Abstract

A new adaptive wave-front control technique and system architectures that offer fast adaptation convergence even for high-resolution adaptive optics is described. This technique is referred to as decoupled stochastic parallel gradient descent (D-SPGD). D-SPGD is based on stochastic parallel gradient descent optimization of performance metrics that depend on wave-front sensor data. The fast convergence rate is achieved through partial decoupling of the adaptive system’s control channels by incorporating spatially distributed information from a wave-front sensor into the model-free optimization technique. D-SPGD wave-front phase control can be applied to a general class of adaptive optical systems. The efficiency of this approach is analyzed numerically by considering compensation of atmospheric-turbulence-induced phase distortions with use of both low-resolution (127 control channels) and high-resolution (256×256 control channels) adaptive systems. Results demonstrate that phase distortion compensation can be achieved during only 10–20 iterations. The efficiency of adaptive wave-front correction with D-SPGD is practically independent of system resolution.

© 2002 Optical Society of America

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References

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  1. Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, New York, 1999), pp. 91–130.
  2. R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).
  3. M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  4. 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]
  5. 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]
  6. T. G. Bifano, J. Perrault, R. Krishnamoorthy Mali, M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
    [CrossRef]
  7. M. K. Lee, W. D. Cowan, B. M. Welsh, V. M. Bright, M. C. Roggemann, “Aberration-correction results from a segmented microelectromechanical deformable mirror and refractive lenslet array,” Opt. Lett. 23, 645–647 (1998).
    [CrossRef]
  8. S. Serati, G. Sharp, R. Serati, D. McKnight, J. Stookley, “128×128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T. Chao, eds., Proc. SPIE2490, 378–387 (1995).
    [CrossRef]
  9. P. Madec, “Control Techniques,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, New York, 1999), pp. 131–154.
  10. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  11. M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, D. G. Voelz, “Adaptive imaging system for phase-distorted extended source/multiple distance objects,” Appl. Opt. 36, 3319–3328 (1997).
    [CrossRef] [PubMed]
  12. M. A. Vorontsov, G. W. Carhart, M. Cohen, G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
    [CrossRef]
  13. T. R. O’Meara, “The multi-dither principle in adaptive optics,” J. Opt. Soc. Am. 67, 306–315 (1977).
    [CrossRef]
  14. M. A. Vorontsov, V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15, 2745–2758 (1998).
    [CrossRef]
  15. J. C. Spall, “A stochastic approximation technique for generating maximum likelihood parameter estimates,” in Proceedings of the American Control Conference (IEEE Press, Piscataway, N.J., 1987), pp. 1161–1167.
  16. J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Autom. Control 37, 332–341 (1992).
    [CrossRef]
  17. G. Cauwenberghs, “A fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems (Morgan Kaufman, Los Altos, Calif., 1993), Vol. 5, pp. 244–251.
  18. M. A. Vorontsov, G. W. Carhart, J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
    [CrossRef] [PubMed]
  19. W. H. Southwell, “Wave-front estimation from slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  20. B. L. Ellerbroek, C. Van Loan, N. P. Pitsianis, R. J. Plemmons, “Optimizing closed-loop adaptive-optics performance with use of multiple control bandwidths,” J. Opt. Soc. Am. A 11, 2871–2886 (1994).
    [CrossRef]
  21. R. A. Muller, A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 64, 1200–1210 (1974).
    [CrossRef]
  22. R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on SiliconG. Cauwenberghs, Magdy A. Bayoumi, eds. (Kluwer Academic, Boston, Mass., 1999), Chap. 1, pp. 359–382.
  23. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
    [CrossRef] [PubMed]
  24. R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
    [CrossRef]
  25. M. A. Vorontsov, E. W. Justh, L. A. Beresnev, “Adaptive optics with advanced phase-contrast technique. I High-resolution wave-front sensing,” J. Opt. Soc. Am. A 18, 1289–1299 (2001).
    [CrossRef]
  26. 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]
  27. D. L. Fried, “Statistics of a geometric representation of wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]

2001 (1)

2000 (1)

1999 (2)

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]

T. G. Bifano, J. Perrault, R. Krishnamoorthy Mali, M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[CrossRef]

1998 (3)

1997 (2)

1994 (1)

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]

J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Autom. Control 37, 332–341 (1992).
[CrossRef]

1980 (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)

1975 (1)

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

1974 (1)

1965 (1)

1955 (1)

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

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]

Beresnev, L. A.

