Abstract

Coherent optical adaptive techniques (COAT) offer promise for overcoming the deleterious effects of phase distortions experienced by optical beams propagating through distorting optics or via a turbulent and absorbing atmosphere. The theory of four classes of such systems, which employ similar multidither principles, is explored. Many modes of operating these systems are briefly reviewed and a detailed analysis of the most widely employed—a glint referencing system with sinusoidal dithers—is developed. A servo signal-to-noise analysis indicates how the optimum choice of dither magnitude depends on the system noise.

© 1977 Optical Society of America

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  1. W. T. Cathey, C. L. Hayes, and W. C. Davis, Appl. Opt. 9, 701 (1970).
    [Crossref] [PubMed]
  2. T. R. O’Meara, W. P. Brown, W. B. Bridges, T. A. Nussmeier, A. D. Jacobson, J. F. Lotspeich, G. M. Janney, and J. Y. Wada, “Coherent Optical Adaptive Techniques,” Hughes Research Laboratories Tech. Rep. RADC-TR (Dec.1970) (unpublished).
  3. R. P. Futrelle, G. E. Meyers, and C. L. Hayes, “Coherent Optical Adaptive Techniques,” Contract F30602-70-C-0298 (April1971) (unpublished).
  4. T. R. O’Meara, U.S. Patent3, 727, 223, “Adaptive Power Redistribution Systems” (April10, 1973) (U. S. Patent Office, Washington, D. C.).
  5. T. R. O’Meara, U.S. Patent3, 731, 103, “Adaptive Arrays” (May1 ,1973) (unpublished).
  6. T. R. O’Meara, U.S. Patent3, 764, 213, “Return-Wave, Phase Controlled Adaptive Array” (October9, 1973) (U. S. Patent Office, Washington, D. C.).
  7. W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, Appl. Opt. 13, 291 (1974).
    [Crossref] [PubMed]
  8. W. B. Bridges, S. Hansen, L. A. Horwitz, S. P. Lazzara, T. R. O’Meara, J. E. Pearson, and T. J. Walsh, J. Opt. Soc. Am. 64, 541 (1975).
  9. J. E. Pearson, W. B. Bridges, L. S. Horwitz, T. J. Walsh, and R. F. Ogrodnik, Paper ThB5, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.
  10. W. B. Bridges and J. E. Pearson, Appl. Phys. Lett. 26, 539 (1975).
    [Crossref]
  11. J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 15, 611 (1976).
    [Crossref] [PubMed]
  12. T. R. O’Meara, “Applications of Laser Radars as Assistors to High-Power Optical Systems (U),” Proceedings of the Sixth DoD Conference on Laser Technology, U.S. Air Force Academy, March 26, 1974, Vol. 1, p. 233.
  13. H. W. Babcock, J. Opt. Soc. Am. 48, 500 (1958).
    [Crossref]
  14. R. A. Muller and A. Buffington, J. Opt. Soc. Am. 64, 1200 (1974).
    [Crossref]
  15. J. Hardy, J. Feinleib, and J. C. Wyant, Paper ThB1, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.
  16. L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Paper ThB2, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.
  17. J. C. Wyant, Appl. Opt. 13, 200 (1974).
    [Crossref] [PubMed]
  18. J. F. Ebersole and J. C. Wyant, Appl. Opt. 13, 1004 (1974).
    [Crossref] [PubMed]
  19. Contracts F30602-73-C-0248 and F30602-75-C-0001, monitored through Rome Air Development Center. See , July1973; , Oct.1973; , Jan.1974; , Apr.1974; , Jan.1975; , Feb.1975; , May1975. All available from Defense Documentation Center.
  20. M. J. Lavan, W. K. Cadwallender, and T. F. DeYoung, “A Visible Wavelength COAT Array,” Opt. Eng. 15, 56 (1976).
    [Crossref]
  21. Insofar as can be determined by us, the first use of the acronym was in the context “coherent optical array techniques,” and the original motivation was for the cophasing of multiple low-power laser oscillators. There has been a tendency in recent years to employ a one-to-one correspondence between specific proprietary adaptive optical systems and acronyms. However, we have always employed the term COAT in the larger or generic sense.
  22. For long-range operation, the phase of the detected signal will be delayed by the round-trip propagation time, and the reference signal must be similarly delayed.
  23. All correctly operating systems ideally introduce the phase conjugate of path distortions. We reserve the term (after microwave terminology) to refer to return-wave systems with heterodyne extraction of phase error information.
  24. T. R. O’Meara, “Stability of an N-loop ensemble-reference phase control system,” J. Opt. Soc. Am. 67, 315–318 (1977) (following paper).
    [Crossref]
  25. The piston elements may be one-dimensional or two-dimensional, commonly square, circular, or hexagonal.
  26. Any tilt component of ϕen which is common to all elements is assumed to be absorbed in a translation of x′.

