Abstract

An analysis of the statistics of laser-induced damage to thin films is carried out for the commonly assumed single-defect model, in which damage is caused by irradiation of identical, randomly distributed defects in the film. The probability for damage due to a single irradiation with a beam of Gaussian spatial profile is calculated, and it is shown that observed variations with beam size of the intensity required to maintain a constant high probability for damage are accounted for by this expression. The multiple-shot damage probability is then calculated, assuming that irradiation is started at a low value of energy and increased stepwise, for two cases, an N-on-1 experiment where the beam irradiates the same site each time and a 1-on-1 experiment where the beam is moved to a new site with each shot. The damage thresholds, defined to be the median values of the distribution functions for these two cases, are compared to one another and to the threshold for a single-shot experiment. Moreover, the dependence of the threshold on the size of the pulse-to-pulse energy increment is determined. Finally, the effect of a second damage mechanism involving damage to the host material is determined by calculating the mean and variance of the probability density function. These results are shown to be in good agreement with prior measurements of the beam-size dependence of threshold.

© 1977 Optical Society of America

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References

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  1. R. W. Hopper, D. R. Uhlmann, J. Appl. Phys. 41, 4023 (1970); R. W. Hopper, C. Lee, D. R. Uhlmann, Damage in Laser Materials, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 341 (U.S. Government Printing Office, Washington, D.C., 1970), p. 55.
    [Crossref]
  2. M. Sparks, C. J. Duthler, J. Appl. Phys. 44, 3038 (1973).
    [Crossref]
  3. N. Bloembergen, Appl. Opt. 12, 661 (1973).
    [Crossref] [PubMed]
  4. L. G. DeShazer, B. E. Newnam, K. M. Leung, Laser Induced Damage in Optical Materials: 1973, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 387 (U.S. Government Printing Office, Washington, D.C., 1973), p. 114.
  5. L. G. DeShazer, B. E. Newnam, K. M. Leung, Appl. Phys. Lett. 23, 607 (1973).
    [Crossref]
  6. D. W. Fradin, D. P. Bua, Appl. Phys. Lett. 24, 555 (1974).
    [Crossref]
  7. D. Milam, R. A. Bradbury, M. Bass, Appl. Phys. Lett. 23, 654 (1973).
    [Crossref]
  8. V. Wang, C. R. Giuliano, B. Garcia, Laser Induced Damage in Optical Materials: 1975, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 435 (U.S. Government Printing Office, Washington, D.C., 1976), p. 216.
  9. K. M. Leung, C. C. Tang, L. G. DeShazer, Thin Solid Films 34, 119 (1976).
    [Crossref]
  10. E. S. Bliss, D. Milam, R. A. Bradbury, Appl. Opt. 12, 677 (1973).
    [Crossref] [PubMed]
  11. M. Bass, K. M. Leung, IEEE J. Quantum Electron. QE-12, 82 (1976).
    [Crossref]
  12. D. Milam, R. A. Bradbury, R. H. Picard, M. Bass, Laser Damage in Dielectric Coatings, Report AFCRL-TR-73-0406 (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass., 1973); Third Conference on High Power Infrared Laser Window Materials, 12–14 November 1973, Vol. 3, C. A. Pitha, H. Posen, A. Armington, eds., Report AFCRL-TR-74-0085 (III) (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass.1974), p. 1011;Laser Induced Damage in Optical Materials: 1974, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 414 (U.S. Government Printing Office, Washington, D.C., 1974), p. 169.
  13. M. Bass, H. H. Barrett, IEEE J. Quantum Electron. QE-8, 338 (1972).
    [Crossref]
  14. If we have no a priori knowledge of Id, we should assume that I1 is the first irradiation level ≥ Id. If Ij is incremented in constant steps, the range of I1 is Id ≤ I1 < Id + ΔI, and the result should be averaged over all possible values of I1 in its range. This will introduce a small correction to the results of this section.
  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 257.
  16. For example, see A. C. Aitken, Statistical Mathematics (Oliver and Boyd, Edinburgh, 1952), p. 27.
  17. We have also seen similar results in our laboratory.
  18. A statement to the contrary was made in an abstract of a paper submitted to the 1976 Optical Society annual meeting [R. H. Picard, D. Milam, R. A. Bradbury, J. Opt. Soc. Am.66, 1119 (1976)]. This statement is incorrect and should be disregarded.
  19. The discrepancy is due to the fact that do is defined in Refs. 4 and 5 to be the mean radius of a spot containing one inclusion only, that is, do≡〈w〉=∫0∞dwwd[1-exp(-πρw2)]/dw=½ρ-1/2. Hence, do is defined to be one-half of the value given by Eq. (2) in this paper. It is stated erroneously in Refs. 4 and 5 that 〈w〉 is also the mean distance between defects; the latter is actually 2〈w〉 = ρ−1/2.
  20. For the reason discussed in Ref. 19, the value do = 50 μm used by DeShazer et al. should be doubled for a proper comparison with our value of do.

