Abstract

Multiterawatt laser systems, particularly the Novette system at the Lawrence Livermore National Laboratory, are simulated using statistical-optics techniques. The results are compared with experimental observations.

© 1985 Optical Society of America

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References

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  1. W. W. Simmons, J. T. Hunt, W. E. Warren, “Light propagation through large laser systems,” IEEE J. Quantum Electron. QE-17, 1727 (1981).
    [CrossRef]
  2. L. Mandel, E. Wolf, eds., Selected Papers on Coherence and Fluctuations of Light (Dover, New York, 1970).
  3. J. W. Goodman, Statistical Optics: An Introduction (Wiley, New York, 1984).
  4. S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353 (1971).
  5. J. B. Trenholme, E. J. Goodwin, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1976).
  6. L. M. Frantz, J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346 (1963).
    [CrossRef]
  7. D. C. Brown, “High-peak-power Nd:glass laser systems,” in Optical Sciences 25, D. L. MacAdam, ed. (Springer-Verlag, New York, 1981), p. 188.
  8. W. Simmons, J. E. Murray, F. Rainer, S. Guch, D. Bubp, R. McWilliams, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1974).,
    [CrossRef]
  9. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Patterns, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1975).
    [CrossRef]
  10. J. F. Holzrichter, D. Eimerl, E. V. George, J. B. Trenholme, W. W. Simmons, J. T. Hunt, “High power glass lasers,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1982).
  11. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), p. 567.
  12. D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
    [CrossRef]

1981 (2)

W. W. Simmons, J. T. Hunt, W. E. Warren, “Light propagation through large laser systems,” IEEE J. Quantum Electron. QE-17, 1727 (1981).
[CrossRef]

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

1971 (1)

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353 (1971).

1963 (1)

L. M. Frantz, J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), p. 567.

Bliss, E. S.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Brown, D. C.

D. C. Brown, “High-peak-power Nd:glass laser systems,” in Optical Sciences 25, D. L. MacAdam, ed. (Springer-Verlag, New York, 1981), p. 188.

Bubp, D.

W. Simmons, J. E. Murray, F. Rainer, S. Guch, D. Bubp, R. McWilliams, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1974).,
[CrossRef]

Eimerl, D.

J. F. Holzrichter, D. Eimerl, E. V. George, J. B. Trenholme, W. W. Simmons, J. T. Hunt, “High power glass lasers,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1982).

Frantz, L. M.

L. M. Frantz, J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

George, E. V.

J. F. Holzrichter, D. Eimerl, E. V. George, J. B. Trenholme, W. W. Simmons, J. T. Hunt, “High power glass lasers,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1982).

Glaze, J. A.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Patterns, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1975).
[CrossRef]

J. W. Goodman, Statistical Optics: An Introduction (Wiley, New York, 1984).

Goodwin, E. J.

J. B. Trenholme, E. J. Goodwin, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1976).

Guch, S.

W. Simmons, J. E. Murray, F. Rainer, S. Guch, D. Bubp, R. McWilliams, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1974).,
[CrossRef]

Herris, J. W.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Holloway, F. W.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Holzrichter, J. F.

J. F. Holzrichter, D. Eimerl, E. V. George, J. B. Trenholme, W. W. Simmons, J. T. Hunt, “High power glass lasers,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1982).

Hunt, J. T.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

W. W. Simmons, J. T. Hunt, W. E. Warren, “Light propagation through large laser systems,” IEEE J. Quantum Electron. QE-17, 1727 (1981).
[CrossRef]

J. F. Holzrichter, D. Eimerl, E. V. George, J. B. Trenholme, W. W. Simmons, J. T. Hunt, “High power glass lasers,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1982).

Johnson, B. C.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Kuizenga, D. J.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

McWilliams, R.

W. Simmons, J. E. Murray, F. Rainer, S. Guch, D. Bubp, R. McWilliams, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1974).,
[CrossRef]

Murray, J. E.

W. Simmons, J. E. Murray, F. Rainer, S. Guch, D. Bubp, R. McWilliams, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1974).,
[CrossRef]

Nodvik, J. S.

L. M. Frantz, J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

Ozarski, R. G.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Patton, H. G.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353 (1971).

Rainer, F.

W. Simmons, J. E. Murray, F. Rainer, S. Guch, D. Bubp, R. McWilliams, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1974).,
[CrossRef]

Rupert, P. R.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Simmons, W.

W. Simmons, J. E. Murray, F. Rainer, S. Guch, D. Bubp, R. McWilliams, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1974).,
[CrossRef]

Simmons, W. W.

