Abstract

We discuss the use of liquid-crystal phase modulators (LCPM’s) both as a repeatable disturbance test source and as an adaptive optics corrector. LCPM’s have the potential to induce controlled, repeatable, dynamic aberrations into optical systems at low cost, low complexity, and high flexibility. Because they are programmable and can be operated as transmissive elements, they can easily be inserted into the optical path of an adaptive optics system and used to generate a disturbance test source. When used as wave-front correctors they act as a piston-only segmented mirror and have a number of advantages. These include low operating power requirements, relatively low cost, and compact size. Laboratory experiments with a Meadowlark LCPM are presented. We first describe use of the LCPM as a repeatable disturbance generator for testing adaptive optics systems. We then describe a closed-loop adaptive optics system using the LCPM as the wave-front corrector. The adaptive optics system includes a Shack–Hartmann wave-front sensor operated with a zonal control algorithm.

© 1998 Optical Society of America

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References

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  1. A. A. Vasil’iev, I. N. Kompanets, A. V. Parfenov, “Progress in the development and applications of optically controlled liquid crystal spatial light modulators,” Sov. J. Quantum Electron. 13, 689–695 (1983).
    [CrossRef]
  2. S. T. Kowel, P. Kornreich, A. Nouhi, “Adaptive spherical lens,” Appl. Opt. 23, 2774–2777 (1984).
    [CrossRef] [PubMed]
  3. A. A. Vasil’ev, A. F. Naumov, V. I. Shmal’gauzen, “Wavefront correction by liquid-crystal devices,” Sov. J. Quantum Electron. 16, 471–474 (1986).
    [CrossRef]
  4. G. D. Love, “Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt. 36, 1517–1524 (1997).
    [CrossRef] [PubMed]
  5. J. Gourlay, G. D. Love, P. M. Birch, R. M. Sharples, A. Purvis, “A real-time closed loop liquid crystal adaptive optics system: first results,” Opt. Commun. 137, 17–21 (1997).
    [CrossRef]
  6. M. Vorontsov, G. Carhart, D. Pruidze, J. Ricklin, D. Voelz, “Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field,” J. Opt. Soc. Am. A 13, 1456–1466 (1996).
    [CrossRef]
  7. D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, V. Biliotti, “Adaptive optics wave front corrector using addressable liquid crystal retarders,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. SPIE1543, 133–143 (1991).
    [CrossRef]
  8. A. Kudryashov, J. Gonglewski, S. Browne, R. Highland, “Liquid crystal phase modulator for adaptive optics. Temporal performance characterization,” Opt. Commun. 141, 247–253 (1997).
    [CrossRef]
  9. S. Restaino, M. Anderson, J. Baker, D. Payne, G. Loos, R. Carreras, “Progress report of USAF Research Laboratory liquid crystal AO program,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998).
    [CrossRef]
  10. J. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985), p. 389.
  11. A. Kolmogorov, “Dissipation of energy in locally isotropic turbulence,” in Turbulence, Classic Papers on Statistical Theory, S. Friedlander, L. Topper, eds., (Wiley-Interscience, New York, 1961).
  12. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  13. F. Roddier, M. Northcott, J. Graves, D. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. 10, 957–965 (1993).
    [CrossRef]
  14. J. Wang, J. Markey, “Modal compensation of atmospheric turbulence phase distortions,” J. Opt. Soc. Am. A 10, 78–87 (1978).
  15. D. Dayton, S. Sandven, J. Gonglewski, “Adaptive optics using a liquid crystal phase modulator in conjunction with a Shack-Hartmann wave-front sensor and zonal control algorithm,” Opt. Exp. 1, 338–346 (1977).
    [CrossRef]
  16. W. Wild, E. Kibblewhite, R. Vuilleumier, “Sparse matrix wave-front estimators for adaptive-optics system for large ground-based telescopes,” Opt. Lett. 20, 955–957 (1995).
    [CrossRef] [PubMed]

1997

G. D. Love, “Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt. 36, 1517–1524 (1997).
[CrossRef] [PubMed]

J. Gourlay, G. D. Love, P. M. Birch, R. M. Sharples, A. Purvis, “A real-time closed loop liquid crystal adaptive optics system: first results,” Opt. Commun. 137, 17–21 (1997).
[CrossRef]

