Abstract

The scaling of laser damage thresholds with irradiated spot size is a well-known effect. When damage is defect dominated, the spot-size scaling can be attributed to the conventional definition of the threshold at the 50% level of damage probability. By redefining the threshold at the 0% level (absolute damage onset), one obtains a result that is spot-size independent. A method is presented here for obtaining the damage onset of optical surfaces subjected to pulsed laser radiation. The method involves weighted least-squares fitting of damage frequency data with a three-parameter distributed ensemble representing inherent defect damage characteristics. Spot-size effects in the data are simulated by a scaling transformation applied to the ensemble before fitting. Direct and inverse transformations are derived for Gaussian and top-hat spatial intensity profiles, and advantages of the latter for testing are pointed out. Three examples of applications to 2.7-μm multilayer coatings are presented, and the general inadequacy of the two-parameter degenerate ensemble is demonstrated. Extraction of defect densities, discrimination of different defect classes, and representation of uncertainty in the onset are discussed.

© 1984 Optical Society of America

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References

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  1. H. E. Bennett, A. H. Guenther, D. Milam, B. E. Newnam, “Laser-Induced Damage in Optical Materials: Thirteenth ASTM Symposium,” Appl. Opt. 22, 3276 (1983).
    [CrossRef] [PubMed]
  2. D. B. Nichols, D. J. Morris, M. P. Bailey, R. B. Hall, “Laser Induced Emission and Laser Damage of Optical Components,” in Proceedings, Fifteenth Annual Symposium on Optical Materials for High Power Lasers, Boulder, Colo., 14–16 Nov. 1983”; in process.
  3. C. D. Marrs, J. O. Porteus, J. R. Palmer, “Defect-Damage precursors in Visible Wavelength Mirrors,” ibid.
  4. S. C. Seitel, J. O. Porteus, “1.06 μm Laser Damage Round-Robin Testing with 13 ns Pulse Duration and 40 μm Spot Size,” Appl. Opt., 23, 3767 (1984).
    [CrossRef] [PubMed]
  5. J. O. Porteus, J. L. Jernigan, W. N. Faith, “Multithreshold Measurement and Analysis of Pulsed Laser Induced Damage in Optical Surfaces,” Natl. Bur. Stand. U.S. Spec. Publ. 509, 507 (1978).
  6. J. O. Porteus, D. L. Decker, J. L. Jernigan, W. N. Faith, M. Bass, “Evaluation of Metal Mirrors for High Power Applications by Multithreshold Damage Analysis,” IEEE J. Quantum Electron. QE-14, 776 (1978).
    [CrossRef]
  7. S. R. Foltyn, “Spotsize Effects in Laser Damage Testing,” Natl. Bur. Stand. U.S. Spec. Publ. 669, 368 (1984).
  8. J. O. Porteus, “Determinations of the Onset of Defect-Driven Pulsed Laser Damage in 2.7 μm Optical Coatings,” High Power Laser Optical Components Topical Meeting, Boulder, Colo., 18, 19 Nov. 1982”; in process.
  9. S. C. Seitel, J. O. Porteus, “Toward Improved Accuracy in Limited-Scale Pulsed Laser Damage Testing Via the Onset Method,” in Proceedings, Fifteenth Annual Symposium on Optical Materials for High Power Lasers, Boulder, Colo., 14–16 Nov. 1983”; in process.
  10. Yu. K. Danileiko, Yu. P. Minaev, V. N. Nikolaev, A. V. Sidorin, “Determination of the Characteristics of Microdefects from Statistical Relationships Governing Laser Damage to Solid Transparent Materials,” Sov. J. Quantum Electron. 11, 1445 (1981).
    [CrossRef]
  11. Correction for the tail or for scattering losses in calibrating axial intensity is a separate problem, which is discussed in Refs. 4 and 17.
  12. L. G. Parratt, Probability and Experimental Errors in Science (Wiley, New York, 1961), Sec. 3.4
  13. Ref. 12, pp. 92–94.
  14. M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (Dover, New York, 1965), pp. 887 and 916.
  15. Ref. 12, pp. 114, 115.
  16. S. C. Seitel, J. B. Franck, C. D. Marrs, G. D. Williams, “Selective and Uniform Laser-Induced Failure of Antireflection-Coated LiNbO3 Surfaces,” IEEE J. Quantum Electron. QE-19, 475 (1983).
    [CrossRef]
  17. J. O. Porteus, D. L. Decker, W. N. Faith, D. J. Grandjean, S. C. Seitel, M. J. Soileau, “Pulsed Laser-Induced Melting of Precision Diamond-Machined Cu, Ag, and Au at Infrared Wavelengths,” IEEE J. Quantum Electron. QE-17, 2078 (1981).
    [CrossRef]