Bifano, T. G.

T. G. Bifano, J. Perrault, R. Krishnamoorthy Mali, M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[CrossRef]

Bright, V. M.

Buffington, A.

Carhart, G. W.

Cauwenberghs, G.

M. A. Vorontsov, G. W. Carhart, M. Cohen, G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
[CrossRef]

G. Cauwenberghs, “A fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems (Morgan Kaufman, Los Altos, Calif., 1993), Vol. 5, pp. 244–251.

R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on SiliconG. Cauwenberghs, Magdy A. Bayoumi, eds. (Kluwer Academic, Boston, Mass., 1999), Chap. 1, pp. 359–382.

Cohen, M.

M. A. Vorontsov, G. W. Carhart, M. Cohen, G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
[CrossRef]

R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on SiliconG. Cauwenberghs, Magdy A. Bayoumi, eds. (Kluwer Academic, Boston, Mass., 1999), Chap. 1, pp. 359–382.

Cowan, W. D.

Edwards, R. T.

R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on SiliconG. Cauwenberghs, Magdy A. Bayoumi, eds. (Kluwer Academic, Boston, Mass., 1999), Chap. 1, pp. 359–382.

Ellerbroek, B. L.

Fried, D. L.

Hardy, J. W.

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

Horenstein, M. N.

T. G. Bifano, J. Perrault, R. Krishnamoorthy Mali, M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[CrossRef]

Justh, E. W.

Krishnamoorthy Mali, R.

T. G. Bifano, J. Perrault, R. Krishnamoorthy Mali, M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[CrossRef]

Lee, M. K.

Madec, P.

P. Madec, “Control Techniques,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, New York, 1999), pp. 131–154.

McKnight, D.

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

Muller, R. A.

O’Meara, T. R.

Perrault, J.

T. G. Bifano, J. Perrault, R. Krishnamoorthy Mali, M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[CrossRef]

Pitsianis, N. P.

Plemmons, R. J.

Pruidze, D. V.

Ricklin, J. C.

Roggemann, M. C.

Serati, R.

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

Serati, S.

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

Sharp, G.

S. Serati, G. Sharp, R. Serati, D. McKnight, J. Stookley, “128×128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T. 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]

Southwell, W. H.

Spall, J. C.

J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Autom. Control 37, 332–341 (1992).
[CrossRef]

J. C. Spall, “A stochastic approximation technique for generating maximum likelihood parameter estimates,” in Proceedings of the American Control Conference (IEEE Press, Piscataway, N.J., 1987), pp. 1161–1167.

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]

Stookley, J.

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

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).

Van Loan, C.

Voelz, D. G.

Vorontsov, M. A.

M. A. Vorontsov, E. W. Justh, L. A. Beresnev, “Adaptive optics with advanced phase-contrast technique. I High-resolution wave-front sensing,” J. Opt. Soc. Am. A 18, 1289–1299 (2001).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, M. Cohen, G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
[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]

M. A. Vorontsov, V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15, 2745–2758 (1998).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, D. G. Voelz, “Adaptive imaging system for phase-distorted extended source/multiple distance objects,” Appl. Opt. 36, 3319–3328 (1997).
[CrossRef] [PubMed]

M. A. Vorontsov, G. W. Carhart, J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
[CrossRef] [PubMed]

R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on SiliconG. Cauwenberghs, Magdy A. Bayoumi, eds. (Kluwer Academic, Boston, Mass., 1999), Chap. 1, pp. 359–382.

Welsh, B. M.

Zernike, F.

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

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

T. G. Bifano, J. Perrault, R. Krishnamoorthy Mali, M. N. Horenstein, “Microelectromechanical deformable mirrors,” IEEE J. Sel. Top. Quantum Electron. 5, 83–89 (1999).
[CrossRef]

IEEE Trans. Autom. Control (1)

J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Autom. Control 37, 332–341 (1992).
[CrossRef]

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 (6)

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]

Opt. Lett. (2)

Proc. IEEE (1)

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

Science (1)

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

Other (8)

R. T. Edwards, M. Cohen, G. Cauwenberghs, M. A. Vorontsov, G. W. Carhart, “Analog VLSI parallel stochastic optimization for adaptive optics,” in Learning on SiliconG. Cauwenberghs, Magdy A. Bayoumi, eds. (Kluwer Academic, Boston, Mass., 1999), Chap. 1, pp. 359–382.