1976 (2)

K. M. Leung, C. C. Tang, L. G. DeShazer, Thin Solid Films 34, 119 (1976).
[Crossref]

M. Bass, K. M. Leung, IEEE J. Quantum Electron. QE-12, 82 (1976).
[Crossref]

1974 (1)

D. W. Fradin, D. P. Bua, Appl. Phys. Lett. 24, 555 (1974).
[Crossref]

1973 (5)

D. Milam, R. A. Bradbury, M. Bass, Appl. Phys. Lett. 23, 654 (1973).
[Crossref]

M. Sparks, C. J. Duthler, J. Appl. Phys. 44, 3038 (1973).
[Crossref]

N. Bloembergen, Appl. Opt. 12, 661 (1973).
[Crossref] [PubMed]

L. G. DeShazer, B. E. Newnam, K. M. Leung, Appl. Phys. Lett. 23, 607 (1973).
[Crossref]

E. S. Bliss, D. Milam, R. A. Bradbury, Appl. Opt. 12, 677 (1973).
[Crossref] [PubMed]

1972 (1)

M. Bass, H. H. Barrett, IEEE J. Quantum Electron. QE-8, 338 (1972).
[Crossref]

1970 (1)

R. W. Hopper, D. R. Uhlmann, J. Appl. Phys. 41, 4023 (1970); R. W. Hopper, C. Lee, D. R. Uhlmann, Damage in Laser Materials, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 341 (U.S. Government Printing Office, Washington, D.C., 1970), p. 55.
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 257.

Aitken, A. C.

For example, see A. C. Aitken, Statistical Mathematics (Oliver and Boyd, Edinburgh, 1952), p. 27.

Barrett, H. H.

M. Bass, H. H. Barrett, IEEE J. Quantum Electron. QE-8, 338 (1972).
[Crossref]

Bass, M.

M. Bass, K. M. Leung, IEEE J. Quantum Electron. QE-12, 82 (1976).
[Crossref]

D. Milam, R. A. Bradbury, M. Bass, Appl. Phys. Lett. 23, 654 (1973).
[Crossref]

M. Bass, H. H. Barrett, IEEE J. Quantum Electron. QE-8, 338 (1972).
[Crossref]

D. Milam, R. A. Bradbury, R. H. Picard, M. Bass, Laser Damage in Dielectric Coatings, Report AFCRL-TR-73-0406 (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass., 1973); Third Conference on High Power Infrared Laser Window Materials, 12–14 November 1973, Vol. 3, C. A. Pitha, H. Posen, A. Armington, eds., Report AFCRL-TR-74-0085 (III) (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass.1974), p. 1011;Laser Induced Damage in Optical Materials: 1974, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 414 (U.S. Government Printing Office, Washington, D.C., 1974), p. 169.

Bliss, E. S.

Bloembergen, N.

Bradbury, R. A.

E. S. Bliss, D. Milam, R. A. Bradbury, Appl. Opt. 12, 677 (1973).
[Crossref] [PubMed]

D. Milam, R. A. Bradbury, M. Bass, Appl. Phys. Lett. 23, 654 (1973).
[Crossref]

D. Milam, R. A. Bradbury, R. H. Picard, M. Bass, Laser Damage in Dielectric Coatings, Report AFCRL-TR-73-0406 (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass., 1973); Third Conference on High Power Infrared Laser Window Materials, 12–14 November 1973, Vol. 3, C. A. Pitha, H. Posen, A. Armington, eds., Report AFCRL-TR-74-0085 (III) (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass.1974), p. 1011;Laser Induced Damage in Optical Materials: 1974, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 414 (U.S. Government Printing Office, Washington, D.C., 1974), p. 169.