1984

S. R. Foltyn, “Spotsize Effects in Laser Damage Testing,” Natl. Bur. Stand. U.S. Spec. Publ. 669, 368 (1984).

S. C. Seitel, J. O. Porteus, “1.06 μm Laser Damage Round-Robin Testing with 13 ns Pulse Duration and 40 μm Spot Size,” Appl. Opt., 23, 3767 (1984).
[CrossRef] [PubMed]

1983

H. E. Bennett, A. H. Guenther, D. Milam, B. E. Newnam, “Laser-Induced Damage in Optical Materials: Thirteenth ASTM Symposium,” Appl. Opt. 22, 3276 (1983).
[CrossRef] [PubMed]

S. C. Seitel, J. B. Franck, C. D. Marrs, G. D. Williams, “Selective and Uniform Laser-Induced Failure of Antireflection-Coated LiNbO3 Surfaces,” IEEE J. Quantum Electron. QE-19, 475 (1983).
[CrossRef]

1981

J. O. Porteus, D. L. Decker, W. N. Faith, D. J. Grandjean, S. C. Seitel, M. J. Soileau, “Pulsed Laser-Induced Melting of Precision Diamond-Machined Cu, Ag, and Au at Infrared Wavelengths,” IEEE J. Quantum Electron. QE-17, 2078 (1981).
[CrossRef]

Yu. K. Danileiko, Yu. P. Minaev, V. N. Nikolaev, A. V. Sidorin, “Determination of the Characteristics of Microdefects from Statistical Relationships Governing Laser Damage to Solid Transparent Materials,” Sov. J. Quantum Electron. 11, 1445 (1981).
[CrossRef]

1978

J. O. Porteus, J. L. Jernigan, W. N. Faith, “Multithreshold Measurement and Analysis of Pulsed Laser Induced Damage in Optical Surfaces,” Natl. Bur. Stand. U.S. Spec. Publ. 509, 507 (1978).

J. O. Porteus, D. L. Decker, J. L. Jernigan, W. N. Faith, M. Bass, “Evaluation of Metal Mirrors for High Power Applications by Multithreshold Damage Analysis,” IEEE J. Quantum Electron. QE-14, 776 (1978).
[CrossRef]

Bailey, M. P.

D. B. Nichols, D. J. Morris, M. P. Bailey, R. B. Hall, “Laser Induced Emission and Laser Damage of Optical Components,” in Proceedings, Fifteenth Annual Symposium on Optical Materials for High Power Lasers, Boulder, Colo., 14–16 Nov. 1983”; in process.

Bass, M.

J. O. Porteus, D. L. Decker, J. L. Jernigan, W. N. Faith, M. Bass, “Evaluation of Metal Mirrors for High Power Applications by Multithreshold Damage Analysis,” IEEE J. Quantum Electron. QE-14, 776 (1978).
[CrossRef]

Bennett, H. E.

Danileiko, Yu. K.

Yu. K. Danileiko, Yu. P. Minaev, V. N. Nikolaev, A. V. Sidorin, “Determination of the Characteristics of Microdefects from Statistical Relationships Governing Laser Damage to Solid Transparent Materials,” Sov. J. Quantum Electron. 11, 1445 (1981).
[CrossRef]

Decker, D. L.