G. Cauwenberghs, “A fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems (Morgan Kaufman, Los Altos, Calif., 1993), Vol. 5, pp. 244–251.

J. C. Spall, “A stochastic approximation technique for generating maximum likelihood parameter estimates,” in Proceedings of the American Control Conference (IEEE Press, Piscataway, N.J., 1987), pp. 1161–1167.

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

P. Madec, “Control Techniques,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, New York, 1999), pp. 131–154.

Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, New York, 1999), pp. 91–130.

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

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Figures (11)

Fig. 1
Fig. 1

Schematics for (a) wave-front phase conjugation and (b) model-free optimization adaptive system types.

Fig. 2
Fig. 2

D-SPGD adaptive system architectures: (a) general schematic, (b) adaptive system with J3 controller (16a). The geometry of the matched wave-front corrector and 127 subaperture sensor is shown at bottom right.

Fig. 3
Fig. 3

Phase contrast wave-front sensor schematic.

Fig. 4
Fig. 4

Input-wave phase distortion/perturbation realizations: (a) atmosphericlike phase aberration pattern superimposed with hexagonal grid of wave-front corrector/sensor subapertures, (b) corrector friendly aberrations φh(r) corresponding to φ(r), (c) phase perturbation δu(r) (Gaussian spectrum), (d) corrector-friendly wave-front perturbation δuh(r) corresponding to δu(r).

Fig. 5
Fig. 5

Simulation results for the low-resolution D-SPGD adaptive system with interferometric wave-front sensor: (a) for corrector friendly aberrations φh(r), (b) for atmospheric-like phase distortions φ(r). Averaged Strehl ratio adaptation curves 1 and 2 and averaged metric curves 4 and 5 corresponding to the D-SPGD algorithms (15a) (J2 curves) and (16a) (J3 curves): 1 and 4 for controller (15a) and 2 and 5 for (16a). The averaged adaptation curves 3 correspond to the conventional SPGD controller (2a). At the bottom-right corner are shown residual phase patterns: (a) δh(r), (b) δ(r). The patterns’ gray-scale dynamical range is 4π rad.

Fig. 6
Fig. 6

Averaged Strehl ratio versus iteration number for the low-resolution D-SPGD adaptive system with different wave-front sensor types: (a) for corrector friendly, (b) for atmosphericlike phase distortions. The adaptation curves 1–5 correspond to 1, interferometer with reference wave; 2 and 5, ZF; 3 and 4, PDI. Labels in parentheses correspond to wave-front sensor type (I is interferometer, etc.) and controller [J2 for (15a) and J3 for (16a)]. The curves 6 correspond to the SPGD controller. Gray-scale images show residual phase patterns corresponding to the D-SPGD system with PDI (dynamical range is 4π rad.).

Fig. 7
Fig. 7

The averaged Strehl ratios StM achieved after M iterations of the adaptation process versus input phase standard deviation σin for the low-resolution D-SPGD adaptive system with PDI and control algorithm (16a) (J3 controller). Numbers in parentheses correspond to Strehl ratio values for σin.

Fig. 8
Fig. 8

Adaptive system with “mismatched” wave-front corrector and sensor: (a) sensor friendly aberration φg(r) superimposed with a hexagonal grid of wave-front sensor subapertures for w=d, (b) pattern of the residual phase aberration δg(r), (c) sensor output intensity Iδ(r); both (b) and (c) are after 60 iterations of the D-SPGD controller (16a); (d) the same as (b) for w=1.2d.

Fig. 9
Fig. 9

Averaged Strehl ratio adaptation curves for corrector friendly aberration compensation with the D-SPGD system and “mismatched” wave-front corrector and sensor. Curves are labeled by the mismatch parameter w/d. Evolution curves for the SPGD controller are indicated by dots. Gray-scale images correspond to the intensity patterns Iδ(r) after 60 iterations: left, for w/d=0.8; right, for w/d=1.2.

