Abstract

Surface and bulk laser damage probability distribution functions are derived for a Gaussian laser beam and a power-law defect damage ensemble. Closed-form solutions are derived and plotted for six and three values of the defect ensemble parameter p in the surface and bulk damage distribution functions, respectively. The derived degenerate (p = 1) and nondegenerate (p = 0) bulk damage equations are least-squares fitted to measured laser damage statistics in polymethyl methacrylate. The results show that the power-law defect ensemble is a reasonable description of the laser-damageable defects in the damage-tested polymethyl methacrylate and that the ensemble is more degenerate in character than uniform.

© 1992 Optical Society of America

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  1. S. R. Foltyn, “Spotsize effects in laser damage testing,” Natl. Bur. Stand. (U.S.) Spec. Publ. 669, 368–377 (1982).
  2. S. C. Seitel, J. O. Porteus, “Toward improved accuracy in limited-scale pulsed laser damage testing via the onset method,” Natl. Bur. Stand. (U.S.) Spec. Publ. 688, 502–512 (1983).
  3. J. O. Porteus, S. C. Seitel, “Absolute onset of optical surface damage using distributed defect ensembles,” Appl. Opt. 23, 3796–3805 (1984).
    [CrossRef] [PubMed]
  4. L. D. Dickson, “Characteristics of a propagating Gaussian beam,” Appl. Opt. 9, 1854–1861 (1970).
    [CrossRef] [PubMed]
  5. J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1989), Chap. 4.
  6. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 17.
  7. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).
  8. S. Nemoto, “Waist shift of a Gaussian beam by plane dielectric interfaces,” Appl. Opt. 27, 1833–1839 (1988).
    [CrossRef] [PubMed]
  9. S. M. Selby, ed., Standard Mathematical Tables, 14th ed. (Chemical Rubber Company, Cleveland, Ohio, 1965).
  10. R. M. O’Connell, “Polymers for high-power laser applications,” in Lasers in Polymer Science and Technology: Applications, J. P. Fouassier, J. F. Rabek, eds. (CRC Press, Boca Raton, Fla., 1990), pp. 97–132.
  11. D. A. Gromov, K. M. Dyumaev, A. A. Manenkov, A. P. Maslyukov, G. A. Matyushin, V. S. Nechitailo, A. M. Prokhorov, “Efficient plastic-host dye lasers,” J. Opt. Soc. Am. B 2, 1028–1031 (1985).
    [CrossRef]
  12. R. M. O’Connell, T. F. Deaton, T. T. Saito, “Single- and multiple-shot laser damage properties of commercial grade PMMA,” Appl. Opt. 23, 682–688 (1984).
    [CrossRef]
  13. R. M. O’Connell, A. B. Romberger, A. A. Shaffer, T. T. Saito, T. F. Deaton, K. E. Siegenthaler, “Improved laser-damage-resistant polymethyl methacrylate,” J. Opt. Soc. Am. B 1, 853–856 (1984).
    [CrossRef]
  14. R. M. O’Connell, R. V. Ellis, A. B. Romberger, T. F. Deaton, K. E. Siegenthaler, A. A. Shaffer, B. W. Mullins, T. T. Saito, “Laser damage studies of several methacrylate polymeric materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 727, 49–58 (1984).

1988

1985

1984

1983

S. C. Seitel, J. O. Porteus, “Toward improved accuracy in limited-scale pulsed laser damage testing via the onset method,” Natl. Bur. Stand. (U.S.) Spec. Publ. 688, 502–512 (1983).

1982

S. R. Foltyn, “Spotsize effects in laser damage testing,” Natl. Bur. Stand. (U.S.) Spec. Publ. 669, 368–377 (1982).

1970

Deaton, T. F.

R. M. O’Connell, T. F. Deaton, T. T. Saito, “Single- and multiple-shot laser damage properties of commercial grade PMMA,” Appl. Opt. 23, 682–688 (1984).
[CrossRef]

R. M. O’Connell, A. B. Romberger, A. A. Shaffer, T. T. Saito, T. F. Deaton, K. E. Siegenthaler, “Improved laser-damage-resistant polymethyl methacrylate,” J. Opt. Soc. Am. B 1, 853–856 (1984).
[CrossRef]

R. M. O’Connell, R. V. Ellis, A. B. Romberger, T. F. Deaton, K. E. Siegenthaler, A. A. Shaffer, B. W. Mullins, T. T. Saito, “Laser damage studies of several methacrylate polymeric materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 727, 49–58 (1984).

