Abstract

Far-field light scattering characteristics from randomly arranged shallow Gaussian-like shaped laser induced pits, found on optics exposed to high energy laser pulses, is studied. Closed-form expressions for the far-field intensity distribution and scattered power are derived for individual pits and validated using numerical calculations of both Fourier optics and FDTD solutions to Maxwell’s equations. It is found that the scattered power is proportional to the square of the pit width and approximately also to the square of the pit depth, with the proportionality factor scaling with pit depth. As a result, the power scattered from shallow pitted optics is expected to be substantially lower than assuming complete scattering from the total visible footprint of the pits.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  8. S. Palmier, I. Tovena, L. Lamaignère, J. L. Rullier, J. Capoulade, B. Bertussi, J. Y. Natoli, and L. Servant, ”Study of laser interaction with aluminum contaminant on fused silica,” Laser-Induced Damage in Optical Materials 2005, Proc. of SPIE Vol. 5991 59910R.
    [Crossref]
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    [Crossref]
  12. Since we assume in this work a known morphology (based on experimental observations), Maxwell’s equations include all the required physics to describe the problem at hand, and its solver sets a sound reference for validating the Fourier propagation calculation. A more detailed study of the shallow pits morphologies, their creation mechanisms and direct scattering measurements are outside the scope of this work and will be the subject for an upcoming manuscript.
  13. L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
    [Crossref]
  14. A more detailed characterization and discussion on the creation mechanism will be covered in a separate manuscript.
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    [Crossref] [PubMed]
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    [Crossref]
  19. A. Chabory, S. Bolioli, and J. Sokoloff, “Novel Gabor-based Guassian Beam expantsion for curved aperture radiation in dimension two,” Progress in Electromagnetics Research, PIER 58, 171–185 (2006).
    [Crossref]
  20. A. Yariv, Optical Electronics, 3rd Ed. (CBS, 1985)
  21. Even though the expression for the cross-section in Eq. (4) includes imaginary terms, the sum of the complementary terms (i.e., when replacing m and l indices) in the series is always real, resulting in a real value for the cross-section.

2013 (2)

M. J. Matthews, N. Shen, J. Honig, J. D. Bude, and A. M. Rubenchik, “Phase modulation and morphological evolution associated with surface-bound particle ablation,” J. Opt. Soc. Am. B 30(12), 3233–3242 (2013).
[Crossref]

L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
[Crossref]

2011 (1)

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

2009 (1)

2007 (1)

2006 (1)

A. Chabory, S. Bolioli, and J. Sokoloff, “Novel Gabor-based Guassian Beam expantsion for curved aperture radiation in dimension two,” Progress in Electromagnetics Research, PIER 58, 171–185 (2006).
[Crossref]

2004 (1)

D. Dinga and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116(3), 1401–1405 (2004).
[Crossref]

2002 (1)

1999 (1)

1983 (1)

Antos, R.

Asmail, C. C.

Auerbach, J. M.

Bertussi, B.

Bolioli, S.

A. Chabory, S. Bolioli, and J. Sokoloff, “Novel Gabor-based Guassian Beam expantsion for curved aperture radiation in dimension two,” Progress in Electromagnetics Research, PIER 58, 171–185 (2006).
[Crossref]

Bowers, M. W.

Bude, J. D.

M. J. Matthews, N. Shen, J. Honig, J. D. Bude, and A. M. Rubenchik, “Phase modulation and morphological evolution associated with surface-bound particle ablation,” J. Opt. Soc. Am. B 30(12), 3233–3242 (2013).
[Crossref]

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Carr, C. W.

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Chabory, A.

A. Chabory, S. Bolioli, and J. Sokoloff, “Novel Gabor-based Guassian Beam expantsion for curved aperture radiation in dimension two,” Progress in Electromagnetics Research, PIER 58, 171–185 (2006).
[Crossref]

Charmasson, L.

L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
[Crossref]

Cormont, P.

Deepak Kallepalli, L. N.

L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
[Crossref]

Delaporte, P.

L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
[Crossref]

Demos, S.

Dinga, D.

D. Dinga and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116(3), 1401–1405 (2004).
[Crossref]

Dixit, S. N.