J. O. Porteus, D. L. Decker, W. N. Faith, D. J. Grandjean, S. C. Seitel, M. J. Soileau, “Pulsed Laser-Induced Melting of Precision Diamond-Machined Cu, Ag, and Au at Infrared Wavelengths,” IEEE J. Quantum Electron. QE-17, 2078 (1981).
[CrossRef]

J. O. Porteus, D. L. Decker, J. L. Jernigan, W. N. Faith, M. Bass, “Evaluation of Metal Mirrors for High Power Applications by Multithreshold Damage Analysis,” IEEE J. Quantum Electron. QE-14, 776 (1978).
[CrossRef]

Faith, W. N.

J. O. Porteus, D. L. Decker, W. N. Faith, D. J. Grandjean, S. C. Seitel, M. J. Soileau, “Pulsed Laser-Induced Melting of Precision Diamond-Machined Cu, Ag, and Au at Infrared Wavelengths,” IEEE J. Quantum Electron. QE-17, 2078 (1981).
[CrossRef]

J. O. Porteus, D. L. Decker, J. L. Jernigan, W. N. Faith, M. Bass, “Evaluation of Metal Mirrors for High Power Applications by Multithreshold Damage Analysis,” IEEE J. Quantum Electron. QE-14, 776 (1978).
[CrossRef]

J. O. Porteus, J. L. Jernigan, W. N. Faith, “Multithreshold Measurement and Analysis of Pulsed Laser Induced Damage in Optical Surfaces,” Natl. Bur. Stand. U.S. Spec. Publ. 509, 507 (1978).

Foltyn, S. R.

S. R. Foltyn, “Spotsize Effects in Laser Damage Testing,” Natl. Bur. Stand. U.S. Spec. Publ. 669, 368 (1984).

Franck, J. B.

S. C. Seitel, J. B. Franck, C. D. Marrs, G. D. Williams, “Selective and Uniform Laser-Induced Failure of Antireflection-Coated LiNbO3 Surfaces,” IEEE J. Quantum Electron. QE-19, 475 (1983).
[CrossRef]

Grandjean, D. J.

J. O. Porteus, D. L. Decker, W. N. Faith, D. J. Grandjean, S. C. Seitel, M. J. Soileau, “Pulsed Laser-Induced Melting of Precision Diamond-Machined Cu, Ag, and Au at Infrared Wavelengths,” IEEE J. Quantum Electron. QE-17, 2078 (1981).
[CrossRef]

Guenther, A. H.

Hall, R. B.

D. B. Nichols, D. J. Morris, M. P. Bailey, R. B. Hall, “Laser Induced Emission and Laser Damage of Optical Components,” in Proceedings, Fifteenth Annual Symposium on Optical Materials for High Power Lasers, Boulder, Colo., 14–16 Nov. 1983”; in process.

Jernigan, J. L.

J. O. Porteus, D. L. Decker, J. L. Jernigan, W. N. Faith, M. Bass, “Evaluation of Metal Mirrors for High Power Applications by Multithreshold Damage Analysis,” IEEE J. Quantum Electron. QE-14, 776 (1978).
[CrossRef]

J. O. Porteus, J. L. Jernigan, W. N. Faith, “Multithreshold Measurement and Analysis of Pulsed Laser Induced Damage in Optical Surfaces,” Natl. Bur. Stand. U.S. Spec. Publ. 509, 507 (1978).

Marrs, C. D.

S. C. Seitel, J. B. Franck, C. D. Marrs, G. D. Williams, “Selective and Uniform Laser-Induced Failure of Antireflection-Coated LiNbO3 Surfaces,” IEEE J. Quantum Electron. QE-19, 475 (1983).
[CrossRef]

C. D. Marrs, J. O. Porteus, J. R. Palmer, “Defect-Damage precursors in Visible Wavelength Mirrors,” ibid.

Milam, D.