A statement to the contrary was made in an abstract of a paper submitted to the 1976 Optical Society annual meeting [R. H. Picard, D. Milam, R. A. Bradbury, J. Opt. Soc. Am.66, 1119 (1976)]. This statement is incorrect and should be disregarded.

Bua, D. P.

D. W. Fradin, D. P. Bua, Appl. Phys. Lett. 24, 555 (1974).
[Crossref]

DeShazer, L. G.

K. M. Leung, C. C. Tang, L. G. DeShazer, Thin Solid Films 34, 119 (1976).
[Crossref]

L. G. DeShazer, B. E. Newnam, K. M. Leung, Appl. Phys. Lett. 23, 607 (1973).
[Crossref]

L. G. DeShazer, B. E. Newnam, K. M. Leung, Laser Induced Damage in Optical Materials: 1973, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 387 (U.S. Government Printing Office, Washington, D.C., 1973), p. 114.

Duthler, C. J.

M. Sparks, C. J. Duthler, J. Appl. Phys. 44, 3038 (1973).
[Crossref]

Fradin, D. W.

D. W. Fradin, D. P. Bua, Appl. Phys. Lett. 24, 555 (1974).
[Crossref]

Garcia, B.

V. Wang, C. R. Giuliano, B. Garcia, Laser Induced Damage in Optical Materials: 1975, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 435 (U.S. Government Printing Office, Washington, D.C., 1976), p. 216.

Giuliano, C. R.

V. Wang, C. R. Giuliano, B. Garcia, Laser Induced Damage in Optical Materials: 1975, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 435 (U.S. Government Printing Office, Washington, D.C., 1976), p. 216.

Hopper, R. W.

R. W. Hopper, D. R. Uhlmann, J. Appl. Phys. 41, 4023 (1970); R. W. Hopper, C. Lee, D. R. Uhlmann, Damage in Laser Materials, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 341 (U.S. Government Printing Office, Washington, D.C., 1970), p. 55.
[Crossref]

Leung, K. M.

M. Bass, K. M. Leung, IEEE J. Quantum Electron. QE-12, 82 (1976).
[Crossref]

K. M. Leung, C. C. Tang, L. G. DeShazer, Thin Solid Films 34, 119 (1976).
[Crossref]

L. G. DeShazer, B. E. Newnam, K. M. Leung, Appl. Phys. Lett. 23, 607 (1973).
[Crossref]

L. G. DeShazer, B. E. Newnam, K. M. Leung, Laser Induced Damage in Optical Materials: 1973, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 387 (U.S. Government Printing Office, Washington, D.C., 1973), p. 114.

Milam, D.

D. Milam, R. A. Bradbury, M. Bass, Appl. Phys. Lett. 23, 654 (1973).
[Crossref]

E. S. Bliss, D. Milam, R. A. Bradbury, Appl. Opt. 12, 677 (1973).
[Crossref] [PubMed]

D. Milam, R. A. Bradbury, R. H. Picard, M. Bass, Laser Damage in Dielectric Coatings, Report AFCRL-TR-73-0406 (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass., 1973); Third Conference on High Power Infrared Laser Window Materials, 12–14 November 1973, Vol. 3, C. A. Pitha, H. Posen, A. Armington, eds., Report AFCRL-TR-74-0085 (III) (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass.1974), p. 1011;Laser Induced Damage in Optical Materials: 1974, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 414 (U.S. Government Printing Office, Washington, D.C., 1974), p. 169.

A statement to the contrary was made in an abstract of a paper submitted to the 1976 Optical Society annual meeting [R. H. Picard, D. Milam, R. A. Bradbury, J. Opt. Soc. Am.66, 1119 (1976)]. This statement is incorrect and should be disregarded.

Newnam, B. E.

L. G. DeShazer, B. E. Newnam, K. M. Leung, Appl. Phys. Lett. 23, 607 (1973).
[Crossref]

L. G. DeShazer, B. E. Newnam, K. M. Leung, Laser Induced Damage in Optical Materials: 1973, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 387 (U.S. Government Printing Office, Washington, D.C., 1973), p. 114.