Erbert, G. V.

Feit, M. D.

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Fujimoto, J.

Germer, T. A.

Grojo, D.

L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
[Crossref]

Haynam, C. A.

Heestand, G. M.

Henesian, M. A.

Herloski, R.

Hermann, M. R.

Honig, J.

Jancaitis, K. S.

Laurence, T. A.

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Legros, P.

Manes, K. R.

Marshall, C. D.

Marshall, S.

Matthews, M. J.

Mehta, N. C.

Menapace, J.

Merlen, A.

L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
[Crossref]

Miller, P. E.

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Minoshima, K.

Monticelli, M. V.

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Moses, E.

Murray, J. R.

Norton, M. A.

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Nostrand, M. C.

Orth, C. D.

Palmier, S.

Patterson, R.

Rubenchik, A. M.

Rullier, J. L.

Sacks, R. A.

Sangar, A.

L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
[Crossref]

Shaw, M. J.

Shen, N.

M. J. Matthews, N. Shen, J. Honig, J. D. Bude, and A. M. Rubenchik, “Phase modulation and morphological evolution associated with surface-bound particle ablation,” J. Opt. Soc. Am. B 30(12), 3233–3242 (2013).
[Crossref]

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Sokoloff, J.

A. Chabory, S. Bolioli, and J. Sokoloff, “Novel Gabor-based Guassian Beam expantsion for curved aperture radiation in dimension two,” Progress in Electromagnetics Research, PIER 58, 171–185 (2006).
[Crossref]

Spaeth, M.

Staggs, M.

Steele, W. A.

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Suratwala, T. I.

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Sutton, S. B.

Torchio, P.

L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
[Crossref]

Utéza, O.

L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
[Crossref]

Van Wonterghem, B. M.

Wegner, P. J.

White, R. K.

Widmayer, C. C.

Williams, W. H.

Wong, L. L.

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

Yang, S. T.

Zhang, Y.

D. Dinga and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116(3), 1401–1405 (2004).
[Crossref]

Appl. Opt. (2)

J. Acoust. Soc. Am. (1)

D. Dinga and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116(3), 1401–1405 (2004).
[Crossref]

J. Am. Ceram. Soc. (1)

T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused Silica optical surfaces,” J. Am. Ceram. Soc. 94(2), 416–428 (2011).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Phys. D Appl. Phys. (1)

L. N. Deepak Kallepalli, D. Grojo, L. Charmasson, P. Delaporte, O. Utéza, A. Merlen, A. Sangar, and P. Torchio, “Long range nanostructuring of silicon surfaces by photonic nanojets from microsphere Langmuir films,” J. Phys. D Appl. Phys. 46(14), 145102 (2013).
[Crossref]

Opt. Express (2)

Progress in Electromagnetics Research, PIER (1)

A. Chabory, S. Bolioli, and J. Sokoloff, “Novel Gabor-based Guassian Beam expantsion for curved aperture radiation in dimension two,” Progress in Electromagnetics Research, PIER 58, 171–185 (2006).
[Crossref]

Other (11)

A. Yariv, Optical Electronics, 3rd Ed. (CBS, 1985)

Even though the expression for the cross-section in Eq. (4) includes imaginary terms, the sum of the complementary terms (i.e., when replacing m and l indices) in the series is always real, resulting in a real value for the cross-section.

Since we assume in this work a known morphology (based on experimental observations), Maxwell’s equations include all the required physics to describe the problem at hand, and its solver sets a sound reference for validating the Fourier propagation calculation. A more detailed study of the shallow pits morphologies, their creation mechanisms and direct scattering measurements are outside the scope of this work and will be the subject for an upcoming manuscript.

A more detailed characterization and discussion on the creation mechanism will be covered in a separate manuscript.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Internal communication with John Trenholme with regard to (circa 2003) Lenslet-Induced intensification in Optics.