A. Kudryashov, J. Gonglewski, S. Browne, R. Highland, “Liquid crystal phase modulator for adaptive optics. Temporal performance characterization,” Opt. Commun. 141, 247–253 (1997).
[CrossRef]

1996

1995

1993

F. Roddier, M. Northcott, J. Graves, D. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. 10, 957–965 (1993).
[CrossRef]

1986

A. A. Vasil’ev, A. F. Naumov, V. I. Shmal’gauzen, “Wavefront correction by liquid-crystal devices,” Sov. J. Quantum Electron. 16, 471–474 (1986).
[CrossRef]

1984

1983

A. A. Vasil’iev, I. N. Kompanets, A. V. Parfenov, “Progress in the development and applications of optically controlled liquid crystal spatial light modulators,” Sov. J. Quantum Electron. 13, 689–695 (1983).
[CrossRef]

1978

J. Wang, J. Markey, “Modal compensation of atmospheric turbulence phase distortions,” J. Opt. Soc. Am. A 10, 78–87 (1978).

1977

D. Dayton, S. Sandven, J. Gonglewski, “Adaptive optics using a liquid crystal phase modulator in conjunction with a Shack-Hartmann wave-front sensor and zonal control algorithm,” Opt. Exp. 1, 338–346 (1977).
[CrossRef]

1976

Anderson, M.

S. Restaino, M. Anderson, J. Baker, D. Payne, G. Loos, R. Carreras, “Progress report of USAF Research Laboratory liquid crystal AO program,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998).
[CrossRef]

Baker, J.

S. Restaino, M. Anderson, J. Baker, D. Payne, G. Loos, R. Carreras, “Progress report of USAF Research Laboratory liquid crystal AO program,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998).
[CrossRef]

Biliotti, V.

D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, V. Biliotti, “Adaptive optics wave front corrector using addressable liquid crystal retarders,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. SPIE1543, 133–143 (1991).
[CrossRef]

Birch, P. M.

J. Gourlay, G. D. Love, P. M. Birch, R. M. Sharples, A. Purvis, “A real-time closed loop liquid crystal adaptive optics system: first results,” Opt. Commun. 137, 17–21 (1997).
[CrossRef]

Bonaccini, D.

D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, V. Biliotti, “Adaptive optics wave front corrector using addressable liquid crystal retarders,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. SPIE1543, 133–143 (1991).
[CrossRef]

Browne, S.

A. Kudryashov, J. Gonglewski, S. Browne, R. Highland, “Liquid crystal phase modulator for adaptive optics. Temporal performance characterization,” Opt. Commun. 141, 247–253 (1997).
[CrossRef]

Brusa, G.

D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, V. Biliotti, “Adaptive optics wave front corrector using addressable liquid crystal retarders,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. SPIE1543, 133–143 (1991).
[CrossRef]

Carhart, G.

Carreras, R.

S. Restaino, M. Anderson, J. Baker, D. Payne, G. Loos, R. Carreras, “Progress report of USAF Research Laboratory liquid crystal AO program,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998).
[CrossRef]

Dayton, D.

D. Dayton, S. Sandven, J. Gonglewski, “Adaptive optics using a liquid crystal phase modulator in conjunction with a Shack-Hartmann wave-front sensor and zonal control algorithm,” Opt. Exp. 1, 338–346 (1977).
[CrossRef]

Esposito, S.

D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, V. Biliotti, “Adaptive optics wave front corrector using addressable liquid crystal retarders,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. SPIE1543, 133–143 (1991).
[CrossRef]

Gonglewski, J.

A. Kudryashov, J. Gonglewski, S. Browne, R. Highland, “Liquid crystal phase modulator for adaptive optics. Temporal performance characterization,” Opt. Commun. 141, 247–253 (1997).
[CrossRef]

D. Dayton, S. Sandven, J. Gonglewski, “Adaptive optics using a liquid crystal phase modulator in conjunction with a Shack-Hartmann wave-front sensor and zonal control algorithm,” Opt. Exp. 1, 338–346 (1977).
[CrossRef]

Goodman, J.

J. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985), p. 389.

Gourlay, J.

J. Gourlay, G. D. Love, P. M. Birch, R. M. Sharples, A. Purvis, “A real-time closed loop liquid crystal adaptive optics system: first results,” Opt. Commun. 137, 17–21 (1997).
[CrossRef]

Graves, J.