W. W. Simmons, J. T. Hunt, W. E. Warren, “Light propagation through large laser systems,” IEEE J. Quantum Electron. QE-17, 1727 (1981).
[CrossRef]

J. F. Holzrichter, D. Eimerl, E. V. George, J. B. Trenholme, W. W. Simmons, J. T. Hunt, “High power glass lasers,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1982).

Speck, D. R.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), p. 567.

Suski, G. J.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Swift, C. D.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353 (1971).

Thompson, C. E.

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Trenholme, J. B.

J. F. Holzrichter, D. Eimerl, E. V. George, J. B. Trenholme, W. W. Simmons, J. T. Hunt, “High power glass lasers,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1982).

J. B. Trenholme, E. J. Goodwin, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1976).

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353 (1971).

Warren, W. E.

W. W. Simmons, J. T. Hunt, W. E. Warren, “Light propagation through large laser systems,” IEEE J. Quantum Electron. QE-17, 1727 (1981).
[CrossRef]

IEEE J. Quantum Electron. (2)

W. W. Simmons, J. T. Hunt, W. E. Warren, “Light propagation through large laser systems,” IEEE J. Quantum Electron. QE-17, 1727 (1981).
[CrossRef]

D. R. Speck, E. S. Bliss, J. A. Glaze, J. W. Herris, F. W. Holloway, J. T. Hunt, B. C. Johnson, D. J. Kuizenga, R. G. Ozarski, H. G. Patton, P. R. Rupert, G. J. Suski, C. D. Swift, C. E. Thompson, “The Shiva laser-fusion facility,” IEEE J. Quantum Electron. QE-17, 1599 (1981).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353 (1971).

J. Appl. Phys. (1)

L. M. Frantz, J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

Other (8)

D. C. Brown, “High-peak-power Nd:glass laser systems,” in Optical Sciences 25, D. L. MacAdam, ed. (Springer-Verlag, New York, 1981), p. 188.

W. Simmons, J. E. Murray, F. Rainer, S. Guch, D. Bubp, R. McWilliams, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1974).,
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Patterns, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1975).
[CrossRef]

J. F. Holzrichter, D. Eimerl, E. V. George, J. B. Trenholme, W. W. Simmons, J. T. Hunt, “High power glass lasers,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1982).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), p. 567.

J. B. Trenholme, E. J. Goodwin, “Laser program annual report,” (Lawrence Livermore National Laboratory, Livermore, Calif., 1976).

L. Mandel, E. Wolf, eds., Selected Papers on Coherence and Fluctuations of Light (Dover, New York, 1970).

J. W. Goodman, Statistical Optics: An Introduction (Wiley, New York, 1984).

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Figures (17)

Fig. 1
Fig. 1

These three beam photographs show the progressive deterioration of the Novette 74-cm-diameter beams as output power is increased. The left-hand photograph shows a 10-TW beam that corresponds to Novette’s routine operating power level per beam. At 11 TW, nonlinear effects are becoming barely visible in most regions of the beam and have eaten away the beam near damage sites on the right. Operation beyond 13 TW (right-hand photograph) is not recommended since small-scale structure is well developed and the damaged regions on the right, although physically no larger, have excited modes that seriously cut into the beam.

Fig. 2
Fig. 2

Propagation involves calculating the mutual intensity in the (x′, y′) plane located a distance z′ − z from the given mutual intensity in the (x, y) plane.

Fig. 3
Fig. 3

(a) Scan taken through the central region of a typical high-power (9-TW) infrared focus measured by one of Novette’s output sensors. This beam is roughly circular, and its dimensions show that the laser system optics have introduced about 0.26 rad/cm of rms phase error in the beam. (b) Measured 25, 50, and 80% included-energy focal-spot radii are compared with statistical-optics estimates over the Novette laser’s operating range.

Fig. 4
Fig. 4

Spatial-filter transmission measurements were made on the Cyclops laser system at LLNL in 1974. This graph is reproduced from Ref. 11 with the theoretical predictions added.

Fig. 5
Fig. 5

This optical schematic shows the relation of Novette’s architecture to earlier LLNL laser systems. The dashed components do not exist in the two Novette chains but will exist in Nova.

Fig. 6
Fig. 6

Novette’s measured performance at (a) 100 psec and (b) 1 nsec is compared in two simulation calculations. The observed saturation flux of 4.05 J/cm2 is consistent with current understanding of the saturation of Nd in LHG-8 and LG 750 phosphate glasses.

Fig. 7
Fig. 7

Given measured small-signal gains, the flux on each optical component is readily calculated and measured with good agreement. These data were obtained from an 11-TW shot.

Fig. 8
Fig. 8

This 11-TW beam was scanned in the crosshatched region to produce the beam profile in (b). The entire beam is scanned to find the experimental distribution functions discussed in the text.