Dickson, L. D.

Dyumaev, K. M.

Ellis, R. V.

R. M. O’Connell, R. V. Ellis, A. B. Romberger, T. F. Deaton, K. E. Siegenthaler, A. A. Shaffer, B. W. Mullins, T. T. Saito, “Laser damage studies of several methacrylate polymeric materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 727, 49–58 (1984).

Foltyn, S. R.

S. R. Foltyn, “Spotsize effects in laser damage testing,” Natl. Bur. Stand. (U.S.) Spec. Publ. 669, 368–377 (1982).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

Gromov, D. A.

Manenkov, A. A.

Maslyukov, A. P.

Matyushin, G. A.

Mullins, B. W.

R. M. O’Connell, R. V. Ellis, A. B. Romberger, T. F. Deaton, K. E. Siegenthaler, A. A. Shaffer, B. W. Mullins, T. T. Saito, “Laser damage studies of several methacrylate polymeric materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 727, 49–58 (1984).

Nechitailo, V. S.

Nemoto, S.

O’Connell, R. M.

R. M. O’Connell, R. V. Ellis, A. B. Romberger, T. F. Deaton, K. E. Siegenthaler, A. A. Shaffer, B. W. Mullins, T. T. Saito, “Laser damage studies of several methacrylate polymeric materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 727, 49–58 (1984).

R. M. O’Connell, T. F. Deaton, T. T. Saito, “Single- and multiple-shot laser damage properties of commercial grade PMMA,” Appl. Opt. 23, 682–688 (1984).
[CrossRef]

R. M. O’Connell, A. B. Romberger, A. A. Shaffer, T. T. Saito, T. F. Deaton, K. E. Siegenthaler, “Improved laser-damage-resistant polymethyl methacrylate,” J. Opt. Soc. Am. B 1, 853–856 (1984).
[CrossRef]

R. M. O’Connell, “Polymers for high-power laser applications,” in Lasers in Polymer Science and Technology: Applications, J. P. Fouassier, J. F. Rabek, eds. (CRC Press, Boca Raton, Fla., 1990), pp. 97–132.

Porteus, J. O.

J. O. Porteus, S. C. Seitel, “Absolute onset of optical surface damage using distributed defect ensembles,” Appl. Opt. 23, 3796–3805 (1984).
[CrossRef] [PubMed]

S. C. Seitel, J. O. Porteus, “Toward improved accuracy in limited-scale pulsed laser damage testing via the onset method,” Natl. Bur. Stand. (U.S.) Spec. Publ. 688, 502–512 (1983).

Prokhorov, A. M.

Romberger, A. B.

R. M. O’Connell, R. V. Ellis, A. B. Romberger, T. F. Deaton, K. E. Siegenthaler, A. A. Shaffer, B. W. Mullins, T. T. Saito, “Laser damage studies of several methacrylate polymeric materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 727, 49–58 (1984).

R. M. O’Connell, A. B. Romberger, A. A. Shaffer, T. T. Saito, T. F. Deaton, K. E. Siegenthaler, “Improved laser-damage-resistant polymethyl methacrylate,” J. Opt. Soc. Am. B 1, 853–856 (1984).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

Saito, T. T.

R. M. O’Connell, T. F. Deaton, T. T. Saito, “Single- and multiple-shot laser damage properties of commercial grade PMMA,” Appl. Opt. 23, 682–688 (1984).
[CrossRef]

R. M. O’Connell, A. B. Romberger, A. A. Shaffer, T. T. Saito, T. F. Deaton, K. E. Siegenthaler, “Improved laser-damage-resistant polymethyl methacrylate,” J. Opt. Soc. Am. B 1, 853–856 (1984).
[CrossRef]

R. M. O’Connell, R. V. Ellis, A. B. Romberger, T. F. Deaton, K. E. Siegenthaler, A. A. Shaffer, B. W. Mullins, T. T. Saito, “Laser damage studies of several methacrylate polymeric materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 727, 49–58 (1984).