F. Roddier, M. Northcott, J. Graves, D. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. 10, 957–965 (1993).
[CrossRef]

Highland, R.

A. Kudryashov, J. Gonglewski, S. Browne, R. Highland, “Liquid crystal phase modulator for adaptive optics. Temporal performance characterization,” Opt. Commun. 141, 247–253 (1997).
[CrossRef]

Kibblewhite, E.

Kolmogorov, A.

A. Kolmogorov, “Dissipation of energy in locally isotropic turbulence,” in Turbulence, Classic Papers on Statistical Theory, S. Friedlander, L. Topper, eds., (Wiley-Interscience, New York, 1961).

Kompanets, I. N.

A. A. Vasil’iev, I. N. Kompanets, A. V. Parfenov, “Progress in the development and applications of optically controlled liquid crystal spatial light modulators,” Sov. J. Quantum Electron. 13, 689–695 (1983).
[CrossRef]

Kornreich, P.

Kowel, S. T.

Kudryashov, A.

A. Kudryashov, J. Gonglewski, S. Browne, R. Highland, “Liquid crystal phase modulator for adaptive optics. Temporal performance characterization,” Opt. Commun. 141, 247–253 (1997).
[CrossRef]

Loos, G.

S. Restaino, M. Anderson, J. Baker, D. Payne, G. Loos, R. Carreras, “Progress report of USAF Research Laboratory liquid crystal AO program,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998).
[CrossRef]

Love, G. D.

G. D. Love, “Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt. 36, 1517–1524 (1997).
[CrossRef] [PubMed]

J. Gourlay, G. D. Love, P. M. Birch, R. M. Sharples, A. Purvis, “A real-time closed loop liquid crystal adaptive optics system: first results,” Opt. Commun. 137, 17–21 (1997).
[CrossRef]

Markey, J.

J. Wang, J. Markey, “Modal compensation of atmospheric turbulence phase distortions,” J. Opt. Soc. Am. A 10, 78–87 (1978).

McKenna, D.

F. Roddier, M. Northcott, J. Graves, D. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. 10, 957–965 (1993).
[CrossRef]

Naumov, A. F.

A. A. Vasil’ev, A. F. Naumov, V. I. Shmal’gauzen, “Wavefront correction by liquid-crystal devices,” Sov. J. Quantum Electron. 16, 471–474 (1986).
[CrossRef]

Noll, R. J.

Northcott, M.

F. Roddier, M. Northcott, J. Graves, D. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. 10, 957–965 (1993).
[CrossRef]

Nouhi, A.

Parfenov, A. V.

A. A. Vasil’iev, I. N. Kompanets, A. V. Parfenov, “Progress in the development and applications of optically controlled liquid crystal spatial light modulators,” Sov. J. Quantum Electron. 13, 689–695 (1983).
[CrossRef]

Payne, D.

S. Restaino, M. Anderson, J. Baker, D. Payne, G. Loos, R. Carreras, “Progress report of USAF Research Laboratory liquid crystal AO program,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998).
[CrossRef]

Pruidze, D.

Purvis, A.

J. Gourlay, G. D. Love, P. M. Birch, R. M. Sharples, A. Purvis, “A real-time closed loop liquid crystal adaptive optics system: first results,” Opt. Commun. 137, 17–21 (1997).
[CrossRef]

Restaino, S.

S. Restaino, M. Anderson, J. Baker, D. Payne, G. Loos, R. Carreras, “Progress report of USAF Research Laboratory liquid crystal AO program,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998).
[CrossRef]

Ricklin, J.

Roddier, D.

F. Roddier, M. Northcott, J. Graves, D. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. 10, 957–965 (1993).
[CrossRef]

Roddier, F.

F. Roddier, M. Northcott, J. Graves, D. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. 10, 957–965 (1993).
[CrossRef]

Salinari, P.

D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, V. Biliotti, “Adaptive optics wave front corrector using addressable liquid crystal retarders,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. SPIE1543, 133–143 (1991).
[CrossRef]

Sandven, S.

D. Dayton, S. Sandven, J. Gonglewski, “Adaptive optics using a liquid crystal phase modulator in conjunction with a Shack-Hartmann wave-front sensor and zonal control algorithm,” Opt. Exp. 1, 338–346 (1977).
[CrossRef]

Sharples, R. M.