Fig. 10
Fig. 10

High-resolution D-SPGD adaptive system adaptation process efficiency for uniform (solid curves) and random (dashed curves) input wave intensity distributions: (a) averaged Strehl ratio 〈St〉 versus iteration number, (b) averaged Strehl ratio achieved after M iterations of the adaptation process StM versus input phase standard deviation σin. Adaptation curves 1–5 correspond to 1, interferometer with reference wave; 2 and 5, ZF; 3 and 4, PDI. Labels in parentheses are the same as in Fig. 5. The standard deviation for intensity scintillations in (a) was σ1=0.6. The D-SPGD system in (b) corresponds to the J3 controller with PDI. A and B correspond to gray-scale images of the input wave intensity patterns in (a) for point σin=1.7 (A) and (b) for point σin=3.4 (B).

Fig. 11
Fig. 11

Averaged Strehl ratio evolution curves for the D-SPGD controller with PDI, perturbations with Gaussian power spectrum, and different values of the correlation radius lp (curve 1 and 2) and for “mixed” perturbations (curve 3): 1, lp=0.07D; 2, lp=0.035D; 3, κ=0.6, lp=0.035D. Gray-scale images represent mixed perturbation patterns δu(r) at n=2 (left) and n=30 (right).

Equations (38)

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

J1s=Iδ2(r, t)d2r.
dul(t)dt=-γJ^l(t),
ul(n+1)=ul(n)-γJ^l(n)(l=1 ,, N)
u(n+1)(r)=u(n)(r)-γδJ(n)δu(n)(r),n=0,1 ,,
 
{J^l(t)}k=1NJ^luk ξk(t)k=1NAk,lξk(t)
dξl(t)dt=-γk=1NAk,lξk(t),
J(t)=l=1Mcljl(t),jl(t)=j(r, t)Zl(r)d2r.
J(t)=l=1Ncl jl(t),jl(t)=Ωlj(r, t)d2r.
J^l=clδulδjl(t)+klNckδulδjk(t).
ul(n+1)=ul(n)-γclδulδjl(n)-γklNckδulδjk(n),
(l=1 ,, N).
ul(n+1)=ul(n)-γclδjl(n)δul,(l=1 ,, N).
klNckδulδjk|clδulδjl|foralll=1 ,, N.
u(n+1)(r)=u(n)(r)-γδj(n)(r)δu(n)(r),n=0, 1 ,.
δjl=I j(r)uIδ(r)Zl(r)δu(r)d2r=k=1Nak,lδuk,
ak,l=I j(r)uIδ(r)Sk(r)Zl(r)d2r.
ul(n+1)=ul(n)-γclk=1Nak,jδulδuk-γk=1NnlNcnak,nδulδuk,(l=1 ,, N).
δJδul=(δul)2clal,l-klNckal,k,
(l=1 ,, N).
klNal,k  |al,l|,(l=1 ,, N).
J1=l=1Njl(1)l=1NΩlIδ2(r)d2r;
J2=l=1Njl(2)l=1N(I¯l)2,I¯l=ΩlIδ(r)d2r,
J3=l=1Njl(3)l=1NI¯l.
ul(n+1)=ul(n)-γI¯l(n)δI¯l(n)δul,(l=1 ,, N),
u(n+1)(r)=u(n)(r)-γIδ(n)(r)δIδ(n)(r)δu(n)(r),
(n=0,1 ,,).
ul(n+1)=ul(n)-γδI¯l(n)δul,(l=1 ,, N).
u(n+1)(r)=u(n)(r)-γδIδ(n)(r)δu(n)(r),
(n=0,1 ,,).
τ dul(t)dt=-γαl(t)bl(t).
Iδ(r, t)=I0(r)+Iin(r)+2μ(r)cos[δ(r, t)+Δ],
Iδ(r)=Iin(r)+2(2πF)2IF(0)-4πFIin1/2(r)IF1/2(0)×{cos[δ(r)-Δ]-sin[δ(r)-Δ]}
 Iδ(r)=Iin(r)+(2πF)2IF(0)-4πFIin1/2(r)IF1/2(0)cos[δ(r)-Δ]
IF(q=0)= Iin1/2(r)exp[iδ(r)]d2r2,
Δ=argIin1/2(r)exp[iδ(r)]d2r.
GA(q)=2π0.033(1.68/r0)5/3(q2+qA2)-11/6exp(-q2/qa2)× [1+1.802(q/qa)-0.254(q/qa)7/6].
σI2=[|A0(r)|2-I¯]2d2r/I¯2,

Metrics