Seitel, S. C.

J. O. Porteus, S. C. Seitel, “Absolute onset of optical surface damage using distributed defect ensembles,” Appl. Opt. 23, 3796–3805 (1984).
[CrossRef] [PubMed]

S. C. Seitel, J. O. Porteus, “Toward improved accuracy in limited-scale pulsed laser damage testing via the onset method,” Natl. Bur. Stand. (U.S.) Spec. Publ. 688, 502–512 (1983).

Shaffer, A. A.

R. M. O’Connell, A. B. Romberger, A. A. Shaffer, T. T. Saito, T. F. Deaton, K. E. Siegenthaler, “Improved laser-damage-resistant polymethyl methacrylate,” J. Opt. Soc. Am. B 1, 853–856 (1984).
[CrossRef]

R. M. O’Connell, R. V. Ellis, A. B. Romberger, T. F. Deaton, K. E. Siegenthaler, A. A. Shaffer, B. W. Mullins, T. T. Saito, “Laser damage studies of several methacrylate polymeric materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 727, 49–58 (1984).

Siegenthaler, K. E.

R. M. O’Connell, R. V. Ellis, A. B. Romberger, T. F. Deaton, K. E. Siegenthaler, A. A. Shaffer, B. W. Mullins, T. T. Saito, “Laser damage studies of several methacrylate polymeric materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 727, 49–58 (1984).

R. M. O’Connell, A. B. Romberger, A. A. Shaffer, T. T. Saito, T. F. Deaton, K. E. Siegenthaler, “Improved laser-damage-resistant polymethyl methacrylate,” J. Opt. Soc. Am. B 1, 853–856 (1984).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 17.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1989), Chap. 4.

Appl. Opt.

J. Opt. Soc. Am. B

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

R. M. O’Connell, R. V. Ellis, A. B. Romberger, T. F. Deaton, K. E. Siegenthaler, A. A. Shaffer, B. W. Mullins, T. T. Saito, “Laser damage studies of several methacrylate polymeric materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 727, 49–58 (1984).

S. R. Foltyn, “Spotsize effects in laser damage testing,” Natl. Bur. Stand. (U.S.) Spec. Publ. 669, 368–377 (1982).

S. C. Seitel, J. O. Porteus, “Toward improved accuracy in limited-scale pulsed laser damage testing via the onset method,” Natl. Bur. Stand. (U.S.) Spec. Publ. 688, 502–512 (1983).

Other

S. M. Selby, ed., Standard Mathematical Tables, 14th ed. (Chemical Rubber Company, Cleveland, Ohio, 1965).

R. M. O’Connell, “Polymers for high-power laser applications,” in Lasers in Polymer Science and Technology: Applications, J. P. Fouassier, J. F. Rabek, eds. (CRC Press, Boca Raton, Fla., 1990), pp. 97–132.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1989), Chap. 4.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 17.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

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Figures (9)

Fig. 1
Fig. 1

Typical plots of the power-law defect damage ensemble [Eqs. (7) and (8)]. Lines for p = 0 and p = 1 represent the uniform and degenerate extremes of the ensemble, respectively.

Fig. 2
Fig. 2

Geometry of a propagating Gaussian beam in the neighborhood of its waist. The on-axis oblong shape represents the above-threshold volume VT), and the transverse circle in the z = 0 plane represents the above-threshold area AT).

Fig. 3
Fig. 3

Plot of Eq. (9) in the z = 0 plane to illustrate the decrease of A(Φ) [from AT) to zero] as a function of beam fluence between ΦT and Φ0.

Fig. 4
Fig. 4

Normalized plots of Eq. (24) (curve a) and Eq. (25) for β = 2, 5, and 10 (curves b, c, and d, respectively).

Fig. 5
Fig. 5

Surface damage probability distributions for the six values of the parameter p (and corresponding expressions for N) given in Table I and for two values each of the parameter Nw.

Fig. 6
Fig. 6

Bulk damage probability distributions for the three values of the parameter p (and corresponding expressions for N) given in Table II and for two values each of the parameter Nw.