J. Gourlay, G. D. Love, P. M. Birch, R. M. Sharples, A. Purvis, “A real-time closed loop liquid crystal adaptive optics system: first results,” Opt. Commun. 137, 17–21 (1997).
[CrossRef]

Shmal’gauzen, V. I.

A. A. Vasil’ev, A. F. Naumov, V. I. Shmal’gauzen, “Wavefront correction by liquid-crystal devices,” Sov. J. Quantum Electron. 16, 471–474 (1986).
[CrossRef]

Stefanini, P.

D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, V. Biliotti, “Adaptive optics wave front corrector using addressable liquid crystal retarders,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. SPIE1543, 133–143 (1991).
[CrossRef]

Vasil’ev, A. A.

A. A. Vasil’ev, A. F. Naumov, V. I. Shmal’gauzen, “Wavefront correction by liquid-crystal devices,” Sov. J. Quantum Electron. 16, 471–474 (1986).
[CrossRef]

Vasil’iev, A. A.

A. A. Vasil’iev, I. N. Kompanets, A. V. Parfenov, “Progress in the development and applications of optically controlled liquid crystal spatial light modulators,” Sov. J. Quantum Electron. 13, 689–695 (1983).
[CrossRef]

Voelz, D.

Vorontsov, M.

Vuilleumier, R.

Wang, J.

J. Wang, J. Markey, “Modal compensation of atmospheric turbulence phase distortions,” J. Opt. Soc. Am. A 10, 78–87 (1978).

Wild, W.

Appl. Opt.

J. Opt. Soc. Am.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
[CrossRef]

F. Roddier, M. Northcott, J. Graves, D. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. 10, 957–965 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J. Gourlay, G. D. Love, P. M. Birch, R. M. Sharples, A. Purvis, “A real-time closed loop liquid crystal adaptive optics system: first results,” Opt. Commun. 137, 17–21 (1997).
[CrossRef]

A. Kudryashov, J. Gonglewski, S. Browne, R. Highland, “Liquid crystal phase modulator for adaptive optics. Temporal performance characterization,” Opt. Commun. 141, 247–253 (1997).
[CrossRef]

Opt. Exp.

D. Dayton, S. Sandven, J. Gonglewski, “Adaptive optics using a liquid crystal phase modulator in conjunction with a Shack-Hartmann wave-front sensor and zonal control algorithm,” Opt. Exp. 1, 338–346 (1977).
[CrossRef]

Opt. Lett.

Sov. J. Quantum Electron.

A. A. Vasil’iev, I. N. Kompanets, A. V. Parfenov, “Progress in the development and applications of optically controlled liquid crystal spatial light modulators,” Sov. J. Quantum Electron. 13, 689–695 (1983).
[CrossRef]

A. A. Vasil’ev, A. F. Naumov, V. I. Shmal’gauzen, “Wavefront correction by liquid-crystal devices,” Sov. J. Quantum Electron. 16, 471–474 (1986).
[CrossRef]

Other

S. Restaino, M. Anderson, J. Baker, D. Payne, G. Loos, R. Carreras, “Progress report of USAF Research Laboratory liquid crystal AO program,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353 (1998).
[CrossRef]

J. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985), p. 389.

A. Kolmogorov, “Dissipation of energy in locally isotropic turbulence,” in Turbulence, Classic Papers on Statistical Theory, S. Friedlander, L. Topper, eds., (Wiley-Interscience, New York, 1961).

D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, V. Biliotti, “Adaptive optics wave front corrector using addressable liquid crystal retarders,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. SPIE1543, 133–143 (1991).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Geometry of the 127-element Meadowlark LCPM. The small circles represent actuator centers for modeling purposes.

Fig. 2
Fig. 2

Fractional error in representing Zernike polynomials on a 127-element LCPM.

Fig. 3
Fig. 3

Phase–shifting interferometer measurements of phase for three Zernike polynomials.

Fig. 4
Fig. 4

Residual mean-square error. (a) Normalized mean-square wave-front contribution for each Zernike mode. (b) Normalized residual error for uncorrected modes >p. Both curves are in units of radians squared and scaled by (D/ r 0)5/3.

Fig. 5
Fig. 5

Structure functions calculated for 36 pixelated and unpixelated Zernikes.