1977 (1)

1976 (2)

M. J. Lavan, W. K. Cadwallender, and T. F. DeYoung, “A Visible Wavelength COAT Array,” Opt. Eng. 15, 56 (1976).
[Crossref]

J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 15, 611 (1976).
[Crossref] [PubMed]

1975 (2)

W. B. Bridges, S. Hansen, L. A. Horwitz, S. P. Lazzara, T. R. O’Meara, J. E. Pearson, and T. J. Walsh, J. Opt. Soc. Am. 64, 541 (1975).

W. B. Bridges and J. E. Pearson, Appl. Phys. Lett. 26, 539 (1975).
[Crossref]

1974 (4)

1970 (1)

1958 (1)

Babcock, H. W.

Bridges, W. B.

J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 15, 611 (1976).
[Crossref] [PubMed]

W. B. Bridges, S. Hansen, L. A. Horwitz, S. P. Lazzara, T. R. O’Meara, J. E. Pearson, and T. J. Walsh, J. Opt. Soc. Am. 64, 541 (1975).

W. B. Bridges and J. E. Pearson, Appl. Phys. Lett. 26, 539 (1975).
[Crossref]

W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, Appl. Opt. 13, 291 (1974).
[Crossref] [PubMed]

J. E. Pearson, W. B. Bridges, L. S. Horwitz, T. J. Walsh, and R. F. Ogrodnik, Paper ThB5, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

T. R. O’Meara, W. P. Brown, W. B. Bridges, T. A. Nussmeier, A. D. Jacobson, J. F. Lotspeich, G. M. Janney, and J. Y. Wada, “Coherent Optical Adaptive Techniques,” Hughes Research Laboratories Tech. Rep. RADC-TR (Dec.1970) (unpublished).

Brown, W. P.

W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, Appl. Opt. 13, 291 (1974).
[Crossref] [PubMed]

T. R. O’Meara, W. P. Brown, W. B. Bridges, T. A. Nussmeier, A. D. Jacobson, J. F. Lotspeich, G. M. Janney, and J. Y. Wada, “Coherent Optical Adaptive Techniques,” Hughes Research Laboratories Tech. Rep. RADC-TR (Dec.1970) (unpublished).

L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Paper ThB2, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Brunner, P. T.

Buffington, A.

Cadwallender, W. K.

M. J. Lavan, W. K. Cadwallender, and T. F. DeYoung, “A Visible Wavelength COAT Array,” Opt. Eng. 15, 56 (1976).
[Crossref]

Cathey, W. T.

Davis, W. C.

DeYoung, T. F.

M. J. Lavan, W. K. Cadwallender, and T. F. DeYoung, “A Visible Wavelength COAT Array,” Opt. Eng. 15, 56 (1976).
[Crossref]

Ebersole, J. F.

Feinleib, J.

J. Hardy, J. Feinleib, and J. C. Wyant, Paper ThB1, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Futrelle, R. P.

R. P. Futrelle, G. E. Meyers, and C. L. Hayes, “Coherent Optical Adaptive Techniques,” Contract F30602-70-C-0298 (April1971) (unpublished).