Fig. 7
Fig. 7

Results of least-squares fitting the degenerate (p = 1) and uniform (p = 0) defect ensemble damage probability models (solid curves) to measured bulk laser damage statistics (circles) from PMMA Sample A (see Table III and the text for a sample description).

Fig. 8
Fig. 8

Results of least-squares fitting the degenerate (p = 1) and uniform (p = 0) defect ensemble damage probability models (solid curves) to measured bulk laser damage statistics (circles) from PMMA Samples B1 (upper plots) and B2 (lower plots) (see Table III and text for sample descriptions).

Fig. 9
Fig. 9

Results of least-squares fitting the degenerate (p = 1) and uniform (p = 0) defect ensemble damage probability models (solid curves) to measured bulk laser damage statistics (circles) from PMMA Samples C1 (upper plots) and C2 (lower plots) (see Table III and text for sample descriptions).

Tables (3)

Tables Icon

Table I Closed-Form Solutions of the Surface Damage Integral N / N w = 1 β u - 1 ( u - 1 ) 1 - p d u in Eq. (33)

Tables Icon

Table II Closed-Form Solutions of the Bulk Damage Integral N / N w = 1 β ( u - 1 ) ( u - 1 ) 1 - p ( β / u - 1 ) 1 / 2 ( β / 2 u + 1 ) d u in Eq. (37)a

Tables Icon

Table III Summary of the Results of Least-Squares Fitting Eq. (3) with Exponents N from Table II for p = 1 (the Degenerate Defect Ensemble) and p = 0 (the Uniform Defect Ensemble)

Equations (43)