Fig. 6
Fig. 6

Experimental and calculated far-field patterns for several different levels of D/ r 0.

Fig. 7
Fig. 7

Far-field intensity patterns for time-correlated turbulence.

Fig. 8
Fig. 8

Power spectrum of Zernike 4 for the unpixelated sequence of phase screens and their pixelated approximations.

Fig. 9
Fig. 9

Optical layout for the LCPM as a disturbance source to test adaptive optics systems.

Fig. 10
Fig. 10

Example of open- and closed-loop focal plane images by use of the LCPM as a disturbance source, D/ r 0) = 6.

Fig. 11
Fig. 11

Residual MTF curves for 36 Zernike mode corrections: (a) pixelated MTF and (b) unpixelated MTF.

Fig. 12
Fig. 12

Hexagonal and square lenslet configurations.

Fig. 13
Fig. 13

Block diagram of the closed-loop control system by use of the interpolated control matrix.

Fig. 14
Fig. 14

Static Zernike mode correction with square lenslet configuration.

Fig. 15
Fig. 15

Spatial light modulator correction with square wave-front sensor; input disturbance, D/ r 0 = 4.8.

Equations (26)

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Φ κ = 0.033 C n 2 κ - 11 / 3 .
ϕ ρ = p   a p Z p ρ .
  d ρ Z p ρ Z p ρ = δ pq
a p =   d ρ Z p ρ ϕ ρ .
ϕ ˜ ρ = k   W k ρ ϕ ρ k ,
W k = 1   if   ρ   lies   within   element   k 0   otherwise ,
ã p =   d ρ Z p ρ ϕ ˜ ρ =   d ρ Z p ρ k   W k ρ ϕ ρ k = k     d ρ Z p ρ W k ρ p   a q Z q ρ k = q   a q k     d ρ W k ρ Z p ρ Z q ρ k .
k     d ρ W k ρ Z p ρ k Z q ρ k δ pq ,
k     d ρ W k ρ Z p ρ Z q ρ k δ pq .
err p = a p - k     d ρ W k ρ Z p ρ Z q ρ k a p .
C pp = 0.15337 - 1 n + n - 2 m / 2 × n + 1 n + 1 1 / 2 Γ 14 / 3 Γ n + n - 5 / 3 / 2 × Γ n - n + 17 / 3 / 2 Γ n + n + 23 / 3 / 2 × Γ n - n + 17 / 3 / 2 - 1 × δ oe , oe δ m , m ,
ϕ r = 4 36   a p Z p r .
σ p 2 = C pp D / r 0 5 / 3 ,
D ϕ ˜ r ¯ ,   ρ ¯ = ϕ ˜ r ¯ - 1 2   ρ ¯ - ϕ ˜ r ¯ - 1 2   ρ ¯ 2 ,
ϕ ˜ 36 ρ ¯ = k   W k ρ ¯ p = 2 36   a p Z p ρ ¯ k .
r 0 = 2.914 / 6.88 k 2 sec   ψ     d hC n 2 h - 3 / 5 , f G = 0.255 k 2 sec   ψ     d hC n 2 h | V h | 5 / 3 3 / 5 .
v = 2.34 f G r 0
Φ ω     ω / v - 17 / 3 .
MTF ρ ¯ = 4 D 2 exp - D ρ ¯ 2 0 π / 2 d θ   0 R   r d r × exp Q ρ ¯ ;   r 2 + S ρ ¯ ;   r ,
D ρ = 6.88 ρ D r 0 5 / 3 ,
Q ρ ¯ ;   r = i = 2 N j = 2 N   a i a j Z i r + ρ ¯ / 2 - Z i r - ρ ¯ / 2 × Zj r + ρ ¯ / 2 - Zj r - ρ ¯ / 2 ,
S ρ ¯ ;   r = i = 2 N j = N + 1   a i a j Z i r + ρ ¯ / 2 - Z i r - ρ ¯ / 2 × Zj r + ρ ¯ / 2 - Zj r - ρ ¯ / 2 ,
Φ hex = Z hex Z sq - 1 H ϕ x 11 ϕ x 12 ϕ y 11 ϕ y 12 .
H = Z hex Z sq - 1 H .
C k = C k - 1 - gH ϕ - ϕ 0 .
S exp - σ ϕ 2 .

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