S. Palmier, I. Tovena, L. Lamaignère, J. L. Rullier, J. Capoulade, B. Bertussi, J. Y. Natoli, and L. Servant, ”Study of laser interaction with aluminum contaminant on fused silica,” Laser-Induced Damage in Optical Materials 2005, Proc. of SPIE Vol. 5991 59910R.
[Crossref]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981)

J. C. Stover, Optical Scattering: Measurement and Analysis, Third Edition (SPIE, 2012)

I. L. Bass, G. M. Guss, M. J. Nostrand, P. L. Wegner, “An improved method of mitigating laser-induced surface damage growth in fused Silica using a rastered, pulsed CO2 laser”, Proc. SPIE 7842, Laser-Induced Damage in Optical Materials: 2010, 784220.

J. A. Folta, M. C. Nostrand, J. Honig, N. N. Wong, F. Ravizza, P. Geraghty, M. G. Taranowski, G. W. Johnson, G. R. Larkin, D. Ravizza, J. E. Peterson, B. J. Welday, P. J. Wegner, “Mitigation of laser damage on NIF optics in volume production,” Proc. SPIE 8885, Laser-Induced Damage in Optical Materials: 2013, 88850Z.

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Figures (6)

Fig. 1
Fig. 1 Scattering from laser induced pits (LSPs) ensemble: (a) Schematic illustration of the studied problem, where an ensemble of LSPs on glass scatter part of the incident beam, resulting in a halo around the far-field spot; (b) Morphology of the LSPs observed at the final optics of the NIF (typical population data, surface morphology of wide LSP as an inset, and two depth profile of wide and narrow LSPs).
Fig. 2
Fig. 2 Comparison of FDTD and paraxial based FFT calculations (near-field magnitude and phase, and far-field magnitude) for (a) narrow and (b) wide LSPs (with the depth profiles given at Fig. 1(b)).
Fig. 3
Fig. 3 Scattering of single LSP: (a) Convergence of the far-field expression (Eq. (2)) to the solution with increasing series length (Gaussian phase object with h = 200 nm, σ = 2 μm). (b) Scattering cross-section (ξ) as function of LSP depth with numerical fit in Eq. (5) and few sample values as an inset table.
Fig. 4
Fig. 4 Far-field intensity of LSP ensemble for the: (a) measured population given in Fig. 1(b) (Gaussian profile assumed) (b) bi-modal population as function of the species ratio (a ‘shoulder’ profile and same depth is assumed for both LSPs).
Fig. 5
Fig. 5 The role of the ‘shoulder’- comparison of the ‘shoulder’ shaped LSP (solid black) with set of Gaussian profiles (dashed colored, and solid red for best fit Gaussian): (a) depth profile of LSP, (b) resulting far-field intensity, the σ multiplier (with respect to arbitrary reference) is denoted on the far-field curves; inset shows the volume difference of a Gaussian LSP and the ‘Shoulder’ shaped LSP as a function of its σ multiplier.
Fig. 6
Fig. 6 Analysis of cusp pit: (a) depth profile of a large cusp pit ; inset shows the top view, (b) FDTD validation of paraxial propagator approach, (c) far-field comparison of the cusp pit numerical calculation with the analytic model (Eq. (4)) for its equivalent Gaussian (having same volume, i.e., σ2 = W2/2); the depths profiles of the two depicted at the inset.

Equations (7)

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

E ( r ) = A N F e i ϕ = A N F { 1 + m = 1 ( i ) m m ! ϕ m }
E F F ( r F F ) = A N F { δ ( r F F = 0 ) + m = 1 A m exp ( α m r F F 2 ) }
P F F m 0 = 2 π η | A N F | 2 0 | m = 1 A m exp ( α m r 2 ) | 2 r d r
P F F m 0 = ( η | A N F | 2 π σ 2 ) ( m = 1 l = 1 i m ( i ) l O P D m + l ( m + l ) m ! l ! )
ξ ~ ( 7.6 × 10 11 [ n m 4 ] h 2 1.8 × 10 8 [ n m 3 ] h + 4.2 × 10 5 [ n m 2 ] ) h 2
Δ P F F m 0 P F F m 0 = 2 Δ σ σ Δ P F F m 0 P F F m 0 h < < 1 μ m 2 Δ h h
r a t i o > > ( h p i t 2 σ 4 ) l arg e p i t s ( h p i t 2 σ 4 ) s m a l l p i t s

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