Picard, R. H.

A statement to the contrary was made in an abstract of a paper submitted to the 1976 Optical Society annual meeting [R. H. Picard, D. Milam, R. A. Bradbury, J. Opt. Soc. Am.66, 1119 (1976)]. This statement is incorrect and should be disregarded.

D. Milam, R. A. Bradbury, R. H. Picard, M. Bass, Laser Damage in Dielectric Coatings, Report AFCRL-TR-73-0406 (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass., 1973); Third Conference on High Power Infrared Laser Window Materials, 12–14 November 1973, Vol. 3, C. A. Pitha, H. Posen, A. Armington, eds., Report AFCRL-TR-74-0085 (III) (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass.1974), p. 1011;Laser Induced Damage in Optical Materials: 1974, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 414 (U.S. Government Printing Office, Washington, D.C., 1974), p. 169.

Sparks, M.

M. Sparks, C. J. Duthler, J. Appl. Phys. 44, 3038 (1973).
[Crossref]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 257.

Tang, C. C.

K. M. Leung, C. C. Tang, L. G. DeShazer, Thin Solid Films 34, 119 (1976).
[Crossref]

Uhlmann, D. R.

R. W. Hopper, D. R. Uhlmann, J. Appl. Phys. 41, 4023 (1970); R. W. Hopper, C. Lee, D. R. Uhlmann, Damage in Laser Materials, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 341 (U.S. Government Printing Office, Washington, D.C., 1970), p. 55.
[Crossref]

Wang, V.

V. Wang, C. R. Giuliano, B. Garcia, Laser Induced Damage in Optical Materials: 1975, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 435 (U.S. Government Printing Office, Washington, D.C., 1976), p. 216.

Appl. Opt. (2)

Appl. Phys. Lett. (3)

L. G. DeShazer, B. E. Newnam, K. M. Leung, Appl. Phys. Lett. 23, 607 (1973).
[Crossref]

D. W. Fradin, D. P. Bua, Appl. Phys. Lett. 24, 555 (1974).
[Crossref]

D. Milam, R. A. Bradbury, M. Bass, Appl. Phys. Lett. 23, 654 (1973).
[Crossref]

IEEE J. Quantum Electron. (2)

M. Bass, H. H. Barrett, IEEE J. Quantum Electron. QE-8, 338 (1972).
[Crossref]

M. Bass, K. M. Leung, IEEE J. Quantum Electron. QE-12, 82 (1976).
[Crossref]

J. Appl. Phys. (2)

R. W. Hopper, D. R. Uhlmann, J. Appl. Phys. 41, 4023 (1970); R. W. Hopper, C. Lee, D. R. Uhlmann, Damage in Laser Materials, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 341 (U.S. Government Printing Office, Washington, D.C., 1970), p. 55.
[Crossref]

M. Sparks, C. J. Duthler, J. Appl. Phys. 44, 3038 (1973).
[Crossref]

Thin Solid Films (1)

K. M. Leung, C. C. Tang, L. G. DeShazer, Thin Solid Films 34, 119 (1976).
[Crossref]

Other (10)

D. Milam, R. A. Bradbury, R. H. Picard, M. Bass, Laser Damage in Dielectric Coatings, Report AFCRL-TR-73-0406 (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass., 1973); Third Conference on High Power Infrared Laser Window Materials, 12–14 November 1973, Vol. 3, C. A. Pitha, H. Posen, A. Armington, eds., Report AFCRL-TR-74-0085 (III) (Air Force Cambridge Research Laboratories, Hanscom AFB, Mass.1974), p. 1011;Laser Induced Damage in Optical Materials: 1974, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 414 (U.S. Government Printing Office, Washington, D.C., 1974), p. 169.

L. G. DeShazer, B. E. Newnam, K. M. Leung, Laser Induced Damage in Optical Materials: 1973, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 387 (U.S. Government Printing Office, Washington, D.C., 1973), p. 114.

V. Wang, C. R. Giuliano, B. Garcia, Laser Induced Damage in Optical Materials: 1975, A. J. Glass, A. H. Guenther, Eds., NBS Special Publication 435 (U.S. Government Printing Office, Washington, D.C., 1976), p. 216.