Hansen, S.

J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 15, 611 (1976).
[Crossref] [PubMed]

W. B. Bridges, S. Hansen, L. A. Horwitz, S. P. Lazzara, T. R. O’Meara, J. E. Pearson, and T. J. Walsh, J. Opt. Soc. Am. 64, 541 (1975).

Hardy, J.

J. Hardy, J. Feinleib, and J. C. Wyant, Paper ThB1, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Hayes, C. L.

W. T. Cathey, C. L. Hayes, and W. C. Davis, Appl. Opt. 9, 701 (1970).
[Crossref] [PubMed]

R. P. Futrelle, G. E. Meyers, and C. L. Hayes, “Coherent Optical Adaptive Techniques,” Contract F30602-70-C-0298 (April1971) (unpublished).

Horwitz, L. A.

W. B. Bridges, S. Hansen, L. A. Horwitz, S. P. Lazzara, T. R. O’Meara, J. E. Pearson, and T. J. Walsh, J. Opt. Soc. Am. 64, 541 (1975).

Horwitz, L. S.

J. E. Pearson, W. B. Bridges, L. S. Horwitz, T. J. Walsh, and R. F. Ogrodnik, Paper ThB5, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Jacobson, A. D.

T. R. O’Meara, W. P. Brown, W. B. Bridges, T. A. Nussmeier, A. D. Jacobson, J. F. Lotspeich, G. M. Janney, and J. Y. Wada, “Coherent Optical Adaptive Techniques,” Hughes Research Laboratories Tech. Rep. RADC-TR (Dec.1970) (unpublished).

Janney, G. M.

T. R. O’Meara, W. P. Brown, W. B. Bridges, T. A. Nussmeier, A. D. Jacobson, J. F. Lotspeich, G. M. Janney, and J. Y. Wada, “Coherent Optical Adaptive Techniques,” Hughes Research Laboratories Tech. Rep. RADC-TR (Dec.1970) (unpublished).

Jenney, J. A.

L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Paper ThB2, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Lavan, M. J.

M. J. Lavan, W. K. Cadwallender, and T. F. DeYoung, “A Visible Wavelength COAT Array,” Opt. Eng. 15, 56 (1976).
[Crossref]

Lazzara, S. P.

W. B. Bridges, S. Hansen, L. A. Horwitz, S. P. Lazzara, T. R. O’Meara, J. E. Pearson, and T. J. Walsh, J. Opt. Soc. Am. 64, 541 (1975).

W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, Appl. Opt. 13, 291 (1974).
[Crossref] [PubMed]

Lotspeich, J. F.

T. R. O’Meara, W. P. Brown, W. B. Bridges, T. A. Nussmeier, A. D. Jacobson, J. F. Lotspeich, G. M. Janney, and J. Y. Wada, “Coherent Optical Adaptive Techniques,” Hughes Research Laboratories Tech. Rep. RADC-TR (Dec.1970) (unpublished).

Meyers, G. E.

R. P. Futrelle, G. E. Meyers, and C. L. Hayes, “Coherent Optical Adaptive Techniques,” Contract F30602-70-C-0298 (April1971) (unpublished).

Miller, L.

L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Paper ThB2, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Muller, R. A.

Nussmeier, T. A.

J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 15, 611 (1976).
[Crossref] [PubMed]

W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, Appl. Opt. 13, 291 (1974).
[Crossref] [PubMed]

T. R. O’Meara, W. P. Brown, W. B. Bridges, T. A. Nussmeier, A. D. Jacobson, J. F. Lotspeich, G. M. Janney, and J. Y. Wada, “Coherent Optical Adaptive Techniques,” Hughes Research Laboratories Tech. Rep. RADC-TR (Dec.1970) (unpublished).

O’Meara, T. R.