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P D = 1 - P ( 0 ) ,
P ( 0 ) = exp ( - N ) ,
P D = 1 - exp ( - N ) ,
N S = Φ T Φ 0 N A ( Φ ) A ( Φ ) d Φ
N B = Φ T Φ 0 N V ( Φ ) V ( Φ ) d Φ
N S = N A A ( Φ T ) ,
N B = N V V ( Φ T ) ,
N S = Φ T Φ 0 d A ( Φ ) d Φ Φ Φ T N A ( Φ ) d Φ d Φ ,
N B = Φ T Φ 0 d V ( Φ ) d Φ Φ Φ T N V ( Φ ) d Φ d Φ .
N ( Φ ) = { 0 for Φ < Φ T , C p ( Φ - Φ T ) - p for Φ Φ T ,
N ( Φ ) = C p δ ( Φ - Φ T ) ,
N S = Φ T Φ 0 N A δ ( Φ - Φ T ) A ( Φ ) d Φ = N A A ( Φ T ) , N B = Φ T Φ 0 N V δ ( Φ - Φ T ) V ( Φ ) d Φ = N V V ( Φ T ) ,
Φ ( r , z ) = [ w 0 w ( z ) ] 2 Φ 0 exp [ - 2 r 2 / w 2 ( z ) ] ,
w 2 ( z ) = w 0 2 ( 1 + z 2 z 0 2 ) ,
z 0 = π w 0 2 n / λ 0 .
A ( Φ T ) = π r 0 2 ( Φ T ) ,
Φ T = Φ 0 exp [ - 2 r 0 2 ( Φ T ) w 0 2 ] ,
r 0 2 ( Φ T ) = w 0 2 2 ln Φ 0 Φ T .
A ( Φ T ) = π w 0 2 2 ln Φ 0 Φ T ,
A ( Φ ) = π w 0 2 2 ln Φ 0 Φ for Φ T Φ Φ 0 .
V ( Φ T ) = 2 0 z m ( Φ T ) π r z 2 ( Φ T ) d z ,
Φ T = [ w 0 w ( z ) ] 2 Φ 0 exp [ - 2 r z 2 ( Φ T ) w 2 ( z ) ] .
r z 2 ( Φ T ) = w 0 2 2 ( 1 + z 2 z 0 2 ) [ ln Φ 0 Φ T - ln ( 1 + z 2 z 0 2 ) ] .
z m ( Φ T ) = z 0 ( Φ 0 Φ T - 1 ) 1 / 2 .
V ( Φ T ) = π w 0 2 0 z 0 ( Φ 0 Φ T - 1 ) 1 / 2 × ( 1 + z 2 z 0 2 ) [ ln Φ 0 Φ T - ln ( 1 + z 2 z 0 2 ) ] d z .
V ( Φ T ) = 4 π 2 w 0 4 n 3 λ 0 × [ ( Φ 0 Φ T - 1 ) 1 / 2 - tan - 1 ( Φ 0 Φ T - 1 ) 1 / 2 + 1 6 ( Φ 0 Φ T - 1 ) 3 / 2 ] .
V ( Φ ) = 4 π 2 w 0 4 n 3 λ 0 [ ( Φ 0 Φ - 1 ) 1 / 2 - tan - 1 ( Φ 0 Φ - 1 ) 1 / 2 + 1 6 ( Φ 0 Φ - 1 ) 3 / 2 ] Φ T Φ Φ 0
r z 2 ( Φ T ) r 0 2 ( Φ T ) = 1 - z 2 z m 2 ( Φ T ) .
r z 2 ( Φ T ) r 0 2 ( Φ T ) = [ 1 + ( β - 1 ) z 2 / z m 2 ( Φ T ) ] × { 1 - ln [ 1 + ( β - 1 ) z 2 / z m 2 ( Φ T ) ] ln β } ,
z p 2 ( Φ T ) z m 2 ( Φ T ) = β exp ( - 1 ) - 1 β - 1 ,
r z p 2 ( Φ T ) r 0 2 ( Φ T ) = β exp ( - 1 ) ln β ,
P D = { 0 for β < 1 , 1 - exp ( - π w 0 2 2 N A ln β ) for β 1 ,
P D = { 0 for β < 1 1 - exp { - 4 π 2 n w 0 4 N V 3 λ 0 [ ( β - 1 ) 1 / 2 - tan - 1 ( β - 1 ) 1 / 2 + 1 6 ( β - 1 ) 3 / 2 ] } for β 1 ,
N = N S = { 0 for Γ < Φ T Φ T Φ 0 d d Φ ( π w 0 2 2 ln Φ 0 Φ ) × Φ Φ T C p ( Φ - Φ T ) - p d Φ d Φ for Φ T Φ Φ 0 .
N = N s = π w 0 2 2 C p 1 - p Φ T Φ 0 ( Φ - Φ T ) 1 - p Φ d Φ .
N = N s = π w 0 2 2 C p 1 - p Φ T 1 - p 1 β u - 1 ( u - 1 ) 1 - p d u .
P D = { 0 for β < 1. 1 - exp [ - π w 0 2 2 C p 1 - p Φ T 1 - p × 1 β u - 1 ( u - 1 ) 1 - p d u ] for β 1 ,             0 p < 1
dV ( Φ ) d Φ = - 2 π 2 n w 0 4 3 λ 0 1 Φ ( Φ 0 Φ - 1 ) 1 / 2 ( Φ 0 2 Φ + 1 ) ,
N = N B = { 0 for Φ < Φ T . 2 π 2 n w 0 4 3 λ 0 C p 1 - p Φ T Φ 0 1 Φ ( Φ 0 Φ - 1 ) 1 / 2 × ( Φ 0 2 Φ + 1 ) ( Φ - Φ T ) 1 - p d Φ for Φ T Φ Φ 0
N = N B = 2 π 2 n w 0 4 3 λ 0 C p 1 - p Φ T 1 - p × 1 β u - 1 ( u - 1 ) 1 - p ( β u - 1 ) 1 / 2 ( β 2 u + 1 ) d u ,
P D = { 0 for β < 1. 1 - exp [ - 2 π 2 n w 0 4 3 λ 0 C p 1 - p Φ T 1 - p × 1 β u - 1 ( u - 1 ) 1 - p ( β u - 1 ) 1 / 2 ( β 2 u + 1 ) d u ] for β 1 , 0 p < 1
d 2 P D ( β ) d β 2 = exp [ - N ( β ) ] [ d 2 N ( β ) d β 2 - ( d N ( β ) d β ) 2 ] .
N V = 3 λ 0 N w 2 π 2 n w 0 4 .

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