Fig. 9
Fig. 9

Laser-power-distribution functions such as this 11-TW example have been measured for several output-power levels. In this case, 20% of the power in the beam was delivered at fluxes below 2.6 GW/cm2, 50% was delivered at less than 3.2 GW/cm2, and 80% was found to be below 3.7 GW/cm2.

Fig. 10
Fig. 10

A typical beam profile presented to Novette’s input sensor diagnostic package is displayed on the right in (a). This beam’s measured power-distribution function is plotted along with the input distribution used for the statistical optics calculation in (b). A consistent feature of the assumed Rician form is that it often overestimates the low-intensity part of the distribution.

Fig. 11
Fig. 11

These two calculated 100-psec power-distribution functions are compared with actual measurements. The double curves on both plots reflect uncertainties in Novette component gains and losses.

Fig. 12
Fig. 12

Given a laser staging configuration, contours of any chosen fractional power level can be predicted. Here 80, 50, and 20% contours are compared with data obtained from 100-psec shots with the indicated infrared-output powers measured at the input to Novette’s frequency-conversion crystal arrays.

Fig. 13
Fig. 13

At 1-nsec pulse duration, operation at or above about 10 TW poses unacceptably large risks to a Novette amplifier chain’s output optics since more than 10% of the laser power will be delivered at intensities exceeding 4 GW/cm2 and some components are expected to tolerate only 5 to 6 GW/cm2.

Fig. 14
Fig. 14

malaprop calculations, the statistical-optics approximation, and measured 9-TW near-field power distributions are compared.

Fig. 15
Fig. 15

Novette’s second-harmonic-generation array is performing very much like a single KDP crystal, as this comparison of measured and predicted conversion efficiency shows. The error bars on input infrared intensity represent 80, 50, and 20% power-distribution intensities for the indicated conversion efficiencies.

Fig. 16
Fig. 16

Target studies must be designed around each laser’s operational capabilities. In Novette’s case, a safe, acceptable risk and a maximum operating energy can be found for each pulse duration. Pushing Novette to its maximum output energy in 1-nsec or longer pulse durations is likely to lead to heavily damaged output optics.

Fig. 17
Fig. 17

These data are reproduced from Fig. 4 of Ref. 8. They show the actual performance of the Shiva laser, Novette’s predecessor. Predictions for low-risk operation from the statistical-optics model discussed here have been added. They conform well to the way Shiva was used for target studies and suggest why Shiva output optics were observed to be damaged on some of the highest-power 0.6- to 0.8-nsec shots.

Tables (2)

Tables Icon

Table 1 Measured Amplifier Gains

Tables Icon

Table 2 Optical Devices in One Novette Chain

Equations (39)