T. R. O’Meara, “Stability of an N-loop ensemble-reference phase control system,” J. Opt. Soc. Am. 67, 315–318 (1977) (following paper).
[Crossref]

W. B. Bridges, S. Hansen, L. A. Horwitz, S. P. Lazzara, T. R. O’Meara, J. E. Pearson, and T. J. Walsh, J. Opt. Soc. Am. 64, 541 (1975).

W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, Appl. Opt. 13, 291 (1974).
[Crossref] [PubMed]

T. R. O’Meara, W. P. Brown, W. B. Bridges, T. A. Nussmeier, A. D. Jacobson, J. F. Lotspeich, G. M. Janney, and J. Y. Wada, “Coherent Optical Adaptive Techniques,” Hughes Research Laboratories Tech. Rep. RADC-TR (Dec.1970) (unpublished).

T. R. O’Meara, U.S. Patent3, 727, 223, “Adaptive Power Redistribution Systems” (April10, 1973) (U. S. Patent Office, Washington, D. C.).

T. R. O’Meara, U.S. Patent3, 731, 103, “Adaptive Arrays” (May1 ,1973) (unpublished).

T. R. O’Meara, U.S. Patent3, 764, 213, “Return-Wave, Phase Controlled Adaptive Array” (October9, 1973) (U. S. Patent Office, Washington, D. C.).

T. R. O’Meara, “Applications of Laser Radars as Assistors to High-Power Optical Systems (U),” Proceedings of the Sixth DoD Conference on Laser Technology, U.S. Air Force Academy, March 26, 1974, Vol. 1, p. 233.

L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Paper ThB2, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Ogrodnik, R. F.

J. E. Pearson, W. B. Bridges, L. S. Horwitz, T. J. Walsh, and R. F. Ogrodnik, Paper ThB5, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Pearson, J. E.

J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, Appl. Opt. 15, 611 (1976).
[Crossref] [PubMed]

W. B. Bridges and J. E. Pearson, Appl. Phys. Lett. 26, 539 (1975).
[Crossref]

W. B. Bridges, S. Hansen, L. A. Horwitz, S. P. Lazzara, T. R. O’Meara, J. E. Pearson, and T. J. Walsh, J. Opt. Soc. Am. 64, 541 (1975).

J. E. Pearson, W. B. Bridges, L. S. Horwitz, T. J. Walsh, and R. F. Ogrodnik, Paper ThB5, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Pedinoff, M. E.

Sanguinet, J. A.

Wada, J. Y.

T. R. O’Meara, W. P. Brown, W. B. Bridges, T. A. Nussmeier, A. D. Jacobson, J. F. Lotspeich, G. M. Janney, and J. Y. Wada, “Coherent Optical Adaptive Techniques,” Hughes Research Laboratories Tech. Rep. RADC-TR (Dec.1970) (unpublished).

Walsh, T. J.

W. B. Bridges, S. Hansen, L. A. Horwitz, S. P. Lazzara, T. R. O’Meara, J. E. Pearson, and T. J. Walsh, J. Opt. Soc. Am. 64, 541 (1975).

J. E. Pearson, W. B. Bridges, L. S. Horwitz, T. J. Walsh, and R. F. Ogrodnik, Paper ThB5, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Wyant, J. C.

J. C. Wyant, Appl. Opt. 13, 200 (1974).
[Crossref] [PubMed]

J. F. Ebersole and J. C. Wyant, Appl. Opt. 13, 1004 (1974).
[Crossref] [PubMed]

J. Hardy, J. Feinleib, and J. C. Wyant, Paper ThB1, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

W. B. Bridges and J. E. Pearson, Appl. Phys. Lett. 26, 539 (1975).
[Crossref]

J. Opt. Soc. Am. (4)

Opt. Eng. (1)

M. J. Lavan, W. K. Cadwallender, and T. F. DeYoung, “A Visible Wavelength COAT Array,” Opt. Eng. 15, 56 (1976).
[Crossref]

Other (15)

Insofar as can be determined by us, the first use of the acronym was in the context “coherent optical array techniques,” and the original motivation was for the cophasing of multiple low-power laser oscillators. There has been a tendency in recent years to employ a one-to-one correspondence between specific proprietary adaptive optical systems and acronyms. However, we have always employed the term COAT in the larger or generic sense.