If we have no a priori knowledge of Id, we should assume that I1 is the first irradiation level ≥ Id. If Ij is incremented in constant steps, the range of I1 is Id ≤ I1 < Id + ΔI, and the result should be averaged over all possible values of I1 in its range. This will introduce a small correction to the results of this section.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 257.

For example, see A. C. Aitken, Statistical Mathematics (Oliver and Boyd, Edinburgh, 1952), p. 27.

We have also seen similar results in our laboratory.

A statement to the contrary was made in an abstract of a paper submitted to the 1976 Optical Society annual meeting [R. H. Picard, D. Milam, R. A. Bradbury, J. Opt. Soc. Am.66, 1119 (1976)]. This statement is incorrect and should be disregarded.

The discrepancy is due to the fact that do is defined in Refs. 4 and 5 to be the mean radius of a spot containing one inclusion only, that is, do≡〈w〉=∫0∞dwwd[1-exp(-πρw2)]/dw=½ρ-1/2. Hence, do is defined to be one-half of the value given by Eq. (2) in this paper. It is stated erroneously in Refs. 4 and 5 that 〈w〉 is also the mean distance between defects; the latter is actually 2〈w〉 = ρ−1/2.

For the reason discussed in Ref. 19, the value do = 50 μm used by DeShazer et al. should be doubled for a proper comparison with our value of do.

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Figures (3)

Fig. 1
Fig. 1

Q-switched ruby laser damage data for a TiO2/SiO2 multilayer dielectric mirror as a function of spot size fit to Eq. (8). Flux levels (○) which produced damage sites having the symmetry of the laser beam, indicating host damage, and levels (△) which caused isolated micropit damage, indicative of defects, are segregated from flux levels (□) which caused no damage by the curve obtained under the assumption that the single-shot damage probability was maintained at the constant high value Γ = 0.9 as the beam size was reduced. The best fit was obtained by using a value of ρ = 1.8 × 104 cm−2 for the defect density, and Id = 12.7 J/cm2 for the defect damage level in Eq. (8). Data points were reproduced from Fig. 18 of Ref. 10.

Fig. 2
Fig. 2

Histogram of multiple-shot damage probability PD,NPD(I) for a 1-on-1 damage experiment (one shot on one site), according to Eq. (21). Probabilities are plotted vs JI/Id for three different increments of energy density in the range from ΔI = 0.05 Id to 0.2 Id.

Fig. 3
Fig. 3

Mean 〈I〉 and standard deviation 〈ΔI21/2 of the damage p.d.f. for Gaussian beams computed from Eqs. (48) and (51). Values ρ = 104 cm−2, Id = 8 J/cm2, and Ih = 25 J/cm2 were obtained by fitting Eq. (48) to the data of Ref. 5, Fig. 2, on Q-switched ruby laser damage to a thin ZrO2 film. The same values were used to compute the standard deviation from Eq. (51).

Tables (1)

Tables Icon

Table I Dependence of 1-on-1 Damage Characteristics on Energy Increment and Corresponding Ratios of N-on-1 to 1-on-1 Damage Thresholds for Multiple-Shot Experiments

Equations (60)