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Γ ( x 1 , y 1 , z 1 , t 1 , x 2 , y 2 , z 2 , t 2 ) = E ( x 1 , y 1 , z 1 , t 1 ) E ( x 2 , y 2 , z 2 , t 2 ) ,
J ( x ¯ 1 , x ¯ 2 ; z ) = ( 0 / 2 ) u ( x ¯ 1 , z ) u * ( x ¯ 2 , z ) .
J ( x ¯ 1 , x ¯ 2 ; z ) = 1 / λ 2 - + - + - + - + × u ( x ¯ 1 , z ) u * ( x ¯ 2 , z ) exp [ - i k ( r 2 - r 1 ) ] d x ¯ 1 / r 1 d x ¯ 2 / r 2 ,
J ( x ¯ 1 , x ¯ 2 ; z ) = 1 / λ 2 - + - + - + - + × J ( x ¯ 1 , x ¯ 2 ; z ) exp [ - i k ( r 2 - r 1 ) ] d x ¯ 1 / r 1 d x ¯ 2 / r 2 ,
( r 2 - r 1 ) = ( x ¯ 2 - x ¯ 2 2 - x ¯ 1 - x ¯ 1 2 ) / [ 2 ( z - z ) ] ,
z - z = n d = ( n 0 + γ I ) d = n 0 d + B ( I ) / k ,
J ( x ¯ 1 , x ¯ 2 ; z ) = [ I + σ A 2 ρ A ( x ¯ 1 , x ¯ 2 ) ] × exp { - [ 1 - ρ p ( x ¯ 1 , x ¯ 2 ) ] σ p 2 } ,
ρ p ( x ¯ 1 , x ¯ 2 ) = exp ( - x ¯ 1 - x ¯ 2 2 / L p 2 ) ~ ( 1 - x ¯ 1 - x ¯ 2 2 / L p 2 )
ρ A ( x ¯ 1 , x ¯ 2 ) = exp ( - x ¯ 1 - x ¯ 2 2 / L A 2 ) .
J ( x ¯ 1 , x ¯ 2 ; z ) = I exp ( - x ¯ 2 / a 2 ) + σ A 2 exp ( - x ¯ 2 / b 2 ) ,
x ¯ = x ¯ 1 - x ¯ 2 ,             a = L p / σ p , b 2 = 1 / ( σ p 2 / L p 2 + 1 / L A 2 ) .
J ( x ¯ 1 , x ¯ 2 ; z ) = λ - 2 ( z - z ) - 2 × - + - + - + - + J ( x ¯ 1 , x ¯ 2 ; z ) × exp [ - i k ( r 2 - r 1 ) ] d x ¯ 1 d x ¯ 2 .
exp [ - i k ( r 2 - r 1 ) - i B ( x ¯ 2 ) + i B ( x ¯ 1 ) ] = exp [ - i k ( r 2 - r 1 ) ] exp { - i [ B ( x ¯ 2 ) - B ( x ¯ 1 ) ] } ,
exp { - i [ B ( x ¯ 2 ) - B ( x ¯ 1 ) ] } = exp { i k γ d [ A 2 ( x ¯ 2 ) - A 2 ( x 1 ) ] } .
A 2 ( x ¯ 2 ) - A 2 ( x ¯ 1 ) ~ 2 A 0 [ A ( x ¯ 2 ) - A ( x ¯ 1 ) ] ,
s 2 = 2 [ 1 - ρ A ( x ¯ 1 , x ¯ 2 ) ] σ A 2 .
exp [ - 2 ( s k γ A 0 d ) 2 ] = exp [ - 2 B 2 ( s / A 0 ) 2 ] .
J ( x ¯ 1 , x ¯ 2 ; d ) = J ( x ¯ 1 , x ¯ 2 ; 0 ) exp [ - 2 B 2 ( s / A 0 ) 2 ] .
J ( x ¯ 1 , x ¯ 2 ; d ) = J ( x ¯ 1 , x ¯ 2 ; 0 ) exp [ - s pe 2 x ¯ 2 - 2 B 2 ( s / A 0 ) 2 ] ,
s 2 ~ 2 x ¯ 2 ( σ A / L A ) 2 ,
a - 2 = a - 2 + s pe 2 + 4 B 2 [ σ A / ( L A A 0 ) ] 2
b - 2 = b - 2 + s pe 2 + 4 B 2 [ σ A / ( L A A 0 ) ] 2 .
( / z + 2 X ¯ x ¯ / i k ) J = - γ k / 4 s 2 J ,
X ¯ = x ¯ 1 + x ¯ 2 ,             x ¯ 1 = x ¯ 1 - x ¯ 2 .
/ z ( I + σ A 2 ) = 0 ,
/ z σ A 2 ( γ k I 2 ) σ A 2
σ A 2 ( B ) σ A 2 ( 0 ) exp ( B / 2 ) ,
B = ( 4 π / 3 ) ( 2 π / λ ) ( 10 - 7 ) ( n 2 / n 0 ) d I ,
a ( z ) = [ 2 z / ( k a ) ] [ 1 + k 2 a 4 ( 1 / z - 1 / f ) 2 / 4 ] 1 / 2 ,
G ( x ¯ , x ¯ ) = ( 1 / M 2 ) 2 J 1 ( k r / f x ¯ + M x ¯ ) k r / f x ¯ + M x ¯ ,
I ( x ) = J ( x 1 , x ¯ 1 ; f ) = I ( k a 2 / 4 f ) 2 exp { - [ k x a / ( 2 f ) ] 2 } + σ A 2 ( k b 2 / 4 f ) 2 exp { - [ k x b / ( 2 f ) ] 2 } .
T ( a ) = 0 r x I ( x ) d x 0 I ( x ) d x ~ 1 - exp { - [ k r a / ( 2 f ) ] 2 }
p ( I ) = { ( 1 / σ A 2 ) exp [ - ( I + I s ) / σ A 2 ] , J 0 ( 2 I I s / σ A 2 ) 0 for I > 0 for I < 0 ,
F ( I ) = ( I s + σ A 2 ) - 1 0 I I p ( I ) d I .
σ I 2 = I 2 - I 2 = σ A 4 + 2 I s σ A 2 ,
σ I / I = ( σ A 2 + 2 I s σ A 2 ) 1 / 2 I s + σ A 2 ( 2 / I s ) 1 / 2 σ A .
η = tanh 2 ( ½ tanh - 1 { sn [ 2 η 0 1 / 2 , ( 1 + δ 2 ) / ( 4 η 0 ) ] } ) ,
I 2 ω I 1 ω = 0 + η p ( I ) d I
risk = 0 + p ( I ) 0 I p D ( I ) d I d I ,

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