For long-range operation, the phase of the detected signal will be delayed by the round-trip propagation time, and the reference signal must be similarly delayed.

All correctly operating systems ideally introduce the phase conjugate of path distortions. We reserve the term (after microwave terminology) to refer to return-wave systems with heterodyne extraction of phase error information.

J. Hardy, J. Feinleib, and J. C. Wyant, Paper ThB1, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Paper ThB2, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

J. E. Pearson, W. B. Bridges, L. S. Horwitz, T. J. Walsh, and R. F. Ogrodnik, Paper ThB5, Proceedings of the OSA Topical Meeting on Optical Propagation through Turbulence, Boulder, Colo., July 1974.

T. R. O’Meara, “Applications of Laser Radars as Assistors to High-Power Optical Systems (U),” Proceedings of the Sixth DoD Conference on Laser Technology, U.S. Air Force Academy, March 26, 1974, Vol. 1, p. 233.

T. R. O’Meara, W. P. Brown, W. B. Bridges, T. A. Nussmeier, A. D. Jacobson, J. F. Lotspeich, G. M. Janney, and J. Y. Wada, “Coherent Optical Adaptive Techniques,” Hughes Research Laboratories Tech. Rep. RADC-TR (Dec.1970) (unpublished).

R. P. Futrelle, G. E. Meyers, and C. L. Hayes, “Coherent Optical Adaptive Techniques,” Contract F30602-70-C-0298 (April1971) (unpublished).

T. R. O’Meara, U.S. Patent3, 727, 223, “Adaptive Power Redistribution Systems” (April10, 1973) (U. S. Patent Office, Washington, D. C.).

T. R. O’Meara, U.S. Patent3, 731, 103, “Adaptive Arrays” (May1 ,1973) (unpublished).

T. R. O’Meara, U.S. Patent3, 764, 213, “Return-Wave, Phase Controlled Adaptive Array” (October9, 1973) (U. S. Patent Office, Washington, D. C.).

The piston elements may be one-dimensional or two-dimensional, commonly square, circular, or hexagonal.

Any tilt component of ϕen which is common to all elements is assumed to be absorbed in a translation of x′.

Contracts F30602-73-C-0248 and F30602-75-C-0001, monitored through Rome Air Development Center. See , July1973; , Oct.1973; , Jan.1974; , Apr.1974; , Jan.1975; , Feb.1975; , May1975. All available from Defense Documentation Center.

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Figures (7)

FIG. 1
FIG. 1

Outgoing-wave multidither or outgoing-wave maximization system. In some applications it is advantageous to split the corrector and dither functions and apply these signals to two separate phase control devices, each tailored for its own function. Coherent (heterodyne) detection may be employed in these systems, but is not commonly required. (a) In the illustrated system, separate sinusoidal tagging frequencies are applied to each element distributed across the feed to a single telescope. (b) In this system variant, laser power amplifiers follow the phase shifter and each element is a separate telescope or beam director.

FIG. 2
FIG. 2

Two-element dithered COAT system. The sinusoidal phase perturbation, applied to the lower element, scans the far-field diffraction pattern back and forth on the glint. This scan generates a corresponding amplitude modulation in the energy scattered back to the receiver.

FIG. 3
FIG. 3

Four types of adaptive optical systems that may be implemented with multidither servo techniques. Diffraction gratings would most commonly be substituted for the beam splitters at high power levels. (a) Multidither outgoing-wave system transmitting optical power to a cooperative target, such as a satellite. (b) Multidither outgoing-wave system operating as an optical radar. (c) Local-loop multidither system that removes the wave-front (phase) distortions from an aberrated source. (d) Multidither return-wave system.