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P ( n n ) = n n exp ( - n ) n ! ,
n ρ A = d o - 2 A
P D , 1 = n = 1 P ( n n ) = 1 - P ( 0 n ) = 1 - exp ( - ρ A ) .
I ( r ) = I o exp ( - 2 r 2 / w 2 ) .
A A ( I o , I d , w ) = π w 2 2 ln ( I o I d ) , I o I d = 0 ,             I o < I d ,
P D , 1 = 1 - exp [ - ( π ρ w 2 / 2 ) ln ( I o / I d ) ]
= 1 - ( I o / I d ) - n o / 2 ,
n o π ρ w 2
I o = I d exp ( k / w 2 ) .
k = 2 π ρ ln ( 1 1 - Γ ) .
A j = π w 2 2 ln I j I d ,             I j I d = 0 ,             I j < I d ,
Δ A j A j - A j - 1 .
P D , N = [ 1 - exp ( - ρ Δ A N ) ] exp ( - ρ A N - 1 ) .
P D , N = J N - 1 - n o / 2 - J N - n o / 2 ,             N 2 = 1 - J 1 - N o / 2 = 0 ,             N = 1 ,
J j I j / I d .
I j = I d + ( j - 1 ) Δ I ,
J j = 1 + ( j - 1 ) Δ J ,
Δ J Δ I / I d .
N = 1 N m P D , N = ½ ,
J N m - n o / 2 = ½
J med J N m = 2 2 / n o
I med = I d · 2 2 / n o .
P D , N = [ 1 - exp ( - ρ A N ) ] j = 1 N - 1 exp ( - ρ A j ) .
P D , N = ( 1 - J N - n o / 2 ) j = 1 N - 1 J j - n o / 2 ,             N 2 = 0 ,             N = 1.
N = 1 N m J N = 2 2 / n o .
Δ J 1 + ( J med - 1 ) / Δ J Γ ( [ J med / Δ J ] + 1 ) Γ ( 1 / Δ J ) = 2 2 / n o .
J med ( ln J med - 1 ) = 2 ln 2 n o Δ J - 1.
J med ( 0 ) = 1 ,
J med ( 1 ) = 1 + 2 ( ln 2 n o Δ J ) 1 / 2 .
P D ( J N , Δ J ) P D , N = ( 1 - J N ) - n o / 2 [ Q ( J N , Δ J ) ] - n o / 2 ,
Q ( J N , Δ J ) Δ J ( J N - 1 ) / Δ J Γ ( J N / Δ J ) Γ ( 1 / Δ J ) J N ( J N / Δ J ) - 1 / 2 exp [ - ( J N - 1 ) / Δ J ]
ln J mode = Δ J 2 J mode ( 1 + J mode - n o / 2 1 - J mode - n o / 2 ) ,
J mode ( 1 ) = 1 + ( 2 n o Δ J ) 1 / 2 .
I mode I d + ( 2 I d n o Δ I ) 1 / 2 ,
R med ( I med ) N - on - 1 ( I med ) 1 - on - 1 ,
R med = 2 2 / n o 1 + 2 ( ln 2 n o Δ J ) 1 / 2
R med exp { 2 ln 2 n o [ 1 - ( n o ln 2 Δ J ) 1 / 2 ] } ,
( I med ) single = I d · 2 2 / n o ( I med ) N - on - 1 ,
Δ I 2 ( I - I ) 2 = I 2 - I 2 .
I ( r ) = I o ,             r w = 0 ,             r > w .
I = I h P ( 0 n ) + I d [ 1 - P ( 0 n ) ] .
I = I d + ( I h - I d ) exp ( - π w 2 / d o 2 ) .
Δ I 2 = ( I h - I d ) 2 exp ( - π w 2 / d o 2 ) [ 1 - exp ( - π w o 2 / d o 2 ) ] .
Δ I 2 max = ( I h - I d ) 2 / 4
w max = ( ln 2 / π ) 1 / 2 d o .
σ R Δ I 2 1 / 2 I = I h - I d I h + I d .
Δ A j 2 π r j Δ r j
Δ r j r j - r j - 1 .
r g = w [ ½ ln ( I h / I d ) ] 1 / 2 .
P D , j 2 π ρ r j Δ r j exp ( - π ρ r j - 1 2 ) ,             2 j < g = 0 ,             j = 1 ,
I = j = 1 g I j P D , j + I h P ( 0 n g ) ,
n g π r g 2 ,
I j = I d exp ( 2 r j 2 / w 2 ) .
I = 2 π ρ I d o r g d r r exp ( 2 r 2 / w 2 - π ρ r 2 ) + I h exp ( - π ρ r g 2 ) .
I = I d 1 - [ 2 / n o ] [ 1 - 2 n o ( I d I h ) n o / 2 - 1 ] ,
lim w / d o I = I d ,
lim w / d o 0 I = I h .
I k = I d k 1 - [ 2 k / n o ] [ 1 - 2 k n o ( I d I h ) n o / 2 - k ]
Δ I 2 = 4 I h 2 ( n o - 4 ) ( n o - 2 ) 2 { n o η - 2 - η - n o / 2 × [ ( n o - 2 ) 2 + ( n o - 4 ) ( η - n o / 2 - n o η ) ] } ,
η I h / I d .

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