Minaev, Yu. P.

Yu. K. Danileiko, Yu. P. Minaev, V. N. Nikolaev, A. V. Sidorin, “Determination of the Characteristics of Microdefects from Statistical Relationships Governing Laser Damage to Solid Transparent Materials,” Sov. J. Quantum Electron. 11, 1445 (1981).
[CrossRef]

Morris, D. J.

D. B. Nichols, D. J. Morris, M. P. Bailey, R. B. Hall, “Laser Induced Emission and Laser Damage of Optical Components,” in Proceedings, Fifteenth Annual Symposium on Optical Materials for High Power Lasers, Boulder, Colo., 14–16 Nov. 1983”; in process.

Newnam, B. E.

Nichols, D. B.

D. B. Nichols, D. J. Morris, M. P. Bailey, R. B. Hall, “Laser Induced Emission and Laser Damage of Optical Components,” in Proceedings, Fifteenth Annual Symposium on Optical Materials for High Power Lasers, Boulder, Colo., 14–16 Nov. 1983”; in process.

Nikolaev, V. N.

Yu. K. Danileiko, Yu. P. Minaev, V. N. Nikolaev, A. V. Sidorin, “Determination of the Characteristics of Microdefects from Statistical Relationships Governing Laser Damage to Solid Transparent Materials,” Sov. J. Quantum Electron. 11, 1445 (1981).
[CrossRef]

Palmer, J. R.

C. D. Marrs, J. O. Porteus, J. R. Palmer, “Defect-Damage precursors in Visible Wavelength Mirrors,” ibid.

Parratt, L. G.

L. G. Parratt, Probability and Experimental Errors in Science (Wiley, New York, 1961), Sec. 3.4

Porteus, J. O.

S. C. Seitel, J. O. Porteus, “1.06 μm Laser Damage Round-Robin Testing with 13 ns Pulse Duration and 40 μm Spot Size,” Appl. Opt., 23, 3767 (1984).
[CrossRef] [PubMed]

J. O. Porteus, D. L. Decker, W. N. Faith, D. J. Grandjean, S. C. Seitel, M. J. Soileau, “Pulsed Laser-Induced Melting of Precision Diamond-Machined Cu, Ag, and Au at Infrared Wavelengths,” IEEE J. Quantum Electron. QE-17, 2078 (1981).
[CrossRef]

J. O. Porteus, D. L. Decker, J. L. Jernigan, W. N. Faith, M. Bass, “Evaluation of Metal Mirrors for High Power Applications by Multithreshold Damage Analysis,” IEEE J. Quantum Electron. QE-14, 776 (1978).
[CrossRef]

J. O. Porteus, J. L. Jernigan, W. N. Faith, “Multithreshold Measurement and Analysis of Pulsed Laser Induced Damage in Optical Surfaces,” Natl. Bur. Stand. U.S. Spec. Publ. 509, 507 (1978).

S. C. Seitel, J. O. Porteus, “Toward Improved Accuracy in Limited-Scale Pulsed Laser Damage Testing Via the Onset Method,” in Proceedings, Fifteenth Annual Symposium on Optical Materials for High Power Lasers, Boulder, Colo., 14–16 Nov. 1983”; in process.

J. O. Porteus, “Determinations of the Onset of Defect-Driven Pulsed Laser Damage in 2.7 μm Optical Coatings,” High Power Laser Optical Components Topical Meeting, Boulder, Colo., 18, 19 Nov. 1982”; in process.

C. D. Marrs, J. O. Porteus, J. R. Palmer, “Defect-Damage precursors in Visible Wavelength Mirrors,” ibid.

Seitel, S. C.