FIG. 4
FIG. 4

Typical control system details for a multidither servo system. A similar block diagram is employed for computer simulations. The clipper is not essential.

FIG. 5
FIG. 5

Operating modes of a multidither outgoint-wave system with piston mirror control. (a) COAT loop inoperative; (b) COAT in glint-tracking mode (glint on central canopy of a plastic model); (c) edge-tracking mode (no glint is present, but beam forms on light/dark boundary); (d) black hole-tracking mode—control signal phase is changed by 180° from (b).

FIG. 6
FIG. 6

Computer simulations of the convergence process in a piston mirror multidither servo system. The wave-front aberrations are piston in nature and fixed (time-invariant) in these runs. (a) The convergence process for an 18-element multidither servo system (S/N = ∞). The dithers are equally spaced and phase locked, which produces the periodic spiking after convergence. (b) The convergence process for an 18-element multidither servo system. Open-loop gain is near the optimum for best convergence (S/N = 20).

FIG. 7
FIG. 7

Optical carrier power required to achieve a specified servo system signal-to-noise ratio.

Equations (51)

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

U n ( x ) = K element area U s ( x ) r exp i ( k x · x r + Φ e n ( x ) ) d x ,
ϕ e n ( x ) = ϕ ¯ e n + ϕ e n ( x ) ,
U n ( x ) U 0 A n e i ( ϕ ¯ e n ) p n ( x ) ,
U 0 = K element area d x r U s
A n = W n S r n ,
S r n = exp - 1 2 ϕ e n 2 ( x ) .
Γ n c = - ϕ n c + ψ 0 sin ω n t .
U t ( x ) = U 0 ( n = 1 N A n e i Γ n p 0 ( x ) ) ,
Γ n = Γ n c + ψ ¯ e n = β n + ψ 0 sin ( ω n t ) ;
β n = ϕ ¯ e n - ϕ c n .
I p = U t 2 = U 0 2 p 0 2 ( x 0 ) | n = 1 N A n e i Γ n | 2 = U 0 2 p 0 2 ( x 0 ) n = 1 N A n 2 + k , l = 1 k l N A k A l cos ( Γ k - Γ l ) .
I p m = U 0 2 p 0 2 ( x 0 ) ( n = 1 N A n 2 + J 0 2 ( ψ 0 ) k , l = 1 N A k A l cos ( β k - β l ) - 4 J 0 ( ψ 0 ) J 1 ( ψ 0 ) A m sin ω m t l = 1 l m N A l sin ( β m - β l ) + 4 J 0 ( ψ 0 ) J 2 ( ψ 0 ) A m cos 2 ω m t l = 1 l m N A l cos ( β m - β l ) ) .
I p m = U 0 2 p 0 2 ( x 0 ) [ A n 2 + H J 0 2 + 4 J 0 J 1 ( l = 1 l m N A l 2 + H m ) 1 / 2 A m sin ( β m - β m c ) sin ω m t + 4 J 0 J 2 ( l = 1 l m N A l 2 + H m ) 1 / 2 A m cos ( β m - β m c ) cos 2 ω m t ] ,
