout(r)Izer(-)(r)=I0(r)+2(2πF)2IF(0)-4πFI01/2(r)IF1/2(0)×{cos[φ˜(r)-Δ]+sin[φ˜(r)-Δ]}.
Idif(r)Izer(+)(r)-Izer(-)(r)=8πFI01/2(r)IF1/2(0)sin[φ˜(r)-Δ],
Aout(q)=A(q)exp[iαIF(q)],
IF(q)=(2πF)-2|A(q)|2.
GK(q)=2π0.033(1.68/r0)5/3q-11/3,
GA(q)=2π0.033(1.68/r0)5/3(q2+qA2)-11/6 exp(-q2/qa2)×[1+1.802(q/qa)-0.254(q/qa)7/6].

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