S. C. Seitel, J. O. Porteus, “1.06 μm Laser Damage Round-Robin Testing with 13 ns Pulse Duration and 40 μm Spot Size,” Appl. Opt., 23, 3767 (1984).
[CrossRef] [PubMed]

S. C. Seitel, J. B. Franck, C. D. Marrs, G. D. Williams, “Selective and Uniform Laser-Induced Failure of Antireflection-Coated LiNbO3 Surfaces,” IEEE J. Quantum Electron. QE-19, 475 (1983).
[CrossRef]

J. O. Porteus, D. L. Decker, W. N. Faith, D. J. Grandjean, S. C. Seitel, M. J. Soileau, “Pulsed Laser-Induced Melting of Precision Diamond-Machined Cu, Ag, and Au at Infrared Wavelengths,” IEEE J. Quantum Electron. QE-17, 2078 (1981).
[CrossRef]

S. C. Seitel, J. O. Porteus, “Toward Improved Accuracy in Limited-Scale Pulsed Laser Damage Testing Via the Onset Method,” in Proceedings, Fifteenth Annual Symposium on Optical Materials for High Power Lasers, Boulder, Colo., 14–16 Nov. 1983”; in process.

Sidorin, A. V.

Yu. K. Danileiko, Yu. P. Minaev, V. N. Nikolaev, A. V. Sidorin, “Determination of the Characteristics of Microdefects from Statistical Relationships Governing Laser Damage to Solid Transparent Materials,” Sov. J. Quantum Electron. 11, 1445 (1981).
[CrossRef]

Soileau, M. J.

J. O. Porteus, D. L. Decker, W. N. Faith, D. J. Grandjean, S. C. Seitel, M. J. Soileau, “Pulsed Laser-Induced Melting of Precision Diamond-Machined Cu, Ag, and Au at Infrared Wavelengths,” IEEE J. Quantum Electron. QE-17, 2078 (1981).
[CrossRef]

Williams, G. D.

S. C. Seitel, J. B. Franck, C. D. Marrs, G. D. Williams, “Selective and Uniform Laser-Induced Failure of Antireflection-Coated LiNbO3 Surfaces,” IEEE J. Quantum Electron. QE-19, 475 (1983).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

S. C. Seitel, J. B. Franck, C. D. Marrs, G. D. Williams, “Selective and Uniform Laser-Induced Failure of Antireflection-Coated LiNbO3 Surfaces,” IEEE J. Quantum Electron. QE-19, 475 (1983).
[CrossRef]

J. O. Porteus, D. L. Decker, W. N. Faith, D. J. Grandjean, S. C. Seitel, M. J. Soileau, “Pulsed Laser-Induced Melting of Precision Diamond-Machined Cu, Ag, and Au at Infrared Wavelengths,” IEEE J. Quantum Electron. QE-17, 2078 (1981).
[CrossRef]

J. O. Porteus, D. L. Decker, J. L. Jernigan, W. N. Faith, M. Bass, “Evaluation of Metal Mirrors for High Power Applications by Multithreshold Damage Analysis,” IEEE J. Quantum Electron. QE-14, 776 (1978).
[CrossRef]

Natl. Bur. Stand. U.S. Spec. Publ.

S. R. Foltyn, “Spotsize Effects in Laser Damage Testing,” Natl. Bur. Stand. U.S. Spec. Publ. 669, 368 (1984).

J. O. Porteus, J. L. Jernigan, W. N. Faith, “Multithreshold Measurement and Analysis of Pulsed Laser Induced Damage in Optical Surfaces,” Natl. Bur. Stand. U.S. Spec. Publ. 509, 507 (1978).

Sov. J. Quantum Electron.

Yu. K. Danileiko, Yu. P. Minaev, V. N. Nikolaev, A. V. Sidorin, “Determination of the Characteristics of Microdefects from Statistical Relationships Governing Laser Damage to Solid Transparent Materials,” Sov. J. Quantum Electron. 11, 1445 (1981).
[CrossRef]

Other

Correction for the tail or for scattering losses in calibrating axial intensity is a separate problem, which is discussed in Refs. 4 and 17.

L. G. Parratt, Probability and Experimental Errors in Science (Wiley, New York, 1961), Sec. 3.4

Ref. 12, pp. 92–94.