β m c = tan - 1 ( l = 1 l m N A l sin β l / l = 1 l m N A l sin β l ) .
H m = k = 1 k l m N l = 1 N A k A l cos ( β k - β l ) ,
H = k = 1 N l = 1 N A k A l cos ( β k - β l ) .
β m - β m c = π .
V AGC = [ ( J 2 ( ψ 0 ) J 1 ( ψ 0 ) S 1 D ) 2 + [ S 2 D ] 2 ] 1 / 2
β k = β l , all k , l
ϕ c n = ϕ ¯ e n , all n
Ī p = F 2 P 2 ( x 0 ) ( n = 1 N A n 2 + J 0 2 ( ψ 0 ) k , l = 1 k l N A k A l ) .
Ī p = F 2 P 2 ( x 0 ) ( n = 1 N W n 2 S r n 2 + J 0 2 k , l = 1 k l N W k S r k W l S r l ) .
n = 1 N W n 2 = const .
( 2 - J 0 2 ) S r m 2 W m + J 0 2 S r m l = 1 N W l S r l + 2 λ W m = 0 ,             m = 1 , 2 , , N .
( W m W k ) opt = S r m S r k 1 + [ ( 2 - J 0 2 ) / J 0 2 ] ( S r m W m / W l S r l ) 1 + [ ( 2 - J 0 2 ) / J 0 2 ] ( S r k W k / W l S r l )             for all m and k .
W l S r l N S ¯ r ,
( W m W k ) opt exp - 1 2 ϕ e m 2 ( x ) exp - 1 2 ϕ e k 2 ( x ) .
( P m / P k ) opt = ρ m / ρ k ,
I p = F 2 P 2 ( x 0 ) A n 2 [ N + ( N 2 - N ) J 0 2 ( ψ 0 ) ] .
( Ī p ) open = F 2 P 2 ( x 0 ) A n 2 .
G A = Ī p / ( Ī p ) open = 1 + ( J 0 2 ( ψ 0 ) k , l = 1 N A k A l ) / ( n = 1 N A n 2 ) .
A k = Ā ( 1 + k ) ,
G A 1 + [ J 0 2 ( ψ 0 ) ( N - 1 ) ] / ( 1 + k 2 ) ,
k 2 = ( k 2 ) / N .
m = I m I 0 = 4 A m J 0 J 1 ( l = 1 , l m N A l 2 + H m ) 1 / 2 ( n = 1 N A n 2 + H J 0 2 ) sin ( β m - β m c ) .
I m I 0 | near convergence 4 A m ( J 1 / J 0 ) [ H m ] 1 / 2 H ( β m - β m c ) sin ω m t .
m 2 = 0 2 4 A m 2 ( J 1 / J 0 ) 2 H m H 2 ( β m - β m c ) 2 .
N 2 = 2 e 0 B N ,
S N 0 = m 2 N 2 = 2 0 e B N A m 2 ( J 1 / J 0 ) 2 H m H 2 ( β m - β m c ) 2 .
P r = A g A d l P 0 2 ( x 0 ) A m 2 + H J 0 2 N 2 P t C ,
0 = e η q h ν A g A d l P 0 2 ( x 0 ) A m 2 + H J 0 2 N 2 P t C ,
S N 0 = N P E ¯ p 0 2 ( x 0 ) A g A d l ( A m J 1 N ) 2 ( β m - β m c ) 2 ,
N P E ¯ = 2 η q P t C / h ν B N .
P p = A p A d l p 0 2 ( x 0 ) A n 2 + H J 0 2 N 2 ,
ϕ e N 2 = ( N P E ¯ p m 2 ( x 0 ) A r A d l ) - 1 ( N A m J 1 ( ψ 0 ) ) 2 .
ϕ e D 2 = ψ 0 2 / 2.
ϕ e T 2 = ϕ e N 2 + ϕ e D 2 = ( N P E ¯ p m 2 ( x 0 ) A r A d l ) - 1 ( N A m J 1 ( ψ 0 ) ) 2 + ψ 0 2 2 .
( ψ 0 ) opt = [ 3 ( N P E ¯ p m 2 ( x 0 ) A r A d l ) - 1 ( N A m ) 2 ] 1 / 4 .
ϕ e T 2 min = [ ( ψ 0 ) opt ] 2 .
N P E ¯ = 2 × 0.2 × 10 - 7 2 × 10 - 20 × 300 6.7 × 10 9 ( photoelectrons ) .
( ψ 0 ) opt = { 8 [ 6.7 × 10 9 × ( 0.3 ) ] - 1 10 4 } 1 / 4 0.08 rad .