M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (Dover, New York, 1965), pp. 887 and 916.

Ref. 12, pp. 114, 115.

J. O. Porteus, “Determinations of the Onset of Defect-Driven Pulsed Laser Damage in 2.7 μm Optical Coatings,” High Power Laser Optical Components Topical Meeting, Boulder, Colo., 18, 19 Nov. 1982”; in process.

S. C. Seitel, J. O. Porteus, “Toward Improved Accuracy in Limited-Scale Pulsed Laser Damage Testing Via the Onset Method,” in Proceedings, Fifteenth Annual Symposium on Optical Materials for High Power Lasers, Boulder, Colo., 14–16 Nov. 1983”; in process.

D. B. Nichols, D. J. Morris, M. P. Bailey, R. B. Hall, “Laser Induced Emission and Laser Damage of Optical Components,” in Proceedings, Fifteenth Annual Symposium on Optical Materials for High Power Lasers, Boulder, Colo., 14–16 Nov. 1983”; in process.

C. D. Marrs, J. O. Porteus, J. R. Palmer, “Defect-Damage precursors in Visible Wavelength Mirrors,” ibid.

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Figures (8)

Fig. 1
Fig. 1

Degenerate defect ensemble (left) and corresponding damage probability distributions (right) for a Gaussian beam.

Fig. 2
Fig. 2

Comparison of damage probability distributions for top-hat and Gaussian beams (right) obtained with a degenerate defect ensemble (left). The integrated intensity is the same for both beams.

Fig. 3
Fig. 3

Distributed (nondegenerate) defect ensembles obtained with the power-law model. Distributions approach degeneracy as p approaches −1.

Fig. 4
Fig. 4

Distributed p = 0 defect ensemble (left) and corresponding probability distributions (right) for Gaussian beams with two different spot sizes and/or defect densities. The parameter Nw is defined by Eq. (22).

Fig. 5
Fig. 5

Comparison of damage probability distributions for top-hat and Gaussian beams (right) obtained with distributed p = 0 defect ensemble (left). The integrated intensity is the same for both beams.

Fig. 6
Fig. 6

Example of damage frequency data and best probability distributions obtained by least-squares fitting of degenerate (dashed curve) and distributed (solid curve) power-law defect ensemble models. The intersection of the curve with the abscissa determines the onset in each case. Intensity units are arbitrary. The uncertainty in the onset is the p-conditional standard deviation (see text).

Fig. 7
Fig. 7

Second example of onset determination with a different coating design. See Fig. 6 and text for the explanation.

Fig. 8
Fig. 8

Third example of onset determination from damage frequency data. This coating design has two nominally different defect ensembles determined by least-squares fitting of power-law ensembles as described in the text. The corresponding probability distributions are indicated here by the solid curves. The dashed curve represents the sum of the two superimposed components. Only the lesser of the two onsets is quoted in the figure. Intensity units are arbitrary.

Tables (2)

Tables Icon

Table I Experimental Parameters

Tables Icon

Table II Summary of Examples

Equations (44)

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d P = [ 1 - P ( t ) ] W [ I ( t ) ] d t .
P = 1 - exp { - 0 τ W [ I ( t ) ] d t } .
P = 1 - exp ( - A d A 0 f ( I d ) × { 1 - exp [ - 0 τ W ( I , I d ) d t ] } d I d ) .
P = 1 - exp [ - A d A 0 I ( x , y ) f ( I d ) d I d ] ,
I ( r ) = I a exp ( - r 2 / w 2 ) ,
1 1 - P d P d I a = 2 π 0 f [ I a exp ( - r 2 / w 2 ) ] r exp ( - r 2 / w 2 ) d r = π w 2 I a - 1 0 I a f ( I ) d I .
P ( I a ) = 1 - exp { - π w 2 0 I a [ I - 1 0 I f ( I ) d I ] d I } .
f ( I a ) = 1 π w 2 ( 1 - P ) [ d P d I a + I a 1 - P ( d P d I a ) 2 + I a d 2 P d I a 2 ] .
f ( I a ) > 0             whenever             d 2 p d I a 2 > 0.
I ( r ) = I a × { 1 ; r w 0 ; r > w ,
1 1 - P d P d I a = 2 π 0 r × { f ( I a ) ; r w 0 ; r > w } d r = π w 2 f ( I a ) .
P ( I a ) = 1 - exp [ - π w 2 0 I a f ( I ) d I ] .
f ( I a ) = 1 π w 2 ( 1 - P ) d P d I a
f ( I ) = N δ ( I - I 0 ) ,
P ( I a : I 0 , N ) = { 0 ; I a < I 0 1 - ( I a I 0 ) - π w 2 N ; I a I 0 .
r < w ( ln I a I 0 ) 1 / 2 ; I a > I 0 .
P ( I a : I 0 , N ) = { 0 ; I a < I 0 1 - exp ( - π w 2 N ) ; I a I 0
r < w ; I a > I 0 .
f ( I ) = ( p + 1 ) N ( I n ) ( I n - I 0 ) p + 1 × { 0 ; I < I 0 ( I - I 0 ) p ; I I 0 ,
0 I n f ( I ) d I = N ( I n ) .
P ( I a : I 0 , N w , p ) = { 0 ; I a < I 0 1 - exp [ - N w 1 I a / I 0 u - 1 ( u - 1 ) p + 1 d u ] ; I a I 0
N w π w 2 N ( I n ) ( I n I 0 - 1 ) p + 1 .
P ( I a : I 0 , N w , p ) = { 0 1 - exp [ - N w ( I a I 0 - 1 ) p + 1 ] ; ; I a < I 0 I a I 0 ,
V ( I 0 , N w , p , F i ) = i = 1 n β i [ ln ( 1 - P i ) - ln ( 1 - F i ) ] 2 .
β i = [ ln ( 1 - F i ) F i σ i ] - 2 = ( 1 - F i ) 2 σ i - 2 .
σ i 2 = F i 2 ( F i - 1 - 1 s i - 1 ) .
V I 0 = 0 ; V N w = 0 ; V p = 0.
V ( I 0 , N w , p , F i ) = - i = 1 n β i [ N w J i + ln ( 1 - F i ) ] 2 ,
J i = 1 I i / I 0 u - 1 ( u - 1 ) p + 1 d u ,
J i = ( I i I 0 - 1 ) p + 1 .
i = 1 n β i J i [ N w J i + ln ( 1 - F i ) ] = 0 ,
J i J i I 0 .
N w = - j = 1 n β j J j ln ( 1 - F j ) j = 1 n β j J j 2 .
G ( I 0 , p , F i ) = 0 ,
G ( I 0 , p , F i ) i = 1 n β i J 1 [ J i j γ i J j j β j J j 2 - ln ( 1 - F i ) ] ,
γ j = β j ln ( 1 - F j ) .
F I 0 = i β i { ( j γ j J j j β J j 2 ) [ J i J i + ( J i ) 2 - 2 J i J i ( j β j J j J j j β j J j 2 ) ] + ( j γ j J j j β j J j 2 ) J i J i - J i ln ( 1 - F i ) }
J 2 J i I 0 2 .
Σ 2 = i = 1 n ( I 0 F i ) 2 σ i 2 ;             σ i F i .
( I 0 F i ) p = - G F i / G I 0 .
G F i = β i J i [ J i ( j γ j J j j β j J j 2 ) - ln ( 1 - F i ) ] + β i J i ( 1 - F i ) + ( j β j J j J j j β j J j 2 ) [ γ i J i - β i J i 2 ( j γ j J j j β j J j 2 ) ] ,
β i i β i F i             and             γ i γ i F i .
I 0 F i = ( I 0 F i ) p + I 0 p p F i ;             σ i F i .
first I i I 0.5 - 3 Δ I 0.5 .

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